{"id":"5179a145-fa11-4295-94e6-312a98e0cc36","arxiv_id":"2506.15605","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For N≥7, the minimal W-algebra of so_N at level -1 equals the Z2-even subalgebra of L_{(N-6)/2}(osp_{1|2})⊗F_{N-4}, yielding strong rationality for even N.","lead":"Large families of symmetries called W-algebras can be built from the orthogonal Lie algebras, and at one special level they had resisted direct analysis. This paper proves that the simplest such family is the same as the even half of a product of a small superalgebra and free fermions, which makes their structure much more tractable and proves a longstanding rationality conjecture.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 rests on the unprinted λ-bracket verification in §2.3: Theorem 2.1 is applied to the even subalgebra without displaying the OPE comparison, so a sign or coefficient error there would invalidate the main claim.","rationale":"The reader identified exactly the same weakest assumption: the proof of Theorem 1.1 depends on the unprinted OPE computation in §2.3. My reading of §2 agrees. The rest of the proof—the central charge comparison, the identification of the W_{i,j} with so_r at level 1, and the use of [30, Theorem 5.1] for a0—is plausibly correct, but none of it replaces the concrete λ-bracket verification needed for Theorem 2.1. I also looked at Appendix A, where the proof of the categorical equivalence Theorem A.1 is terse and possibly incomplete; however that appendix is not needed for the central isomorphism Theorem 1.1 or for Corollary 1.2 itself, so the most load-bearing concern remains the omitted OPE check. The correct response is to keep the reader's CONDITIONAL verdict rather than to reject or accept outright: if the bracketed identities are supplied and verified, the main theorem follows; until then it is a proof gap, not a demonstrated falsehood.","tokens_in":15027,"tokens_out":18868,"duration_ms":195632,"concrete_test":"Compute the full λ-bracket table for the generators S in Vℓ(r) using the standard Leibniz/OPE rules (or an OPE computer-algebra package) and compare term-by-term with the minimal W-algebra brackets of W_{−1}(so_N,O_min) from [30, Theorem 5.1]. In particular verify: (1) [W_{i,j}_λ W_{k,l}] reproduces the affine so_r level-1 brackets; (2) [W_{i,j}_λ X_m], [W_{i,j}_λ Y_m] give the standard so_r representation and [e_λ X_m]=0, [e_λ Y_m]=X_m, [f_λ X_m]=Y_m, [f_λ Y_m]=0, [h_λ X_m]=X_m, [h_λ Y_m]=−Y_m; and (3) the leading λ^2 coefficient of [X_m_λ Y_n] equals (1/6)a0(−1)(e,[u_m,v_n]) with a0(−1)=−3(N−3), and the matrix (X_m)_{(2)}Y_n is nondegenerate. If all coefficients match, Theorem 2.1 applies and Theorem 1.1 is sound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 uses the uniqueness criterion Theorem 2.1 on the simple quotient of (Vℓ(osp1|2)⊗F_r)_0. To invoke it, the candidate generators must satisfy (M1)–(M3) exactly: the r(r−1)/2 fields W_{i,j} must generate L_1(so_r); the 2r fields X_m,Y_m must transform as the standard representation of so_r⊕sl2 under the λ-brackets with W_{i,j} and e,f,h; and the remaining brackets [X_m_λ Y_n] (and the X–X, Y–Y brackets) must reproduce the minimal W-algebra λ-brackets with the correct leading coefficient a0(−1)=−3(N−3), the correct first-order and constant terms, and a nondegenerate form (A,B)↦A_(2)B on the weight-3/2 subspace. The text says only 'the calculation of the λ-brackets show...' and does not display any of these comparisons. Lemma 2.2 establishes strong generation, not the OPE structure. Since Theorem 2.1 is a uniqueness theorem, every one of these identities is load-bearing: a single sign error (for example, in the relative sign of the x–y contraction and the ψ_m–ψ_n contraction, or in the so_r action on X_m,Y_m) would violate (M3) or change a0, and Theorem 1.1 would not follow. This is not an internal inconsistency—the asserted isomorphism may well be true—but the central claim is currently conditional on an unverified computation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves, for N≥7, that the simple minimal W-algebra of so_N at level −1 is isomorphic to the even subalgebra of the tensor product of the