{"id":"f92297f1-8a44-49f9-ae81-432a2550bd07","arxiv_id":"2608.01031","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Double super Yangians for arbitrary 0^m1^n sequences are defined, their two presentations are shown isomorphic, and level-1 bosonizations are constructed.","lead":"This paper builds double super Yangians for all parity sequences 0^m1^n, proves that their R-matrix and Drinfeld presentations agree, and constructs level-1 bosonic representations. It gives a uniform framework for super Yangian doubles that previously existed only for the standard parity order.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.6 transfers the linear-independence proof of [3, Thm 2.6] to arbitrary parity sequences and a new ordering in one unproved sentence; this PBW basis underlies the classical limit and the injectivity of Theorem 5.1, so the central isomorphism rests on an unverified step.","rationale":"The reader’s CONDITIONAL verdict is appropriate, and the weakest point identified is indeed the most load-bearing one. The PBW linear-independence transfer in Lemma 3.6 is a single unproved assertion, yet it supports Proposition 3.9 and the injectivity half of Theorem 5.1, the paper’s central isomorphism. I considered whether the incomplete verification of the general-N current relations in Theorem 4.10 or the reliance on the unpublished preprint [38] in Section 7 could be even more serious, but those are downstream or supplementary: Theorem 4.10’s completeness matters only after the PBW basis and the Gaussian-correspondence are established, and Section 7 concerns the sl-case extension, not the core gl-isomorphism. The paper has real scaffolding—R-matrix formalism, quasideterminant Gauss decomposition, and known results for the standard sequence—so the gap is probably fillable, but the manuscript as written does not close it. The verdict therefore stays CONDITIONAL: the paper should be accepted only after the linear-independence claim is either proved for the new ordering and arbitrary s or replaced by an explicit argument via Lemma 3.4 transporting the standard PBW basis.","tokens_in":31298,"tokens_out":11676,"duration_ms":117789,"concrete_test":"For a small non-standard case, e.g. N=3 with s=010 (gl_{1|2}), explicitly compute the commutation relations (3.14)–(3.16) and calculate the dimension of the span of the ordered monomials B of total degree 2 in DY^0_h(gl^s_{m|n}). Compare this dimension with the corresponding filtered piece of U(\\widehat{gl}^s_{1|2}) given by the PBW theorem for affine gl_{1|2}. If the dimensions agree, the transferred independence claim is plausible; if they disagree, Lemma 3.6 fails and Theorem 5.1’s injectivity proof collapses. Alternatively, write out the proof of [3, Theorem 2.6] line by line and check whether any step uses the standard parity signs or the specific generator order rather than only the filtration degrees (r−1 and −r).","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Lemma 3.6, after a long spanning argument, linear independence is dispatched with: “since the proof of linear independence in [3, Theorem 2.6] does not depend on the generator ordering or the parity sequence, the same argument establishes that B is linearly independent.” This is load-bearing. The PBW basis of Theorem 3.7 is used in Proposition 3.9 to prove the classical-limit isomorphism, and Theorem 5.1’s injectivity proof invokes Proposition 3.9 together with the PBW theorem for U(\\widehat{gl}^s_{m|n}) to assert that ordered Gaussian monomials form a basis of DY_h(gl^s_{m|n}). If the independence argument in [3] implicitly uses the standard ordering’s convexity or the specific d_i signs, the new order ≺ defined by (j−i, i, r) could allow extra linear dependencies, breaking the PBW basis and hence the injectivity of the Drinfeld-to-R-matrix map. Remark 3.8 concedes that the PBW basis here differs from [3]’s even for the standard sequence, so the transfer is not a mere restatement. No computation or auxiliary argument is given. While Lemma 3.4 might provide an alternative route by transporting the standard-case PBW basis, the paper does not use that route; as written, the proof of Theorem 5.1 depends on an unproved assertion.