{"id":"f9786bf4-e43c-4f3d-bdef-af88e65af4cb","arxiv_id":"2511.22822","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Residues and special values of supersymmetric zeta functions (built from single-particle indices) are conjectured to reproduce anomaly coefficients, Casimir energies, and central charges across 2d/4d/6d.","lead":"Supersymmetric zeta functions — Mellin transforms of (plethystic-log) superconformal indices — are proposed as new spectral tools whose residues and special values encode Cardy-like asymptotics, Casimir energies, and central charges. The paper catalogs 2d, 4d, and 6d examples; the headline relations hold in examples but are proposed, not derived.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central bridge E_SUSY = ½Z(−1,0) is only checked for free fields; its use on non-Lagrangian SCFTs makes the 'converse' claim a conjecture, not a derivation.","rationale":"I read the paper as proposing a new spectral tool, with the central claim that residues and special values of supersymmetric zeta functions are universally tied to anomaly coefficients and central charges. The Mellin-transform machinery and the Cardy-like expansions are internally coherent, and the free-field checks are clean. The weakest point is the conversion of zeta values into physical central charges: eq. (2.76) is explicitly called 'highly non-trivial' by the authors and is verified only for free theories. For the non-Lagrangian AD theories in §5.2.5–5.2.8, the paper applies it without a derivation, and the input single-particle indices are themselves conjectural (e.g., eq. 5.259 from [137]). The fact that the output matches previously known central charges is encouraging but does not establish the 'conversely' direction as an independent route. The residue-to-anomaly formulas are likewise imported from known Cardy-limit results and anomaly-polynomial computations, so the universal statements are proposals supported by examples rather than theorems. I therefore agree with the reader's weakest_assumption. A concrete finite-N interacting test, such as N=4 SU(N) SYM, would directly test whether eq. (2.76) survives beyond free fields. Until such a test is done, the conditional verdict is appropriate.","tokens_in":85814,"tokens_out":7865,"duration_ms":74172,"concrete_test":"Compute Z(−1,0) for 4d N=4 SU(N) SYM from the plethystic logarithm of its finite-N superconformal index (all fugacities set equal, same R-twist as §5.1) and compare with the independent localization/anomaly-polynomial result E_SUSY from Bobev–Bullimore–Kim [66] using the known central charges a=c=(N²−1)/4. Run this for N=2 and N=3. If ½Z(−1,0) differs from the [66] value, eq. (2.76) is a free-field artifact and the claimed central-charge extraction from zeta values has no demonstrated foundation for interacting SCFTs; if it matches, the working assumption is strongly supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The headline claim that anomaly coefficients and central charges can be read off from supersymmetric zeta functions depends on the identification E_SUSY = ½ Z(−1,0) (eq. 2.76), introduced in §2.4.4. For free theories this is equivalent to Kim's regulated trace (eq. 2.78), but for interacting and non-Lagrangian SCFTs there is no derivation: the single-particle index is a plethystic-log generating function, not the spectrum of one-particle states, and the vacuum energy is not obviously its first zeta-regularized moment. The paper uses precisely this assumption in §5.2.5–5.2.8 to convert Z_Schur(−1,0) (and analogues) into c4d and a4d for AD theories. Moreover, the residue-to-anomaly identities (5.8), (5.140), (6.7)–(6.9), and (6.10) are imported from earlier Cardy-limit/anomaly computations and are only 'observed' here, so the asserted converse is currently a consistency check against known central charges, not an independent spectral extraction. The relation could fail for interacting theories even if all free-field examples work. A further self-admitted caveat is the constant-term ambiguity in the degeneracy formula (§2.4.3, §4.1.1), which complicates extracting special values from growth data, though the main blocker is the unproven bridge (2.76).