simple affine vertex superalgebra L_{−1+(N−4)/2}(osp_{1|2}) and N−4 free fermions. The proof uses the uniqueness criterion of Arakawa–Creutzig–Kawasetsu–Linshaw for minimal W-algebras, after identifying the candidate strong generators in the orbifold and matching their conformal weights, central charge, and affine subalgebras. The paper also classifies the simple ordinary modules of this W-algebra, realizes them through quantum Hamiltonian reduction, and derives categorical consequences for the categories of ordinary modules of L_{−1}(so_N) and of the W-algebra. The main theorem is used to settle the Arakawa–Moreau rationality conjecture for even N.","tokens_in":1961,"tokens_out":4486,"duration_ms":122307,"significance":"If the proof is completed, the result is significant: it gives a new non-admissible-level example of a minimal W-algebra realized as a simple-current orbifold of a tensor product of an affine vertex superalgebra and free fermions, yields strong rationality for the D-type minimal W-algebras at level −1, and provides a complete classification of simple ordinary modules together with tensor-categorical consequences. The approach via the uniqueness theorem and the matching of central charge and level is conceptually clean and involves no fitted parameters. The paper also connects the result to truncation curves of W-infinity algebras, which is an attractive larger picture. However, the central proof currently rests on a λ-bracket verification that is asserted rather than displayed; because the uniqueness theorem is a rigidity statement, any sign or coefficient error in that verification would invalidate the main isomorphism.","major_comments":[{"comment":"The proof applies Theorem 2.1 to the simple quotient of V^ℓ(r) after stating that 'the calculation of the λ-brackets show...', but none of the load-bearing λ-bracket identities are displayed. In particular, for the generators X_m^{0,0}, Y_m^{0,0}, W_{i,j}^{0,0}, and Z_{x,y}^{0,0}, the paper must verify: the λ-brackets of X_m and Y_m with the affine generators e,f,h and W_{i,j} realize the standard representation of sl_2 ⊕ so_r; the bracket [X_m _λ Y_n] has leading coefficient a0(−1)=−3(N−3) together with the correct first-order and constant terms matching equation (2.3); the X–X and Y–Y brackets are correct; and the bilinear form (A,B)↦A_{(2)}B on the weight-3/2 subspace is nondegenerate. Since Theorem 2.1 is a uniqueness theorem, every sign and coefficient is load-bearing; for example, a relative sign error between the osp_{1|2} contraction x–y and the free-fermion contraction ψ_m–ψ_n would change a0 or violate (M3). I recommend that the authors provide a complete λ-bracket table for all strong generators, including normal-ordered products and all central terms, or an explicit auxiliary computation with a reproducible derivation.","section":"§2.3, proof of Theorem 1.1"},{"comment":"After identifying W_{i,j}^{0,0} with E_{i,j}−E_{j,i}, the text says that 'comparing the λ-brackets' shows the W_{i,j} generate a quotient of V^1(so_r), and that since they generate the simple vertex algebra F_r^0, they generate the simple quotient L_1(so_r). This contains two unstated claims: first, the λ-bracket comparison itself is not shown, and second, the equality of the even subalgebra of r free fermions with L_1(so_r) (not merely with a subalgebra or a different extension) is not proved or referenced. This equality is needed for condition (M1) with k^♯_2=1. The statement is standard and likely true, but as written it is an additional gap in the verification of Theorem 2.1.","section":"§2.3, identification of the affine subalgebra L_1(so_r)"},{"comment":"Corollary 1.4 asserts that the category of ordinary L_{−1}(so_N)-modules is a semisimple, finite ribbon category, and it is said to follow from Theorem 1.3 'together with some categorical considerations presented in Appendix A'. However, Corollary A.3 is itself conditional: it proves ribbon structure only under the assumption that C(g,k) is finite and semisimple. The paper does not prove these assumptions for C(so_N,−1); Theorem 1.3 classifies simple ordinary W-modules, and Theorem A.1 gives a categorical equivalence