{"id":"27fb6c96-a4f4-4e4d-a517-907b1efa6cfc","arxiv_id":"1908.00529","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Integral representations are constructed for solutions of the non-stationary elliptic Calogero-Sutherland equation, yielding elliptic analogues of Jack polynomials for non-integer coupling values.","lead":"This paper gives explicit integral formulas for exact solutions of the elliptic version of the Calogero-Sutherland quantum many-body equation, including non-integer coupling values. The formulas produce elliptic generalizations of Jack polynomials, and for one coupling choice the family is complete.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1 rests on Lemma 4.1 from [11], but the paper's kernel differs from the proven one by a branch-phase factor whose benign nature is asserted rather than demonstrated on the complex integration cycles used in Lemma 4.2.","rationale":"The reader identified the reliance on the kernel identity from [11] as the weakest assumption; I agree that this is the most load-bearing step. However, I do not regard the concern as decisive: the phase-factor argument in the paper is plausible, and the p→0 limit plus the internal consistency of the eigenvalue recurrences provide strong independent support. The proof relies on the kernel identity only through Lemma 4.2, and no internal inconsistency was found in the analyticity, boundary term, or eigenvalue computations. The one point that is genuinely not demonstrated is that the locally constant phase claim survives on the complex shifted integration cycles and for all branch choices implied by the integral transform. Since this is exactly what connects the explicit integral representation to the differential equation, a targeted check is warranted, but nothing in the paper suggests the claim is false. I therefore recommend no change to the reader's ACCEPT verdict, while noting that the verification step would further de-risk the central claim. Agreement is partial because the reader framed the issue as a general phase factor concern, whereas I focus on the specific need for the phase to be locally constant on the complex cycles used in the induction, which is the precise condition that must hold for (27)-(28) to transfer unchanged.","tokens_in":21705,"tokens_out":35003,"duration_ms":306300,"concrete_test":"Independently verify Lemma 4.1 for the modified kernel by direct substitution for the smallest nontrivial case N=2, M=1. Compute K_{2,1}(x_1,x_2;y) = Ψ_2(x)Ψ_1(y)[ϑ(x_1-y)ϑ(x_2-y)]^g with Ψ_2(x) = (ϑ(x_1-x_2)^2)^{g/2} and Ψ_1(y)=1, and check that (27) D_2(x)K + D_1(y)K = 0 and (28) (g i/π ∂_τ + H_2(x) − H_1(y) − c_{2,1})K = 0 hold exactly with c_{2,1} as in (39). Repeat for complex y with Im(y) = −ε and for generic non-integer g (e.g., g = 3/4, g = 5/3) using a computer algebra system that tracks branches of ϑ^g. Any nonzero residual, or any dependence of the residual on ε, would show the phase factor is not benign and the induction in Section 4.3 needs modification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, Theorem 3.1(b), is proven by induction in Section 4.3. At each step, Lemma 4.2 converts a solution for M particles into one for N = M + k particles, and the proof of Lemma 4.2 uses the functional identities (27)-(28) of Lemma 4.1, which is quoted from Ref. [11] and not proved in the present paper. The paper notes a difference: the kernel K_{NM} in (38) uses Ψ_N(x) = (∏_{j<k} ϑ(x_j-x_k)^2)^{g/2}, whereas [11] proves the identities for ∏_{j<k} ϑ(x_j-x_k)^g. The authors state that these differ by a locally constant phase, so the identities are unaffected. This is plausible for real x because ϑ^2 ≥ 0 and (ϑ^2)^{g/2} = |ϑ|^g, but in Lemma 4.2 the integrations are over shifted complex contours y ∈ [−π−iε, π−iε]^M, and the input ψ_M involves Ψ_M(y) with y complex. On these contours the ratio of the two functions is a phase that is locally constant only if the branch of ϑ^g is chosen consistently with the product representation (36); a mismatch of branch cuts could produce additional terms under the derivatives in (27)-(28), changing the eigenvalue computation and invalidating (33). Since the induction is the only mechanism connecting the explicit integral (21a) to the non-stationary eCS equation, an unverified phase-factor modification is the least secure link in the proof. If the modified kernel does not satisfy (27)-(28), Theorem 3.1(b) could fail even though the p→0 limit (Part (c)) is independently supported by the Jack polynomial integrals.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs explicit solutions of the non-stationary elliptic Calogero-Sutherland (eCS) equation for coupling constant κ = kg with g > 1/2 and integer k ≥ 1. The main result, Theorem 3.1, states that for L ≥ 1, integer vectors r, and block sizes s = (s1, k^{L-1}) with s1 = 1 for k = 1 and s1 ∈ {1, k} for k ≥ 2, the function ψ_{r,s,L}(x;τ) = (∏_{j≠k} θ(z_j/z_k;p))^{g/2} P_{r,s,L}(z;p), with P_{r,s,L} defined by the multiple integral (21a), satisfies the non-stationary eCS equation (18) with eigenvalue (22) and momentum eigenvalue (23). The integrals are shown to be analytic in the annulus (21b), and in the limit p→0 they reduce, up to explicit nonzero constants, to the Awata-Matsuo-Odake-Shiraishi integral representations of Jack polynomials when r is ordered; for unordered r the limit vanishes. The proof is by induction, using a generalized kernel identity from Ref. [11] (Langmann 2006), with base cases for one and k particles and an integral transform (Lemma 4.2) that