{"id":"53d44f07-4107-4a71-b310-ee1325e48dba","arxiv_id":"2502.01766","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every p ≥ 2, a half-integer graded simple vertex algebra C_p is constructed with modular characters, and for p=3,4,5, C_p is isomorphic to affine W-algebras of types G2, F4, and E8.","lead":"The authors construct a family of new vertex algebras C_p for each integer p ≥ 2, and show that for p=3,4,5 the members coincide with affine W-algebras of types G2, F4, and E8, respectively. The algebras have finite-dimensional graded pieces and modular characters, yielding a non-rational but controllable family that supports the quasi-lisse conjecture.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1's decomposition of D^ch into U_p rests on complete reducibility of KL at levels -2±1/p and -2-1/p, which is asserted but not proved; if false, the C_p construction and all later identifications lose support.","rationale":"The reader's weakest assumption and mine coincide. I considered the omitted uniqueness proof behind Theorems 6.5/6.9 and the unspecified congruence subgroup in Theorem 9.3; both are genuine gaps, but they concern identifications of C_4/C_5 and the modularity wording, whereas without the Theorem 3.1 decomposition there is no family C_p at all. I also note independent checks in the paper: the p=3 and p=4 special cases are proved by explicit character identities and inverse QHR, and the modularity expression in Proposition 9.2 gives a concrete route to Theorem 9.3. This supports a conditional verdict rather than rejection. The concrete test on Ext^1 would settle whether the foundational semisimplicity premise is valid or whether Theorem 3.1 needs a new proof.","tokens_in":27579,"tokens_out":21690,"duration_ms":205162,"concrete_test":"Use the classification/resolution results for admissible sl2 modules (for example the V(p)/R(p) categories of [13]) to compute Ext^1 in KL_{-2+1/p}(sl2) and in KL_{-2-1/p}(sl2) between the two lowest simple modules appearing in U_p, i.e. L_{-(2+1/p)Λ_0} and L_{-(3+1/p)Λ_0+Λ_1} (resp. the corresponding pair at level -2-1/p). If any such Ext^1 is nonzero, the complete reducibility invoked in Theorem 3.1 is false for the relevant subcategory and the proof fails; if all vanish, the decomposition is supported. A secondary check is to verify the character of D^ch_{-2+1/p} equals the character sum in Theorem 3.1 for p=3 and p=4 to high q-order.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Theorem 3.1. The proof that D^ch_{SL2,-2+1/p} is isomorphic to U_p invokes 'complete reducibility of the categories KL_{-2+1/p}(sl2) and KL_{-2-1/p}(sl2)'. For generic level this is standard, but the levels here are not generic: -2+1/p is admissible but non-rational, and -2-1/p is a negative admissible level. Complete reducibility of KL is not automatic for such levels, and the paper gives no proof and no precise reference for this exact statement. The entire later construction of V_p, then C_p via inverse QHR, and the p=3,4,5 isomorphisms presuppose this direct-sum decomposition. A nonzero extension between any two simple modules appearing in U_p would invalidate the claimed isomorphism D^ch ≅ U_p, hence Theorem 1.1. This is not a disagreement with consensus; it is an unsecured premise on which the central family is built.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for each positive integer p, a simple half-integer graded vertex algebra C_p obtained by inverse quantum Hamiltonian reduction from the algebra of chiral differential operators on SL_2 at level -2+1/p. The main structural result, Theorem 1.1, states that L_{-2+1/p}(sl_2) ⊗ L_{-2-p}(sl_2) is conformally embedded in C_p with an explicit decomposition into tensor products of simple sl_2-modules. The paper then identifies C_3 with L_{-5/3}(G_2), C_4 with W_{-23/4}(F_4, A_1+\\tilde A_1), and C_5 with W_{-119/5}(E_8, A_4+A_2); shows that these three algebras are quasi-lisse via admissibility; computes explicit modular linear differential equations for p=2,3,4,5; and derives a closed character formula from which modularity of ch[C_p] is