for the W-algebra side, but the semisimplicity and finiteness of C(so_N,−1) are not derived explicitly. This is a logical gap in the statement of Corollary 1.4 as currently worded.","section":"§3.2 and Appendix A, Corollary 1.4"}],"minor_comments":[{"comment":"The symbol V^ℓ(r) is first defined as the even subalgebra of the universal tensor product (V^ℓ(osp_{1|2})⊗F_r)_0, but in the proof of Theorem 1.1 the same symbol is used for the simple quotient (L^ℓ(osp_{1|2})⊗F_r)_0. Please introduce separate notation for the universal and simple objects.","section":"§2.3, notation"},{"comment":"The formula for a0(k) is quoted from [30, Theorem 5.1] as a scalar, but the paper does not give the explicit expression for a0(k) in general. Since the subsequent proof needs the specific value a0(−1)=−3(N−3), it would help the reader to display the general formula and then specialise.","section":"§2.2, equation (2.3)"},{"comment":"Some displayed formulas in the proof of Lemma 2.2 are visually overloaded, especially the case distinction for W_{a,0}^{i,j}W_{0,0}^{j,m} and the formula for W_{a+1,0}^{i,i}. The notation is understandable but would benefit from clearer line breaks and parentheses.","section":"Lemma 2.2"},{"comment":"The sentence 'we prove the conjecture of Arakawa and Moreau ... for arbitrary n≥4' is only literally true for the D-type family so_{2n}; for odd N the paper proves a different, related statement. Please clarify the wording to avoid ambiguity.","section":"§1, Introduction"},{"comment":"Table 1 is informative but the entries such as ρ_{λω}, ρ_{(λ−1)ω}, and ρ_{(λ+1)ω}⊕ρ_{(λ−1)ω} are not explicitly defined in the text before the table; a one-sentence explanation of the notation ρ_{μω} would improve readability.","section":"§3.2, Table 1"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is plausible and the overall strategy is sound, but the proof as written is conditional on a substantial λ-bracket computation that is not shown. I do not see grounds for rejection because the gap is of the kind that can be filled by displaying the computation or by providing an ancillary verification. I recommend requesting that the authors supply the complete OPE table and clarify the identification of the affine subalgebra before the paper is accepted. The module-category claims also need a tighter logical presentation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know about this paper if you care about W-algebras at non-admissible levels: it gives a uniform orbifold realization of W_{-1}(so_N, O_min) as the even subalgebra of L_ℓ(osp_{1|2}) ⊗ F_{N-4}, and settles the Arakawa–Moreau strong rationality conjecture for even N in one stroke. That is a real result, not a repackaging. Prior work only had scattered cases (so_8, Deligne series), and the W-infinity truncation-curve discussion in the introduction is explicitly heuristic.\n\nWhat it does well: the proof strategy is intelligent. Instead of a case-by-case BRST analysis, they invoke the uniqueness theorem [10, Thm 3.2] for minimal W-algebras. They identify the candidate strong generators, check the affine subalgebra structure, match central charge, and then the theorem gives the isomorphism. The module classification in §3 is a clean consequence of the orbifold picture, and Appendix A gives a plausible categorical framework for the ribbon category statement.\n\nThe soft spot is exactly the one the stress-test flags: the λ-bracket comparison in §2.3 is asserted, not displayed. The text says \"the calculation of the λ-brackets show...\" and moves on. For a uniqueness theorem, every coefficient in (M1)–(M3) is load-bearing, including the leading a0(−1)=−3(N−3) and the non-degeneracy on the weight-3/2 subspace. A sign or coefficient error there would invalidate the isomorphism. I don't see any reason to suspect an error—the structure is natural and the claimed central charge matches—but as written the central claim is conditional on an unprinted computation. That is a fixable but real gap.