{"id":"5682b831-7b34-4444-b7bd-565cdb4abe65","arxiv_id":"2508.19981","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Carrollian limit of CFT3 operator products yields towers of sl(2,C) blocks that reproduce conformally soft photon and graviton theorems and the S and w1+∞ celestial symmetry algebras.","lead":"When you squash the time direction of a 3D conformal field theory, its operator products split into infinite towers of 2D symmetry modes that match the celestial symmetry algebras of photons and gravitons in flat space. This suggests those infinite symmetries may be universal consequences of 3D conformal kinematics rather than special features of scattering.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed celestial match relies on an unexplained residue renormalization (eq. 70); without it, the derived OPE coefficients diverge at the physical soft dimensions, so the central claim is not yet established.","rationale":"The reader identified the renormalization in eq. (70) as the weakest assumption, and I agree. The paper has real strengths: the derivation is detailed, the reduction of spinning correlators to scalar ones via weight-shifting operators is explicit, and the scalar decomposition of Section 4 is a non-trivial structural result. However, the advertised conclusions—the towers of soft theorems and the S and w1+∞ algebras—require the OPE blocks (68), (78), (89), and (97) to match the celestial blocks of [9,45], and that match only occurs after the residue renormalization (70). The authors candidly state that they do not understand why this renormalization is needed. That is a direct admission that the logical chain from the Carrollian limit to the celestial algebras has a missing link. The concern is specific, localized, and testable: it is not a disagreement with the outside consensus but a question of whether the paper's own derivation, without an extra input, produces the claimed result. I also considered whether the generalized use of the distributional identity (44) is more load-bearing; it is important, but it is a known technique in the celestial literature and the paper flags it with a reference to [70]. The renormalization (70) is the point where the derivation's output diverges at the physical dimensions, so it is the most direct threat to the central claim. The proposed test—reversing the order of limits—would settle whether the poles are genuine or artifacts; the paper's own discussion in Section 4 makes this test well-posed. Because the reader's verdict (CONDITIONAL) already reflects this concern, my read does not change the verdict.","tokens_in":35217,"tokens_out":13312,"duration_ms":125934,"concrete_test":"Keep c finite in the integrand of (37), evaluate the u_i integrals (11) exactly, and only then take c→0 with the normalization (9). This is the opposite order of limits from the one used in the paper, which the authors acknowledge changes the result. Compute the resulting 2d current-scalar OPE block and examine its pole structure in s1 and δ2. If the poles at integer s1,δ2 are absent, or their residues differ from the prediction of eq. (69), then the residue renormalization (70) is an artifact of the paper's order of limits and the match with [45] is not established. If the poles and residues are reproduced, the renormalization is robust and the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that Carrollian limits of CFT3 current and stress-tensor OPE blocks reproduce the celestial OPE blocks underlying the S and w1+∞ algebras—depends on the renormalization in eq. (70). The derived 2d OPE coefficients (69), (79), (90), (98) contain poles at s1,δ2∈Z, i.e. exactly at the negative-integer dimensions of the conformally soft modes, so the naive blocks are divergent where they must be finite. The authors remove these poles by replacing the operators with their residues, e.g. J^+_{1-s1} → Res_{δ1=1-s1}(δ1-1+s1)J^+_{δ1}, and then claim perfect agreement with the celestial OPE blocks of [45] and [9]. However, the authors explicitly state (Section 5.1) that they 'do not fully understand why the operators we obtain upon dimensional reduction do not exactly coincide with the operators appearing in celestial CFT.' If this renormalization is an ad hoc prescription rather than a consequence of the construction, then the derivation does not independently establish the celestial OPE: the final answer is effectively inserted by hand. The paper's own caution about the order of limits in Section 4 makes this especially acute: the poles may be artifacts of taking c→0 before evaluating the u_i integrals. This is the most load-bearing weak point because it sits precisely between the 3d input and the 2d output that the paper advertises.