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces 'supersymmetric zeta functions' Z(s,z) and 'supersymmetric determinants' D(z) associated with the plethystic logarithm (single-particle index) of supersymmetric indices. It derives Mellin-transform relations between the zeta functions and the Cardy-like asymptotics of indices, and between the determinants and the constant term in that asymptotics. It then proposes two universal claims: (i) residues and special values of these zeta functions are determined by anomaly coefficients/central charges of superconformal field theories, and (ii) conversely, anomaly coefficients and central charges can be read off from the zeta functions, using the relation E_SUSY = 1/2 Z(-1,0) (Eq. 2.76). The bulk of the paper computes these zeta functions and determinants for a large number of 2d, 4d, and 6d theories, including free multiplets, SYM, minimal models, and Argyres-Douglas theories, and checks the proposed relations against known central charges.","tokens_in":86259,"tokens_out":3719,"duration_ms":35686,"significance":"If the proposed relations hold, the paper offers a new spectral-language bridge between BPS counting data and conformal anomaly data, with potential applications to Cardy-like limits, Casimir energies, and the 'spectral extraction' of central charges for non-Lagrangian theories. The paper is explicit and computationally rich: it provides concrete zeta functions, determinants, residues, and zeta values for dozens of theories, and in all examples where the comparison is possible it reproduces previously known central charges and anomaly coefficients. These checks give the conjectures nontrivial support. However, the core universal relations are presented as proposals or observations rather than derivations, and the argument that the 'converse' extraction is actually independent of the input data is not fully established.","major_comments":[{"comment":"The central bridge E_SUSY = 1/2 Z(-1,0) is introduced as a 'non-trivial relation' and checked explicitly only for free theories via Kim's regularized trace (2.77)-(2.78). For interacting and non-Lagrangian theories, including the AD theories in §5.2.5-5.2.8, this relation is used as a working assumption to convert zeta values into central charges. But for such theories the single-particle index is a plethystic-logarithm generating function, not literally the one-particle spectrum, so the zeta-regularized first moment is not manifestly the supersymmetric Casimir energy. This is load-bearing for the paper's 'converse' claim: without a derivation or an independent check of (2.76) beyond free fields, the extraction of central charges from zeta values rests on an unproved conjecture. Please either provide a derivation (e.g., from the anomaly polynomial / equivariant integral framework) or exp","section":"§2.4.4, Eq. (2.76)"},{"comment":"The universal residue-to-anomaly identities (5.8), (5.140), (6.7)-(6.9) and the special-value formula (6.10) are not derived in this paper; they are imported from earlier Cardy-limit / anomaly computations (di Pietro-Komargodski, etc.) or 'observed' from the examples. The 6d formula (6.10) in particular is checked only for free hyper/tensor/vector multiplets. Since the headline claim includes the converse statement that anomaly coefficients can be 'calculated from' the zeta functions, the paper should state clearly that the zeta functions reproduce, rather than independently derive, these anomaly relations. As written, the converse is a rearrangement of previously known input, not a new spectral derivation. Please either prove the relevant universal relations from the zeta-function formalism or temper the converse claim accordingly.","section":"§5.1, Eq. (5.8); §5.2, Eq. (5.140); §6, Eqs. (6.7)-(6.10)"},{"comment":"The asymptotic degeneracy formula (2.64) contains a constant term Z(0,0)' = log D(0) whose determination from growth data is ambiguous, as the authors themselves note in §2.4.3 and as the numerical example in §4.1.1 shows (the ratio of asymptotic to exact value approaches 2, not 1, for large n). This does not affect the leading poles, but it means that special values such as Z(0,0)' and D(0) cannot be robustly extracted from the asymptotic growth of coefficients without additional input. Since the paper emphasizes the spectral origin of the zeta values, the authors should either resolve this ambiguity or clearly mark it as an obstruction to the 'read off from growth data' aspect of their program.","section":"§2.4.3 and §4.1.1"}],"minor_comments":[{"comment":"The Mellin-transform derivation uses both q and t as integration variables, which can confuse; please use a single dummy variable and clarify the contour/regularization of the exchange of sum and integral.","section":"§2.2, Eq. (2.7)"},{"comment":"The asymptotic formula contains a factor sin(pi r/2) even though it is then evaluated at r=1; please indicate whether this factor is meant to be kept for general r or is a typo.","section":"§4.1.1, Eq. (4.32)"},{"comment":"The table would be easier to read if the columns 'c_4d' and 'a_4d' included the numerical values for the smallest k, as the corresponding table in §5.2.7 does; currently only the closed forms are given.","section":"§5.2.5, table (5.270)"},{"comment":"The paper is very long and contains many examples; a summary table or flowchart collecting the universal formulas (residue/anomaly relations, E_SUSY = 1/2 Z(-1,0)) and the status of each (derived vs. observed vs. conjectured) would greatly improve readability and would make the paper's foundational assumptions more transparent.