\n\nTwo minor notes. First, the paper cites [1], an \"in preparation\" article, for the W_{-2}(so_n) result; that is background context, not load-bearing for this paper, but a published reference would be better. Second, the categorical appendix is terse; Corollary A.3 relies on finiteness and semisimplicity of the ordinary module category, which is not derived in detail here.\n\nWho is this for? Vertex algebra people, especially those working on rationality and orbifold realizations. It deserves a serious referee: the result is significant and the missing computation can be checked. My own verdict is conditional—I would not desk-reject, and I'd send it to review, asking the authors to move the λ-bracket verification from \"calculation shows\" to an appendix.\n\nBest,","headline":"A significant uniform orbifold realization of W_{-1}(so_N) that likely resolves the Arakawa–Moreau conjecture, but the central claim is conditional on an unprinted OPE check.","tokens_in":15912,"tokens_out":2228,"would_cite":true,"duration_ms":22141,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17B69","17B67","81R10"],"pacs":[],"model":"deepseek-v4-flash","headline":"At level -1, the minimal W-algebra of so_N is exactly the even half of an osp(1|2) affine superalgebra tensored with free fermions.","keywords":["minimal W-algebra","affine vertex superalgebra","osp(1|2)","free fermions","orbifold construction","strongly rational vertex algebra","module classification","level -1"],"falsifier":"For the smallest new case, $N=7$, write the even subalgebra $(L_{1/2}(\\mathfrak{osp}_{1|2})\\otimes\\mathcal{F}_3)^{\\mathbb{Z}_2}$ explicitly and compute the $\\lambda$-brackets of the six weight-$3/2$ fields with each other; if the leading $\\lambda^2$ coefficient differs from $-12$, or the bilinear form $(A,B)\\mapsto A_{(2)}B$ on that six-dimensional space is degenerate, Theorem 1.1 fails. A second check is to count simple ordinary modules for $N=8$: the classification predicts eight modules, and a mismatch with the BRST-reduction list would disprove the classification.","tokens_in":14848,"feed_emoji":"","tokens_out":14153,"duration_ms":122045,"temperature":0.7,"pith_summary":"At level minus one, the minimal W-algebra of the orthogonal Lie algebra $\\mathfrak{so}_N$ is not a new vertex algebra but the $\\mathbb{Z}_2$-invariant (even) subalgebra of a tensor product of a simple affine vertex superalgebra of $\\mathfrak{osp}_{1|2}$ and $N-4$ free fermions. The paper proves this as an explicit isomorphism of vertex algebras for every $N\\ge 7$. The payoff is that for even $N$ the W-algebra is strongly rational, meaning lisse and rational, which settles an open conjecture. The same isomorphism yields a complete list of simple ordinary modules and shows that the category of ordinary modules of $L_{-1}(\\mathfrak{so}_N)$ is a semisimple finite ribbon category.","feed_headline":"Level -1 W-algebra of so(N) is the even half of osp(1|2) plus fermions","feed_subtitle":"For N≥7 this identity makes the even-N W-algebra strongly rational, settling an open conjecture.","key_machinery":"The load-bearing mechanism is the uniqueness theorem for minimal W-algebras: a simple vertex algebra that is strongly generated by fields with the conformal weights, affine subalgebra, Virasoro subalgebra, and weight-$3/2$ transformation properties of $\\mathcal{W}_k(\\mathfrak{g},\\mathcal{O}_{\\min})$, with the leading $\\lambda^2$ coefficient nonzero and a nondegenerate bilinear form on the weight-$3/2$ space, is isomorphic to the minimal W-algebra. The paper constructs the even subalgebra $(L_\\ell(\\mathfrak{osp}_{1|2})\\otimes \\mathcal{F}_r)^{\\mathbb{Z}_2}$ with $r=N-4$ and $\\ell=-1+\\frac{r}{2}$, and checks these data: the even fields $e,f,h$ generate an $\\mathfrak{sl}_2$ affine subalgebra at level $\\ell$; the fermion bilinears $W_{i,j}$ generate $\\mathfrak{so}_r$ at level $1$; the $2r$ fields $X_m,Y_m$ carry the standard representation of $\\mathfrak{sl}_2\\oplus\\mathfrak{so}_r$ at weight $3/2$; and the single weight-2 field $Z_{x,y}$ gives the conformal vector with the correct central charge. The decisive check is that the $\\lambda$-brackets of these fields match the minimal W-algebra operator products, including the leading coefficient $a_0(-1)=-3(N-3)\\neq 0$ and nondegeneracy of the weight-$3/2$ pairing.","core_discovery":"The