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces double super Yangians DY_h(gl^s_{m|n}) and DY_h(sl^s_{m|n}) attached to arbitrary 0^m1^n parity sequences s, and claims three main results: (i) an explicit isomorphism between the R-matrix presentation and a Drinfeld-type current presentation of DY_h(gl^s_{m|n}); (ii) a quantum Berezinian construction that yields the R-matrix presentation of DY_h(sl^s_{m|n}) and its equivalence with a Drinfeld presentation; (iii) explicit level-1 bosonic Fock representations of both algebras. The proofs follow the strategy of Yang–Jing for type A double Yangians and Gow/Peng for super Yangians with arbitrary parity sequences. The main technical engine is a Poincaré–Birkhoff–Witt type basis for DY_h(gl^s_{m|n}) (Theorem 3.7) and the resulting classical-limit isomorphism (Proposition 3.9), which are then used to prove injectivity of the Drinfeld-to-R-matrix map in Theorem 5.1.","tokens_in":31648,"tokens_out":7020,"duration_ms":73154,"significance":"If the results are correct, the paper resolves the isomorphism problem between R-matrix and Drinfeld presentations for type A double super Yangians for all parity sequences, extending the standard-sequence results of Bagnoli–Kožić and Zhang. It also provides concrete bosonic modules, which are valuable for applications in representation theory and mathematical physics. The paper is well organized and the overall strategy is coherent. Its main strengths are the explicit Gauss-decomposition formulas, the uniform treatment of all parity sequences, and the explicit formulas for the quantum Berezinian and the level-1 bosonization. However, several load-bearing steps are either asserted without proof or outsourced to unpublished/non-verifiable references; these gaps need to be addressed before the results can be considered established.","major_comments":[{"comment":"Linear independence of the set B is dispatched with the sentence: \"the proof of linear independence in [3, Theorem 2.6] does not depend on the generator ordering or the parity sequence.\" This is a load-bearing assertion: Theorem 3.7 (PBW basis), Proposition 3.9 (classical limit isomorphism), and the injectivity part of Theorem 5.1 all rely on it. Moreover, Remark 3.8 explicitly states that the total order used here differs from that in [3] even for the standard sequence, so the transfer is not a mere restatement. The order (j−i, i, r) is new, and the signs d_i are precisely what varies with s. Please supply a full proof of linear independence for this ordering and parity sequence, or alternatively prove injectivity by transporting the standard-sequence PBW basis through Lemma 3.4.","section":"§3.2, Lemma 3.6"},{"comment":"Theorem 4.10 lists relations for all i, including the non-adjacent relations (4.56) and (4.60) and the Serre relations (4.61) and (4.62). The proof for N≥4 only verifies relations between the two endpoint sets {k_1,e_1,f_1} and {k_N,e_{N-1},f_{N-1}} and then asserts that induction and Lemma 4.5 give the rest. The embedding argument is not written out; in particular, Lemma 4.9 proves the Serre relation (4.39) only for a single i. If any of the relations in Theorem 4.10 is not actually implied by the checked cases, then dDY_h(gl^s_{m|n}) in Theorem 5.1 may be strictly larger than the image of DY_h(gl^s_{m|n}), breaking the claimed isomorphism. Please spell out the induction/embedding procedure for every family of relations.","section":"§4.3.3, Lemma 4.9 / Theorem 4.10"},{"comment":"Section 7 relies on the authors' unpublished preprint [38] for several key statements: Lemma 7.2 is proved by reference to [38, Lemma 4.4], Lemma 7.4 by reference to [38, Lemma 4.7], and Proposition 7.3 follows the argument of [38]. Since [38] is not publicly available, the referee cannot verify these steps. These lemmas are load-bearing for the entire construction of DY_h(sl^s_{m|n}) and its R-matrix presentation. Please either include full proofs in the paper or replace the references to [38] with a published source or with an appendix containing the needed results.","section":"§7, Lemmas 7.2 and 7.4; Proposition 7.3"},{"comment":"The injectivity of φ: dDY_h(sl^s_{m|n}) → DY_h(sl^s_{m|n}) is dismissed with \"The same arguments as in the proof of Theorem 5.1 show that φ is also injective.\" Those arguments require a PBW-type spanning set for dDY_h(sl^s_{m|n}) and linear independence of the corresponding ordered monomials inside DY_h(sl^s_{m|n}). The paper does not construct such a basis for the sl-type algebra, nor does it prove that the currents H_i, E_i, F_i admit an ordered monomial basis in the subalgebra DY_h(sl^s_{m|n}). Without this, the claimed isomorphism in Theorem 7.6 is not established. Please provide the missing basis argument.","section":"§7, Theorem 7.6"},{"comment":"The proof of Theorem 8.2 explicitly verifies only the relation (4.57) among the currents, and ends with \"The remaining relations in Theorem 4.10 follow by similar calculations.