raises the particle number by k. Analyticity and boundary-term arguments are deferred to Appendices B and C.","tokens_in":22049,"tokens_out":26365,"duration_ms":239666,"significance":"If correct, Theorem 3.1 provides explicit integral representations of elliptic generalizations of Jack polynomials for continuous coupling g > 1/2, extending earlier representation-theoretic constructions that required integer g. The formulas are explicit and parameter-free, involving only theta functions and contour integrals; the p→0 limit recovers the known Jack-integral formulas of Awata et al. For κ = g the construction is complete (all integer vectors λ), while for k ≥ 2 it gives a natural subfamily. The proof is self-contained except for one quoted kernel identity from the published paper [11], for which the authors supply a translation table. The paper also re-proves the Awata et al. integral formulas in Appendix B and discusses a concrete test of Shiraishi's conjecture on non-stationary eCS functions.","major_comments":[],"minor_comments":[{"comment":"The remark that the difference between Ψ_N in (37) and ∏_{j<k} ϑ(x_j-x_k)^g is a locally constant phase is correct, but it is terse. I verified that on the domain used in Lemma 4.2 the issue is benign: the differences y_j-y_k are real on the integration contour, and the arguments x_j-y_k avoid the zero set of ϑ, so a compatible branch choice makes the phase factor exactly constant and the identities of Ref. [11] apply verbatim. A sentence making this branch choice explicit would preempt any concern about extra terms under the derivatives in (27)-(28).","section":"§4.3.1, Lemma 4.1"},{"comment":"There are typos in the abstract: 'represenations' should be 'representations' and 'polyomials' should be 'polynomials'; in the keywords, 'f unction' should be 'function'.","section":"Abstract and keywords"},{"comment":"In Eq. (21a), the notation is dense; a short parenthetical identifying the case L = n, s1 = 1, k = 1 as the elliptic generalization of the Jack integral (12a) would help readers navigate between the two formulas.","section":"§2.2 and §3"},{"comment":"The limit 'lim_{p→0} P_{r,s,L}(z;p)' is stated without specifying the mode of convergence. A brief note that the convergence is uniform on compact subsets of the annulus (by dominated convergence, given the contour conditions in (21b)) would make the statement precise.","section":"Theorem 3.1(c)"}],"recommendation":"minor_revision","confidential_remarks":"The paper's central construction depends on a kernel identity from a co-author's earlier paper [11]. This is a standard and acceptable practice; the identity is published and the translation table is helpful. The stress-test concern about the phase factor does not, on close reading, land: on the integration cycles used in Lemma 4.2 the differences y_j-y_k are real and the factors ϑ(x_j-y_k) avoid zero, so the phase can be chosen constant and the quoted identities apply. The remaining issues are local and presentational. The paper is within the scope of the journal and the novelty is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear X,\n\nThe thing to know: this paper constructs explicit integral solutions of the non-stationary elliptic Calogero-Sutherland equation for non-integer coupling g > 1/2 and kappa = kg, and for kappa = g it gives a complete family of elliptic generalizations of Jack polynomials. That genuinely extends earlier representation-theoretic work by Etingof-Kirillov and Felder-Varchenko, which required integer g. The n = 2 case was already known, but the general result and the completeness statement are new.\n\nThe paper does a lot of things right. The proof of Theorem 3.1 is detailed, with appendices for analyticity and boundary terms. The p -> 0 limit reduces to the known Awata-Matsuo-Odake-Shiraishi integral representation of Jack polynomials, which is a strong consistency check. The authors also openly state that the central kernel identities are quoted from Langmann’s 2006 paper, and they provide a translation table. This is honest scholarship.\n\nThe soft spot is exactly the one the stress-test note flags. The kernel K_NM in (38) differs from the one proved in [11] by a phase factor: here one uses (prod \\vartheta^2)^{g/2}, while [11] proves the identities for (prod \\vartheta)^g. The authors assert the difference is a locally constant phase and does not affect the identities. That is plausible for real x, but Lemma 4.2 uses these identities on shifted complex contours, and the branch structure of the g-th powers is not analyzed carefully enough to rule out extra terms under differentiation. This is asserted, not demonstrated. I don’t think it is fatal, but a referee should ask the authors to close that gap, either by re-proving the kernel identities for the modified kernel on the relevant cycles or by citing a version that already covers it.\n\nThe intended audience is people working on elliptic KZB equations, Calogero-Sutherland models, and special functions. The paper is worth their time. It deserves a serious referee: I would send it to peer review, with the phase-factor issue as the main request for revision.","headline":"New integral solutions for non-stationary eCS at non-integer coupling, with a genuinely useful construction and a small but real gap in the kernel-identity argument.","tokens_in":22604,"tokens_out":8060,"would_cite":true,"duration_ms":80633,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-14T15:48:45.258235+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}