paper's central discovery is the identity\n$$\\mathcal{W}_{-1}(\\mathfrak{so}_N,\\mathcal{O}_{\\min}) \\simeq \\left(L_{-1+\\frac{N-4}{2}}(\\mathfrak{osp}_{1|2})\\otimes \\mathcal{F}_{N-4}\\right)^{\\mathbb{Z}_2}$$\nfor $N\\in\\mathbb{Z}_{\\ge 7}$; here the right-hand side is the even subalgebra of the tensor product of the simple affine vertex superalgebra of $\\mathfrak{osp}_{1|2}$ at level $\\ell=-1+\\frac{N-4}{2}$ and the vertex superalgebra of $N-4$ free fermions, fixed by the parity involution. The isomorphism is proved by showing that the even subalgebra is strongly generated by fields satisfying exactly the defining data of the minimal W-algebra, so a uniqueness theorem forces the identification. For even $N$, the affine superalgebra factor is known to be strongly rational, and strong rationality is preserved under this $\\mathbb{Z}_2$-orbifold identification; hence the W-algebra is strongly rational, confirming the conjecture the paper set out to prove. The paper also classifies all simple ordinary modules of $\\mathcal{W}_{-1}(\\mathfrak{so}_N,\\mathcal{O}_{\\min})$ in both the $D$-type and $B$-type cases, and derives the semisimple ribbon-category statement for $L_{-1}(\\mathfrak{so}_N)$-modules.","pith_inferences":["The same orbifold mechanism is likely to produce isomorphisms at other non-admissible levels: the truncation-curve picture sketched in the introduction predicts pointwise coincidences whenever two such curves intersect, so checking the next predicted case would test whether this is the beginning of a family rather than a single coincidence.","Because the right-hand side is a tensor product of a well-studied affine $\\mathfrak{osp}_{1|2}$ superalgebra and free fermions, the fusion rules and modular data of the W-algebra can be extracted from the known category of weight modules of $\\mathfrak{osp}_{1|2}$; the paper notes that a ribbon-category description is in preparation, but the isomorphism makes this computation a direct consequence.","For odd $N$ the isomorphism does not by itself establish strong rationality, because the $\\mathfrak{osp}_{1|2}$ factor then sits at half-integer level; deciding rationality for odd $N$ would require a separate argument about that factor's category."],"forward_implications":["For even $N=2n$, the minimal W-algebra $\\mathcal{W}_{-1}(\\mathfrak{so}_{2n},\\mathcal{O}_{\\min})$ is strongly rational, so its representation category is a modular tensor category.","The simple ordinary modules of $\\mathcal{W}_{-1}(\\mathfrak{so}_N,\\mathcal{O}_{\\min})$ are completely classified: in the $D$-type case they are $L^{\\pm}(\\lambda)$ and $L^{\\pm}_R(\\lambda)$ for $0\\le\\lambda\\le n-3$, and in the $B$-type case they are $L^{\\pm}(\\lambda)$ for $0\\le\\lambda\\le 2n-3$ together with $L^-_R(\\lambda)$ for $0\\le\\lambda\\le 2n-4$.","Each of these modules is realized by the zero-th BRST cohomology from a simple highest-weight module of the affine Lie algebra of $\\mathfrak{so}_N$ at level $-1$, giving a concrete dictionary between the orbifold and quantum-Hamiltonian-reduction pictures.","The category of ordinary modules of $L_{-1}(\\mathfrak{so}_N)$ is a semisimple finite ribbon category, and it is tensor equivalent to the underlying (even-morphism) category of ordinary and Ramond-twisted modules of $L_{-1+\\frac{N-4}{2}}(\\mathfrak{osp}_{1|2})\\otimes\\mathcal{F}_{N-4}$."],"supporting_citations":[{"why":"The quantum Hamiltonian reduction paper that sets up the structural description of minimal W-algebras, including the formula for $a_0(k)$ used in the uniqueness check.","marker":"[30]"},{"why":"Supplies the uniqueness theorem for minimal W-algebras used as Theorem 2.1 to force the isomorphism.","marker":"[10]"},{"why":"Gives the lisse criterion for these W-algebras at level $-1$ and states the rationality conjecture that the paper proves.","marker":"[12]"},{"why":"Provides the BRST reduction theorem classifying simple ordinary modules of minimal W-algebras, used to identify orbifold modules with $H^0_{\\mathcal{O}}(L_{-1}(\\Lambda))$.","marker":"[6]"},{"why":"Classifies ordinary and Ramond-twisted modules of the affine vertex superalgebra of $\\mathfrak{osp}_{1|2}$, used