\" To define a module over DY_h(gl^s_{m|n}), all relations (4.48)–(4.63) must be checked, including the k–X commutations (4.52)–(4.56), the mixed X^+–X^- relation (4.63), and the Serre relations (4.61)–(4.62). The associated operator product expansions are nontrivial and sign-sensitive because of the parity sequence. Please include the full set of checks, or at least the key OPEs, so that the Fock-space module structure is actually proven.","section":"§8.1, Theorem 8.2"}],"minor_comments":[{"comment":"The notation \"0^m1^n--sequence\" in the abstract and in Section 2 is typeset inconsistently (e.g., \"0m1n--sequences\" in the abstract). Use a consistent math mode notation.","section":"Abstract and throughout"},{"comment":"The text says \"Given a 01–sequence s = ...\"; it should say \"Given a 0^m1^n-sequence s = ...\" for clarity.","section":"Definition 6.4"},{"comment":"The delta symbol δ(v/u) is used before being defined in the same line. It would help to define it just before first use, e.g., in the N=2 subsection.","section":"Eq. (4.16) and (4.63)"},{"comment":"The proof of independence of the parity sequence is short but correct; however, it would be helpful to state explicitly that the permutation σ is chosen so that d_{σ(i)} = d_i, since this is what makes the relabeling preserve the R-matrix relations.","section":"Lemma 3.4"},{"comment":"The remark that brackets are ordinary is useful, but it would be even clearer to say explicitly that the super signs in the commutation relations of the Drinfeld currents are already encoded in the relations being verified, and the bosonic operators are even.","section":"Remark 8.1"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a natural and timely question, and the overall structure is plausible. The main risk is the unproved transfer of the PBW linear-independence argument in Lemma 3.6, which is the cornerstone of the isomorphism proof. The heavy reliance on the authors' own unpublished preprint [38] in Section 7 is also a concern for a journal publication. I would recommend major revision, with the authors asked either to supply full proofs for these steps or to restructure the paper so that the central claims do not depend on unverifiable assertions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this one. First, it does something genuinely new: prior double super Yangian papers treat only the standard 0^m1^n parity sequence, and here the authors write down explicit Drinfeld-type presentations, a quantum Berezinian, and level-1 bosonic modules for every parity sequence. Second, the main structural theorem—isomorphism of the Drinfeld and R-matrix presentations—currently leans on a single sentence in Lemma 3.6 that transfers the PBW linear-independence proof from [3] to the new ordering and arbitrary parity, with no auxiliary argument. That sentence is load-bearing: the PBW basis is used in Proposition 3.9 and again in the injectivity half of Theorem 5.1. Remark 3.8 even admits the order differs from [3] in the standard case, so the transfer is not a formality. The stress-test note got this right. There is a possible easy fix—Lemma 3.4 shows the algebras for different sequences are isomorphic, so one could transport the standard-case PBW basis—but the paper does not take that route. As written, the proof of the central theorem has a real hole. That is the main soft spot, and it is moderate, not fatal. The other weaknesses are smaller and more standard: the full set of Gaussian relations in Theorem 4.10 is extended from N=2,3 by induction with the dirty work summarized, the sl-section leans on the authors' unpublished preprint [38], and the bosonic module verification in Theorems 8.2/8.3 ends with \"similar calculations.\" None of these are obviously wrong; a competent referee could probably fill them in, but they should be asked to. What the paper does well: the strategy is coherent, the Gauss decomposition is handled cleanly, the quantum Berezinian is extended via simple reflections in a believable way, and the bosonization formulas are explicit enough to be checked by direct computation. The citation pattern is mostly healthy—[2] and [3] are the natural references for the standard case, and the self-citation to [38] is worth flagging but not disqualifying. So who gets value from this? People working on Yangians, double Yangians, and representations of superalgebras, especially those interested in parity-dependent presentations and Bethe-ansatz-type applications. It is a subfield-level contribution, not a breakthrough. My recommendation: send it to peer review. A serious referee should ask for a complete proof of the PBW linear independence (or an explicit use of Lemma 3.4 to bypass it), fuller details on the induction in Theorem 4.10, and either a proof or a reference for the sl-case arguments currently outsourced to [38]. With those fixed, this would be a useful, citable paper.","headline":"Solid extension of double super Yangian technology to all parity sequences, but the central isomorphism proof rests on a one-sentence