claimed for all p.","tokens_in":27780,"tokens_out":7885,"duration_ms":75264,"significance":"If the central identifications hold, this is a valuable family of non-rational, potentially quasi-lisse vertex algebras with finite-dimensional graded pieces and computable characters. The explicit isomorphisms with W-algebras of Deligne-series type, the MLDEs, and the closed Appell-Lerch expression for the character are concrete and falsifiable, and the inverse-QHR framework is clearly presented. The paper is also honest about the conjectural nature of quasi-lissness for general p. The main risk is that the proof infrastructure is uneven: Theorem 3.1 relies on an unproved semisimplicity assertion, and the C_4 and C_5 identifications depend on omitted uniqueness arguments and on a GAP computation reported without data. These points are load-bearing and need to be addressed before the paper's principal claims can be regarded as fully established.","major_comments":[{"comment":"The proof of Theorem 3.1 asserts 'complete reducibility of the categories KL_{-2+1/p}(sl_2) and KL_{-2-1/p}(sl_2)' without a proof or a precise reference. These are non-generic levels, and complete reducibility of KL at admissible non-rational levels is not automatic; a nonzero extension between any two simple modules appearing in the direct sum would invalidate the isomorphism D^ch_{SL_2,-2+1/p} ≅ U_p and hence Theorem 1.1 and all later identifications of C_p. Please supply a proof or an exact reference for semisimplicity at these specific levels.","section":"Section 3, Theorem 3.1"},{"comment":"The identification C_4 ≅ W_{-23/4}(F_4, A_1+\\tilde A_1) depends on the uniqueness result Proposition 6.4, but its proof is replaced by 'Using the same proof as in [7]', and Theorem 6.5 then says 'It is easy to check that our vertex algebra V satisfies the same conditions'. The omitted verification is load-bearing because the isomorphism is obtained by matching strong generators, gradings, and OPEs; without it the C_4 identification is not substantiated. Please include the proof or a precise reference to the exact statement used.","section":"Section 6.2, Proposition 6.4 and Theorem 6.5"},{"comment":"The E_8 computation of the graded pieces of g^f for f=f_{A_4+A_2} is reported as 'performed with GAP [20]', but no data, script, or output is provided, and Theorem 6.9 is then obtained by 'similar arguments as in the proof of Theorem 6.5'. Since the C_5 identification depends on this decomposition and on the uniqueness/extension argument, the computation should be made reproducible (for example, in an appendix with explicit module data or GAP code) or replaced by a proof.","section":"Section 6.3, Theorem 6.9"},{"comment":"The modularity claim is not established by the text. In Proposition 9.2 the specialized expression has ϑ_{n,n}(1)=0, so the evaluation at x=y=1 requires a limit, and derivatives of Appell-Lerch series are not modular termwise. The sentence 'An(τ) is clearly modular' after the identity is insufficient; please provide the modular transformation law, or a reference, for the specialized second derivative, and specify the congruence subgroup on which An is modular.","section":"Section 9, Theorem 9.3"}],"minor_comments":[{"comment":"The abstract says p ≥ 2 while Theorem 1.1 states p ≥ 1; please clarify the intended range and check that all formulas in Section 8 are stated consistently.","section":"Abstract and Theorem 1.1"},{"comment":"The notation V is used in Theorem 6.5 without being defined in this section; it should refer explicitly to C_4 from Theorem 5.3 or be redefined.","section":"Section 6.2, Theorem 6.5"},{"comment":"The citation 'Proposition 6.3 in [6]' has the same number as the current proposition, which is confusing; please cite as [6, Prop. 6.3] and verify the numbering.","section":"Proposition 6.3"},{"comment":"The condition 'ℓ ∈ Z≥0, ℓ ≥ 2' is not meaningful as written because the displayed character formulas depend on i; please correct the index and the range.","section":"Lemma 4.1"},{"comment":"There are several typos and grammatical slips, for example 'respond.' for 