in the orbifold module classification.","marker":"[16]"},{"why":"Establishes strong rationality of the affine $\\mathfrak{osp}_{1|2}$ vertex superalgebra at the relevant level for even $N$, used to conclude the W-algebra is strongly rational.","marker":"[26]"},{"why":"Provides the vertex tensor category framework used in the appendix to turn finiteness and semisimplicity into the ribbon-category statement.","marker":"[24]"}],"fun_headline_variants":["so_N level -1 W-algebra: even half of osp(1|2) plus N-4 fermions","Minimal W-algebra of so_N at -1 is even subalgebra of osp(1|2) plus fermions","Conjecture confirmed: so_N level -1 W-algebra is even half of osp(1|2) plus fermions","For even N, so_N level -1 W-algebra is strongly rational via osp(1|2) + fermions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on a comparison, done in the text but not displayed, of the $\\lambda$-bracket identities of the even subalgebra with those of the minimal W-algebra: the leading coefficient must be $-3(N-3)$ and the pairing on the weight-$3/2$ fields must be nondegenerate; a wrong sign or coefficient there would break the isomorphism.","fun_headline_variants_meta":{"raw":{"variants":["so_N level -1 W-algebra: even half of osp(1|2) plus N-4 fermions","Minimal W-algebra of so_N at -1 is even subalgebra of osp(1|2) plus fermions","Conjecture confirmed: so_N level -1 W-algebra is even half of osp(1|2) plus fermions","For even N, so_N level -1 W-algebra is strongly rational via osp(1|2) + fermions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001875,"raw_usage":{"total_tokens":7357,"prompt_tokens":942,"completion_tokens":6415,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":558,"completion_tokens_details":{"reasoning_tokens":6292}},"tokens_in":558,"tokens_out":6415,"duration_ms":38855,"temperature":1.0,"reasoning_tokens":6292,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:35:16.647189+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the smallest new case, $N=7$, write the even subalgebra $(L_{1/2}(\\mathfrak{osp}_{1|2})\\otimes\\mathcal{F}_3)^{\\mathbb{Z}_2}$ explicitly and compute the $\\lambda$-brackets of the six weight-$3/2$ fields with each other; if the leading $\\lambda^2$ coefficient differs from $-12$, or the bilinear form $(A,B)\\mapsto A_{(2)}B$ on that six-dimensional space is degenerate, Theorem 1.1 fails. A second check is to count simple ordinary modules for $N=8$: the classification predicts eight modules, and a mismatch with the BRST-reduction list would disprove the classification.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The quantum Hamiltonian reduction paper that sets up the structural description of minimal W-algebras, including the formula for $a_0(k)$ used in the uniqueness check."},{"cited_title":"Arakawa, T","cited_arxiv_id":null,"evidence_quote":"Supplies the uniqueness theorem for minimal W-algebras used as Theorem 2.1 to force the isomorphism."},{"cited_title":"Arakawa, A","cited_arxiv_id":null,"evidence_quote":"Gives the lisse criterion for these W-algebras at level $-1$ and states the rationality conjecture that the paper proves."},{"cited_title":"Arakawa, Representation theory of superconformal algebras and the Kac–Roan–Wakimoto conjecture, Duke Math","cited_arxiv_id":null,"evidence_quote":"Provides the BRST reduction theorem classifying simple ordinary modules of minimal W-algebras, used to identify orbifold modules with $H^0_{\\mathcal{O}}(L_{-1}(\\Lambda))$."},{"cited_title":"Creutzig, N","cited_arxiv_id":null,"evidence_quote":"Classifies ordinary and Ramond-twisted modules of the affine vertex superalgebra of $\\mathfrak{osp}_{1|2}$, used in the orbifold module classification."},{"cited_title":"Frenkel, Y","cited_arxiv_id":null,"evidence_quote":"Establishes strong rationality of the affine $\\mathfrak{osp}_{1|2}$ vertex superalgebra at the relevant level for even $N$, used to conclude the W-algebra is strongly rational."},{"cited_title":"Creutzig, J","cited_arxiv_id":null,"evidence_quote":"Provides the vertex tensor category framework used in the appendix to turn finiteness and semisimplicity into the ribbon-category statement."}],"review_version":2}