transfer of a PBW independence argument; worth refereeing, but the gaps need to be filled.","tokens_in":32171,"tokens_out":2925,"would_cite":true,"duration_ms":33144,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17B37","17B67"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every $0^m1^n$ parity sequence, the double super Yangian has two matching presentations and explicit level-1 boson modules.","keywords":["double super Yangian","Drinfeld presentation","R-matrix presentation","quantum Berezinian","PBW basis","bosonic representation","parity sequence","level-1 module"],"falsifier":"Compute the determinant of the linear map from R-matrix monomials to the ordered PBW monomials for a non-standard sequence such as $s=1010$ at low order in $h$; a nontrivial kernel would refute Lemma 3.6. Alternatively, plug the bosonic assignment of Theorem 8.2 into the Serre relation (4.62) for a sequence with $|\\alpha_i|=1$ and check that the coefficient of every non-symmetric monomial vanishes.","tokens_in":31145,"feed_emoji":"🧮","tokens_out":6389,"duration_ms":61443,"temperature":0.7,"pith_summary":"This paper extends the double super Yangian construction, previously known for the standard parity sequence, to every $0^m1^n$ sequence that labels the even and odd basis directions in $\\mathfrak{gl}(m|n)$. It proves that for each such sequence the algebra defined by R-matrix relations is isomorphic to the algebra presented by Drinfeld currents $k_i^\\pm(u), e_j^\\pm(u), f_j^\\pm(u)$, so the two presentations describe the same object. It also constructs a quantum Berezinian for arbitrary parity and uses it to define and identify the $\\mathfrak{sl}$ version, then gives explicit level-1 Fock-space (bosonic) representations for both algebras. If correct, this means the representation theory of type-A double super Yangians is independent of the choice of parity sequence, and the bosonized modules provide concrete infinite-dimensional representations for every sequence.","feed_headline":"All parity sequences yield one double super Yangian","feed_subtitle":"Explicit level-1 boson modules exist for every gl^s(m|n) and sl^s(m|n), extending the standard type-A story.","key_machinery":"The load-bearing tools are: (1) the Gauss decomposition of the generating matrix $T^\\pm(u)$ into quasideterminantal series ('Gaussian generators') $k_i^\\pm(u)$, $e_j^\\pm(u)$, $f_j^\\pm(u)$, which yields the Drinfeld currents; (2) a PBW-type basis with a new total order, whose associated graded algebra is the enveloping algebra of the affine Lie superalgebra $\\widehat{\\mathfrak{gl}}^s_{m|n}$; (3) the quantum Berezinian, a multiplicative power series in the center that separates the $\\mathfrak{gl}$ and $\\mathfrak{sl}$ double super Yangians.","core_discovery":"The paper's central claim is that for fixed $(m,n)$ and any parity sequence $s$, the R-matrix double super Yangian $\\mathrm{DY}_h(\\mathfrak{gl}^s_{m|n})$ is isomorphic to the Drinfeld-current algebra generated by the coefficients of $k_i^\\pm(u)$, $e_j^\\pm(u)$, $f_j^\\pm(u)$ with the relations of Theorem 4.10; the analogous isomorphism holds for $\\mathrm{DY}_h(\\mathfrak{sl}^s_{m|n})$ using currents $H_i^\\pm(u)$, $E_i(u)$, $F_i(u)$ with the relations of Lemma 7.5. The proof routes through a PBW-type basis for the R-matrix algebra, its classical limit to $U(\\widehat{\\mathfrak{gl}}^s_{m|n})$, and the Gauss decomposition of the generator matrix into quasideterminants. It then produces level-1 modu","pith_inferences":["The same PBW-transfer strategy may prove analogous presentation isomorphisms for double super Yangians of orthosymplectic type with arbitrary parity data; this extension is not stated in the paper.","In the specialization $h\\to 0$, the level-1 bosonic modules should recover standard level-1 modules of the affine Lie superalgebra $\\widehat{\\mathfrak{gl}}^s_{m|n}$; a character comparison would test this consequence.","The $h$-shifts inside the quantum Berezinian suggest a link to shifted Yangian structures, where the shift parameters are determined by the parity sequence; the paper does not discuss this."],"forward_implications":["For any fixed $(m,n)$, the double super Yangians for different parity sequences $s$ are isomorphic, so odd reflections do not produce new algebras.","The Drinfeld presentation is available for arbitrary $s$, making current-based methods such as Bethe ansatz and vertex-operator constructions applicable for any parity sequence.","The quantum Berezinian gives central elements and, for $m\\neq n$, a tensor decomposition $\\mathrm{DY}_h(\\mathfrak{gl}^s_{m|n}) \\cong Z \\otimes \\mathrm{DY}_h(\\mathfrak{sl}^s_{m|n})$.","The explicit level-1 bosonic modules give concrete infinite-dimensional representations