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper is a substantial compendium of explicit computations and is likely to be of interest to the hep-th community. The main concern is that the central conjecture (2.76) is used as a derivation tool for the 'converse' claim, yet it is only free-theory checked. The residue/anomaly relations are also imported rather than derived. I recommend major revision: the authors should either prove or clearly demarcate these assumptions, or soften the abstract/introduction so that the claims are presented as a conjectural framework with supporting evidence rather than an established derivation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper defines supersymmetric zeta functions by Mellin transforming the plethystic log of supersymmetric indices and then catalogues their residues and special values across many 2d, 4d and 6d theories. The genuinely new part is the systematic framework: explicit zeta functions, determinants, vacuum exponents, and proposed universal formulas—(5.141)-(5.143) for Macdonald/Schur indices, (6.10) for 6d—that tie residues and special values to anomaly coefficients and central charges. The example computations are explicit and the consistency checks against known central charges are extensive. For free theories, the relation E_SUSY = 1/2 Z(-1,0) reduces to Kim's regulated trace and checks out.\n\nThe soft spot is exactly what the authors themselves flag. The bridge E_SUSY = 1/2 Z(-1,0) is proposed, not derived. In §2.4.4 they call it non-trivial, then in §5.2.5-5.2.8 they use it to convert Schur zeta values into central charges for Argyres-Douglas theories. That is not an independent extraction; it is a consistency check against known anomaly computations, since the single-particle index already encodes the same anomalies. The residue-to-anomaly relations are also mostly imported from earlier Cardy-limit work. So the headline converse claim—central charges can be calculated from the zeta functions—is currently a conjecture, and the load-bearing assumption (2.76) has no derivation for interacting SCFTs. The constant-term ambiguity in the degeneracy asymptotics is another caveat, and the R-symmetry mixing parameters are chosen post hoc.\n\nNone of this kills the paper. The authors are explicit about proposal versus check. The framework reorganizes a lot of known material in one spectral language, and the explicit formulas will be cited. A good referee will push on (2.76) and the universality of the residue formulas, but there is real content.\n\nWho is it for: people working on superconformal indices, Cardy limits, Casimir energy, and non-Lagrangian SCFTs. I would send it to a serious referee. My own verdict is conditional rather than accept.\n\nRecommendation: engage; ask for a derivation of (2.76), or at least a clean statement separating what is proven for free fields from what is assumed.","headline":"Systematic zeta-function reformulation of index asymptotics; the new central-charge extraction rests on an unproven but explicitly flagged identification.","tokens_in":86729,"tokens_out":3003,"would_cite":true,"duration_ms":28496,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Supersymmetric zeta functions turn BPS spectral data into anomaly coefficients and central charges.","keywords":["supersymmetric zeta function","supersymmetric determinant","supersymmetric index","Cardy-like limit","supersymmetric Casimir energy","central charges","anomaly coefficients","BPS degeneracy"],"falsifier":"Take a single-particle index of a known interacting superconformal field theory, compute Z(−1,0) from its zeta function, and compare ½ Z(−1,0) with the supersymmetric Casimir energy obtained by independent localization or anomaly-polynomial methods; any mismatch beyond the free-field cases would disprove the universal bridge.","tokens_in":127,"feed_emoji":"🧮","tokens_out":1915,"duration_ms":84409,"temperature":0.7,"pith_summary":"The paper introduces supersymmetric zeta functions and supersymmetric determinants, new spectral objects built from supersymmetric indices via Mellin and Laplace transforms. It claims that the residues and special values of these zeta functions are universally determined by the anomaly coefficients of superconformal field theories, and conversely that those coefficients can be extracted from the zeta functions alone. The central bridge