'respectively', 'In particilar' for 'In particular', 'we we would like' for 'we would like', and 'this results was obtained' for 'this result was obtained'.","section":"Throughout"},{"comment":"The statement about strong generation is unclear: it is not specified whether the 'six generators of conformal weight 1' are distinct from the generators of \\widehat{sl_2}\\times\\widehat{sl_2}, and the type notation '(16,(p-1)/2)^{2p+2}' should be explained.","section":"Corollary 3.5"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the journal's scope and the main ideas are promising. The heavy use of self-citations [9], [12], [13] is not by itself problematic, but the refereeing would be easier if the authors identified exactly which external statements are needed for Theorem 3.1 and Proposition 6.4. The omitted GAP data and the terse modularity argument should also be addressed; none of these appear to be fatal, but they are central enough to require a revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you the short version. This paper constructs an infinite family C_p of half-integer graded simple vertex algebras, proves explicit isomorphisms to affine W-algebras for p=3,4,5, and shows the characters are modular for all p. The family is new, the p=3,4,5 identifications are non-obvious, and the q-series work in Section 7 plus the Appell-Lerch argument in Section 9 are real computations. If the structural assumptions hold, this is a solid contribution to the quasi-lisse program.\n\nThe soft spots are real, and they are concentrated where the reader's report says. Theorem 3.1 is load-bearing: it identifies D^ch at level -2+1/p with the direct sum Up, and the proof cites \"complete reducibility of the categories KL_{-2+1/p}(sl2) and KL_{-2-1/p}(sl2)\" without a proof or a precise reference. These are not generic levels, and complete reducibility is not automatic there. The rest of the paper—construction of V_p, inverse QHR, and the p=3,4,5 isomorphisms—presupposes this decomposition. The stress-test is right to call it an unsecured premise. An expert may be able to supply the argument from [3] or [13], but it is not in the paper.\n\nSmaller issues: Proposition 6.4 says \"using the same proof as in [7]\" without spelling out the adaptation; Theorem 6.5 has an \"it is easy to check\" that is doing actual work; the GAP computation in Section 6.3 is reported without data or code; and Theorem 9.3 says \"on a certain congruence group\" without naming it. All of these are fixable, but each is a place where a referee has to take something on faith.\n\nOn citations: self-citations are frequent, but the central results are not circular. The p=2 case is explicitly credited to [12], the p=3,4,5 isomorphisms are new, and the modularity theorem does not assume quasi-lisse. I see no integrity problem.\n\nBottom line: this deserves a serious referee. The conditional verdict is fair. A referee should ask for (a) a proof or exact citation for complete reducibility in Theorem 3.1, (b) details for Proposition 6.4 and the \"easy to check\" step, (c) the GAP input, and (d) the congruence subgroup. If those are supplied, I expect this will be a high-citation paper. The audience is vertex algebra and W-algebra people; bring it to a reading group and assign the open items as homework.","headline":"New family C_p of vertex algebras with real content; the main structural premise (complete reducibility in Thm 3.1) is asserted rather than proved, and a few W-algebra identifications lean on 'easy to check' claims, but the character and modularity work is substantive.","tokens_in":28376,"tokens_out":3811,"would_cite":true,"duration_ms":36569,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17B69","17B20","17B67"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs simple vertex algebras $\\mathcal{C}_p$, $p\\ge 1$, with modular characters, and identifies $\\mathcal{C}_3,\\mathcal{C}_4,\\mathcal{C}_5$ with exceptional affine vertex algebras and $W$-algebras.","keywords":["vertex algebra","affine vertex algebra","W-algebra","quasi-lisse","conformal embedding","chiral