for all $\\mathfrak{gl}$ and $\\mathfrak{sl}$ double super Yangians in type A for every parity sequence."],"supporting_citations":[{"why":"Supplies the strategy for proving the Drinfeld-to-R-matrix isomorphism in the non-super double Yangian setting, which the paper adapts.","marker":"[39]"},{"why":"Provides the PBW basis and double Yangian construction for the standard parity sequence that the paper extends.","marker":"[3]"},{"why":"Gauss decomposition of the super Yangian, used to define the Gaussian generators.","marker":"[17]"},{"why":"Quantum Berezinian for the standard double super Yangian, generalized here to arbitrary parity.","marker":"[2]"},{"why":"Parabolic presentations of super Yangians for arbitrary 01-sequences, used for embeddings and parity independence.","marker":"[33]"},{"why":"Bosonic representations of Yangian doubles for $\\mathfrak{gl}_N$ and $\\mathfrak{sl}_N$, the template for the level-1 construction.","marker":"[18]"},{"why":"Yangian doubles of classical types and vertex representations, providing the level-1 framework for other types.","marker":"[21]"},{"why":"Drinfeld's new realisation of Yangians, the origin of the current presentation.","marker":"[9]"}],"fun_headline_variants":["Every parity sequence yields a double super Yangian","Level-1 boson modules for every gl^s(m|n) and sl^s(m|n)","Drinfeld and R-matrix presentations unified for double super Yangians","Double super Yangians exist for any 0^m 1^n sequence","Bosonic representations for all double super Yangians in type A"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The PBW linear-independence argument is taken from the standard-parity case and assumed to transfer unchanged to every parity sequence and to the new generator ordering; that transfer is not proved in detail and is load-bearing for the classical-limit and isomorphism theorems.","fun_headline_variants_meta":{"raw":{"variants":["Every parity sequence yields a double super Yangian","Level-1 boson modules for every gl^s(m|n) and sl^s(m|n)","Drinfeld and R-matrix presentations unified for double super Yangians","Double super Yangians exist for any 0^m 1^n sequence","Bosonic representations for all double super Yangians in type A"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001079,"raw_usage":{"total_tokens":4388,"prompt_tokens":820,"completion_tokens":3568,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":564,"completion_tokens_details":{"reasoning_tokens":3470}},"tokens_in":564,"tokens_out":3568,"duration_ms":26697,"temperature":1.0,"reasoning_tokens":3470,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T00:33:42.774019+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the determinant of the linear map from R-matrix monomials to the ordered PBW monomials for a non-standard sequence such as $s=1010$ at low order in $h$; a nontrivial kernel would refute Lemma 3.6. Alternatively, plug the bosonic assignment of Theorem 8.2 into the Serre relation (4.62) for a sequence with $|\\alpha_i|=1$ and check that the coefficient of every non-symmetric monomial vanishes.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the strategy for proving the Drinfeld-to-R-matrix isomorphism in the non-super double Yangian setting, which the paper adapts."},{"cited_title":"Bagnoli, S","cited_arxiv_id":null,"evidence_quote":"Provides the PBW basis and double Yangian construction for the standard parity sequence that the paper extends."},{"cited_title":"Gow, Gauss decomposition of the YangianY(gl m|n), Comm","cited_arxiv_id":null,"evidence_quote":"Gauss decomposition of the super Yangian, used to define the Gaussian generators."},{"cited_title":"Bagnoli, S","cited_arxiv_id":null,"evidence_quote":"Quantum Berezinian for the standard double super Yangian, generalized here to arbitrary parity."},{"cited_title":"Peng, Parabolic presentations of the super YangianY(gl M|N ) associated with arbi- trary 01-sequences, Comm","cited_arxiv_id":null,"evidence_quote":"Parabolic presentations of super Yangians for arbitrary 01-sequences, used for embeddings and parity independence."},{"cited_title":"Iohara, Bosonic representations of Yangian doubleDY ℏ(g) withg=gl N ,sl N , J","cited_arxiv_id":null,"evidence_quote":"Bosonic representations of Yangian doubles for $\\mathfrak{gl}_N$ and $\\mathfrak{sl}_N$, the template for the level-1 construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Yangian doubles of classical types and vertex representations, providing the level-1 framework for other types."},{"cited_title":"Drinfeld, A new realization of Yangians and of quantum affine algebras, Dokl","cited_arxiv_id":null,"evidence_quote":"Drinfeld's new realisation of Yangians, the origin of the current presentation."}],"review_version":1}