is the proposed identity E_SUSY = ½ Z(−1,0), which connects a zeta value to the supersymmetric Casimir energy. The authors verify the proposal across many free, interacting, and non-Lagrangian examples in two, four, and six dimensions.","feed_headline":"Zeta functions read central charges off BPS spectra","feed_subtitle":"New spectral tool ties index residues to anomalies and Casimir energies across 2d, 4d, and 6d.","key_machinery":"The central object is the supersymmetric zeta function Z(s,z), defined as a Dirichlet series over BPS charges with a shift z, together with its associated supersymmetric determinant D(z)=exp[∂_s Z(s,z)|_{s=0}]. Residues of Z at positive integers s=k encode the leading 1/β^k terms in the Cardy-like limit of the index, while the derivative at s=0 gives the constant term that appears in both the index asymptotics and the degeneracy growth. The identity E_SUSY = ½ Z(−1,0) is the load-bearing mechanism that turns special zeta values into Casimir energies.","core_discovery":"The paper defines Z(s,z) = Tr(−1)^F (Δ+z)^{−s}, summed over BPS states, and D(z)=exp[∂_s Z(s,z)|_{s=0}], and shows how they encode the spectrum of a supersymmetric theory. It proposes that poles of Z(s,z) at positive integers give the Cardy-like growth of the supersymmetric index, with residues fixed by anomaly coefficients, and that special values at negative integers give regularized energy moments. The main claim is that Z(s,z) provides a new spectral-language route to previously known universal data: residues at s=1,2,… match 't Hooft anomaly coefficients, Z(−1,0) is twice the supersymmetric Casimir energy, and Z(0,0) gives the logarithmic correction in the index. These relations let cen","pith_inferences":["The identity E_SUSY = ½ Z(−1,0) is verified only for free theories; extending it to all interacting and non-Lagrangian theories is the crucial step that would make central-charge extraction from spectral data fully rigorous.","Since Z(s,z) is defined from the plethystic logarithm of the index, the approach could be adapted to flavored or higher-sheet indices, potentially giving a spectral derivation of black-hole entropy counting.","The connection between Z(−1,0) and Casimir energy suggests that other negative-integer values, such as Z(−2,0), might encode higher-order scheme-independent energy moments that have not yet been explored.","A direct numerical test on a known interacting SCFT with independently computed Casimir energy would settle whether the factor of 1/2 is universal or an artifact of the free-field checks."],"forward_implications":["If the proposed relations hold, central charges and anomaly coefficients of any SCFT can be read off directly from its single-particle index, bypassing localization or Lagrangian descriptions.","The Cardy-like limits of supersymmetric indices become consequences of the analytic structure of Z(s,z), offering a unified explanation across 2d, 4d, and 6d.","The supersymmetric determinant provides a new probe of vacuum stability: a vanishing or divergent vacuum exponent D(0) is proposed to signal supersymmetry breaking or absence of a normalizable vacuum."],"fun_headline_variants":["Supersymmetric zeta functions decode spectral data","Zeta poles expose anomalies in 2d, 4d, and 6d","New determinant measures BPS spectra's hidden order","Spectral tool links indices to Casimir energies"],"cache_read_input_tokens":87936,"weakest_assumption_plain":"The claim that the supersymmetric Casimir energy equals half of the zeta value Z(−1,0) is assumed to hold for interacting and non-Lagrangian theories, even though it is only explicitly checked in free theories.","fun_headline_variants_meta":{"raw":{"variants":["Supersymmetric zeta functions decode spectral data","Zeta poles expose anomalies in 2d, 4d, and 6d","New determinant measures BPS spectra's hidden order","Spectral tool links indices to Casimir energies"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000267,"raw_usage":{"total_tokens":1386,"prompt_tokens":616,"completion_tokens":770,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":360,"completion_tokens_details":{"reasoning_tokens":714}},"tokens_in":360,"tokens_out":770,"duration_ms":7009,"temperature":1.0,"reasoning_tokens":714,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T19:40:43.245590+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a single-particle index of a known interacting superconformal field theory, compute Z(−1,0) from its zeta function, and compare ½ Z(−1,0) with the supersymmetric Casimir energy obtained by independent localization or anomaly-polynomial methods; any mismatch beyond the free-field cases would disprove the universal bridge.","supporting_citations":[],"review_version":1}