differential operators","regular representation","modular characters"],"falsifier":"A decisive numerical check would be to expand both sides of the character formula in Theorem 8.1 to order $q^{10}$ for $p=5$ and compare with the character of $W_{-30+31/5}(E_8,A_4+A_2)$ computed from the strong-generator description; a mismatch would disprove Theorem 6.9. Structurally, the identification $\\mathcal{C}_4\\cong W_{-23/4}(F_4,A_1+\\tilde A_1)$ depends on the omitted 'same proof as in [7]' uniqueness statement, so exhibiting another simple vertex algebra satisfying the three conditions of Proposition 6.4 but not isomorphic to the $F_4$ $W$-algebra would decide that case. At the base, a single non-semisimple module in the category $KL_{-2+1/p}(\\mathfrak{sl}_2)$ (or $KL_{-2-1/p}$) at the relevant negative level would falsify Theorem 3.1.","tokens_in":27354,"feed_emoji":"","tokens_out":17272,"duration_ms":140814,"temperature":0.7,"pith_summary":"The paper constructs, for every integer $p\\ge 1$, a simple vertex algebra $\\mathcal{C}_p$ that contains $L_{-2+1/p}(\\mathfrak{sl}_2)\\otimes L_{-2-p}(\\mathfrak{sl}_2)$ as a conformal subalgebra and decomposes as an explicit infinite direct sum of irreducible modules. The construction is a deformation of the regular representation of $\\mathfrak{sl}_2$, or equivalently of the chiral differential operators on $SL_2$ at level $-2+1/p$, but with the second level chosen as $-2-p$ rather than the generic dual level. For small $p$ the algebra becomes a known object: $\\mathcal{C}_3\\cong L_{-5/3}(\\mathfrak{g}_2)$, $\\mathcal{C}_4\\cong W_{-23/4}(F_4,A_1+\\tilde A_1)$, and $\\mathcal{C}_5\\cong W_{-30+31/5}(E_8,A_4+A_2)$, giving explicit decompositions of certain conformal embeddings into exceptional affine $W$-algebras. The paper also proves that every $\\mathcal{C}_p$ is half-integer graded with finite-dimensional graded pieces and that its character is modular of weight zero. The modularity result is offered as evidence for the conjectural quasi-lisse property of these algebras.","feed_headline":"Simple vertex algebras tie SL2 to G2, F4, and E8","feed_subtitle":"For p=3,4,5 the algebra C_p becomes an exceptional affine vertex algebra or W-algebra; all C_p have modular characters.","key_machinery":"The engine is inverse quantum Hamiltonian reduction (inverse QHR), built from the Drinfeld-Sokolov reduction functor $H_{\\mathrm{DS},f}$. Starting from $U_p$, the vertex algebra of chiral differential operators on $SL_2$ at level $-2+1/p$, the paper reduces in one $\\mathfrak{sl}_2$ direction to obtain $V_p$, a tensor product of a Virasoro vertex algebra and $L_{-2+1/p}(\\mathfrak{sl}_2)$. Tensoring with the lattice/Heisenberg vertex algebra $\\Pi(0)_{1/2}$ and taking the maximal integrable part for $\\mathfrak{sl}_2\\times\\mathfrak{sl}_2$ converts the Virasoro modules back into $L_{-2-p}(\\mathfrak{sl}_2)$ modules; this is the step that produces $\\mathcal{C}_p$. The generation and simplicity of $\\mathcal{C}_p$ are proved using Virasoro fusion rules and the fusion rules of $L_{-2+1/p}(\\mathfrak{sl}_2)$-modules, together with the screening-operator realization from Proposition 2.1. For $p=4,5$, the same vertex algebras are independently realized as simple quotients of affine $W$-algebras at admissible levels, with strong generation by low-weight fields, and the isomorphisms are then proven by comparing extensions of the same conformal subalgebra.","core_discovery":"The central object is a simple vertex algebra $\\mathcal{C}_p$ defined for every integer $p\\ge 1$ as an extension of $L_{-2+1/p}(\\mathfrak{sl}_2)\\otimes L_{-2-p}(\\mathfrak{sl}_2)$, with decomposition $\\mathcal{C}_p = \\bigoplus_{\\ell\\ge 0} L_{\\widehat{\\mathfrak{sl}}_2}(-(2+p+p\\ell)\\Lambda_0+p\\ell\\Lambda_1)\\otimes L_{\\widehat{\\mathfrak{sl}}_2}(-(2-1/p+\\ell)\\Lambda_0+\\ell\\Lambda_1)$. $\\mathcal{C}_p$ is built by taking the maximal integrable part, for $\\mathfrak{sl}_2\\times\\mathfrak{sl}_2$, of $V_p\\otimes \\Pi(0)_{1/2}$, where $V_p$ is the Drinfeld-Sokolov reduction of $U_p$, the vertex algebra of chiral differential operators on $SL_2$ at level $-2+1/p$. The paper proves that this algebra is simple and generated by its lowest two graded pieces, that its graded pieces are finite-dimensional, and that its character is modular of weight zero. In small cases the construction lands on known exceptional vertex algebras: $\\mathcal{C}_3\\cong L_{-5/3}(\\mathfrak{g}_2)$, $\\mathcal{C}_4\\cong W_{-23/4}(F_4,A_1+\\tilde A_1)$, and $\\mathcal{C}_5\\cong W_{-30+31/5}(E_8,A_4+A_2)$. This resolves the decomposition of certain conformal embeddings $\\mathfrak{sl}_2\\times\\mathfrak{sl}_2$ into these exceptional algebras and gives explicit modular linear differential equations satisfied by the characters for $p=2,3,4,5$.","pith_inferences":["Beyond the paper: the same inverse-QHR mechanism is sketched in Section 10 for higher rank simple Lie algebras; if it works, there should be analogues $\\mathcal{C}_{p,\\mathfrak{g}}$ for every simple $\\mathfrak{g}$, with $\\mathcal{C}_{2,\\mathfrak{sl}_3}$ expected to match an $F_4$ algebra from the exceptional series.","Beyond the paper: the appearance of $G_2$, $F_4$, and $E_8$ for $p=3,4,5$ suggests that the family may interpolate along the exceptional series; the paper explicitly does not expect affine $W$-algebra isomorphisms for $p\\ge 6$, so a natural test is whether weaker relations (cosets, subalgebras, or identical modular data) persist.","Beyond the paper: because the modularity proof passes through Appell-Lerch series and indefinite theta functions, the characters of $\\mathcal{C}_p$ are natural candidates for mock theta-function interpretations, and checking whether $\\mathcal{C}_6$ satisfies the MLDE predicted by Conjecture 8.2 is a direct numerical test."],"forward_implications":["Every $\\mathcal{C}_p$ has finite-dimensional $\\frac12\\mathbb{Z}_{\\ge 0}$-graded pieces and a convergent character, so the family gives non-rational vertex algebras whose characters are modular rather than merely formal series.","For $p=2,3,4,5$, the algebras are realized as $M(3)$, $L_{-5/3}(\\mathfrak{g}_2)$, $W_{-23/4}(F_4,A_1+\\tilde A_1)$, and $W_{-30+31/5}(E_8,A_4+A_2)$, giving explicit decompositions of the corresponding conformal embeddings into these exceptional algebras.","$\\mathcal{C}_3$, $\\mathcal{C}_4$, and $\\mathcal{C}_5$ are quasi-lisse, and their characters satisfy explicit modular linear differential equations.","For all $p\\ge 2$, $\\eta(\\tau)^6 \\mathrm{ch}[\\mathcal{C}_p]$ is a modular form of weight 3, so $\\mathrm{ch}[\\mathcal{C}_p]$ is modular of weight zero.","The isomorphisms resolve the problem of decomposing certain conformal embeddings of affine vertex algebras into affine $W$-algebras."],"supporting_citations":[{"why":"Supplies $U_p$ as the vertex algebra of chiral differential operators on $SL_2$ at level $-2+1/p$, including its decomposition and simplicity, which is the starting point for $\\mathcal{C}_p$.","marker":"[8]"},{"why":"Provides the complete reducibility and simplicity argument (Section 9.1.6) cited for the key decomposition theorem.","marker":"[3]"},{"why":"Gives the inverse QHR realization of $L_k(\\mathfrak{sl}_2)$ as the integrable part of a Virasoro $\\otimes$ $\\Pi(0)_{1/2}$ module, the central mechanism of the construction.","marker":"[9]"},{"why":"Provides the fusion rules for $L_{-2+1/p}(\\mathfrak{sl}_2)$-modules used to prove simplicity and generation of $\\mathcal{C}_p$.","marker":"[13]"},{"why":"Establishes that quasi-lisse vertex algebras have modular characters satisfying MLDEs, used to conclude quasi-lisse property and to motivate Conjecture 8.2.","marker":"[6]"},{"why":"The uniqueness theorem for minimal $W$-algebras cited by 'same proof as in [7]' in the identification of $\\mathcal{C}_4$.","marker":"[7]"},{"why":"Supplies the quantum reduction theory and strong generation statements for affine $W$-algebras used for $V_p$ and for the $p=4,5$ cases.","marker":"[26]"},{"why":"Provides the Verma-module resolutions used to compute characters of the $\\widehat{\\mathfrak{sl}}_2$ modules appearing in the decomposition.","marker":"[29]"},{"why":"Supplies q-series identities, including the Legendre four-triangular-numbers identity and Appell-Lerch function identities, used in character and modularity proofs.","marker":"[27]"},{"why":"Proves the base case $\\mathcal{C}_2\\cong M(3)$ and gives character formulas used for comparison across the family.","marker":"[12]"}],"fun_headline_variants":["Vertex algebras tie SL2 to G2, F4, and E8","Simple vertex algebras connect SL2 to exceptional groups","New C_p algebras realize exceptional vertex algebras","Regular SL2 representations spawn exceptional vertex algebras"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction rests on cited complete reducibility (semisimplicity) of the module categories for $L_{-2+1/p}(\\mathfrak{sl}_2)$ and $L_{-2-1/p}(\\mathfrak{sl}_2)$ at these specific levels, plus an omitted proof by analogy for the uniqueness of the $F_4$ $W$-algebra; if those cited facts do not hold, the decomposition theorem and the $p=4$ isomorphism lose their support, even though $\\mathcal{C}_p$ would still exist as a graded vertex algebra.","fun_headline_variants_meta":{"raw":{"variants":["Vertex algebras tie SL2 to G2, F4, and E8","Simple vertex algebras connect SL2 to exceptional groups","New C_p algebras realize exceptional vertex algebras","Regular SL2 representations spawn exceptional vertex algebras"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00078,"raw_usage":{"total_tokens":3555,"prompt_tokens":1160,"completion_tokens":2395,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":776,"completion_tokens_details":{"reasoning_tokens":2344}},"tokens_in":776,"tokens_out":2395,"duration_ms":19414,"temperature":1.0,"reasoning_tokens":2344,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T14:34:32.484976+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive numerical check would be to expand both sides of the character formula in Theorem 8.1 to order $q^{10}$ for $p=5$ and compare with the character of $W_{-30+31/5}(E_8,A_4+A_2)$ computed from the strong-generator description; a mismatch would disprove Theorem 6.9. Structurally, the identification $\\mathcal{C}_4\\cong W_{-23/4}(F_4,A_1+\\tilde A_1)$ depends on the omitted 'same proof as in [7]' uniqueness statement, so exhibiting another simple vertex algebra satisfying the three conditions of Proposition 6.4 but not isomorphic to the $F_4$ $W$-algebra would decide that case. At the base, a single non-semisimple module in the category $KL_{-2+1/p}(\\mathfrak{sl}_2)$ (or $KL_{-2-1/p}$) at the relevant negative level would falsify Theorem 3.1.","supporting_citations":[{"cited_title":"Arakawa and A","cited_arxiv_id":null,"evidence_quote":"Provides the complete reducibility and simplicity argument (Section 9.1.6) cited for the key decomposition theorem."},{"cited_title":"Arakawa and K","cited_arxiv_id":null,"evidence_quote":"Establishes that quasi-lisse vertex algebras have modular characters satisfying MLDEs, used to conclude quasi-lisse property and to motivate Conjecture 8.2."},{"cited_title":"Arakawa, T","cited_arxiv_id":null,"evidence_quote":"The uniqueness theorem for minimal $W$-algebras cited by 'same proof as in [7]' in the identification of $\\mathcal{C}_4$."},{"cited_title":"Kac and M","cited_arxiv_id":null,"evidence_quote":"Supplies the quantum reduction theory and strong generation statements for affine $W$-algebras used for $V_p$ and for the $p=4,5$ cases."},{"cited_title":"Malikov, Verma modules over Kac–Moody algebras of ra nk 2, Algebra i Analiz, 1990, Volume 2, Pages 65–84","cited_arxiv_id":null,"evidence_quote":"Provides the Verma-module resolutions used to compute characters of the $\\widehat{\\mathfrak{sl}}_2$ modules appearing in the decomposition."},{"cited_title":"Kac and M","cited_arxiv_id":null,"evidence_quote":"Supplies q-series identities, including the Legendre four-triangular-numbers identity and Appell-Lerch function identities, used in character and modularity proofs."},{"cited_title":"Adamovi´ c, V","cited_arxiv_id":null,"evidence_quote":"Proves the base case $\\mathcal{C}_2\\cong M(3)$ and gives character formulas used for comparison across the family."}],"review_version":1}