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Carrollian limit of three-dimensional conformal field theory OPE blocks, defined by taking t = c u and c → 0, and shows that scalar, current, and stress-tensor blocks decompose into towers of sl(2,C) blocks with negative integer scaling dimensions. The main applications are that the current-scalar-scalar and stress-tensor-scalar-scalar blocks reproduce the towers of conformally soft photon and graviton theorems, while the current-current and stress-tensor-stress-tensor blocks reproduce the celestial S and w_{1+∞} algebras. The derivation proceeds by first writing CFT3 three-point functions in an integral representation, taking the Carrollian limit inside that representation, and then applying integral transforms to extract 2d conformal primary modes. Spinning correlators are reduced to scalar correlators via explicit weight-shifting operators, and the final OPE blocks are compared with known celestial OPE blocks from the literature.","tokens_in":35573,"tokens_out":3207,"duration_ms":32960,"significance":"If the construction is fully justified, the paper provides a concrete derivation of celestial CFT structures, including the S and w_{1+∞} algebras, from a controlled limit of ordinary CFT3 data. This would be a significant conceptual unification: the infinite towers of soft theorems and the associated 2d symmetry algebras would follow from the decomposition of specific unitary so(3,2) representations rather than being imposed as separate axioms. The paper is explicit and self-contained in most of its technical steps: the weight-shifting operators are written out, the dimensional reduction of the scalar block is performed in detail, and the final OPE coefficients are not fitted but compared with independent computations in refs. [9], [44], and [45]. The main open issues concern the legitimacy of two limiting procedures, the generalized distributional identity in eq. (44) and the residue renormalization in eq. (70), both of which are load-bearing for the advertised conclusions.","major_comments":[{"comment":"The 2d OPE coefficients derived from dimensional reduction, e.g. c^{2d}_{JO}(s_1, δ_2) in eq. (69), contain poles precisely at the integer dimensions of the conformally soft modes, and the paper removes these poles by the residue renormalization in eq. (70). The authors state on page 15 that they do not fully understand why the dimensionally reduced operators do not coincide with those of celestial CFT. Since the claimed perfect agreement with the celestial OPE blocks of ref. [45] and the subsequent derivation of the soft-theorem towers depend on this renormalization, the central identification is not yet established: without eq. (70), the naive blocks are divergent exactly where the celestial blocks are finite. I ask the authors to derive eq. (70) from a well-defined operator normalization or from a specific correlation-function limit, or to prove that the same result follows from an associativity or OPE-convergence requirement, rather than presenting it as a prescription adopted because it gives the expected answer.","section":"§5.1, eqs. (69)–(70)"},{"comment":"The identity lim_{c→0} c^{-ν} = 2πν δ(ν) is used for real values of ν, for example in eq. (45) where it produces the dimension-conserving delta function that projects the λ integral in eq. (50) onto δ_3 = δ_1 + δ_2 − 2. The paper notes that the identity strictly holds only for ν ∈ iR and defers rigor to ref. [70]. This is a load-bearing step because it is the mechanism by which the 3d OPE block is reduced to a single 2d exchange with the dimension predicted by collinear factorization. Please either provide a rigorous justification for the generalized use of eq. (44) in this context, or reformulate the projection as a limit of finite-c expressions that yields the same δ_3 condition without relying on a distributional identity outside its proven domain.","section":"§4, eq. (44)"},{"comment":"The extraction of magnetic Carroll sector correlators involves discarding distributional terms that arise in the c→0 limit, and the paper itself notes that in ref. [32] the electric (distributional) sector was found to dominate at the same order in c. The choice to keep only the power-law magnetic branch is motivated by the goal of recovering standard 2d CFT OPE blocks, but this is close to assuming the desired output. The authors should state a selection rule, observable, or symmetry criterion that uniquely picks out the magnetic Carroll subsector before the distributional terms are discarded, and should show that this criterion is not equivalent to postulating the celestial OPE blocks that are later compared with the literature.","section":"§4, discussion after eq. (52) and §5–6"}],"minor_comments":[{"comment":"The normalization in eq. (9) appears to contain a typographical error in the factor involving ℓ; the expression '∆−1 ℓ + ∆−1Γ(∆−1/2)' is not typeset as a well-formed product of factors and should be corrected.","section":"§2, eq. (9)"},{"comment":"The definitions of β_{12} and β_{13} are written identically; presumably β_{13} should be (Δ_1 + Δ_3 − Δ_2)/2, and this typo should be fixed because these exponents are used throughout the paper.","section":"§4, eq. (34)"},{"comment":"The factor δ(−s_1 + δ_2 + δ_3 − 2)(−s_1 + δ_2 + δ_3 − 2) is written in an unusual order and without parentheses around the distribution argument; making the argument of the delta function explicit would improve readability.","section":"§5.1, eq. (65)"},{"comment":"The notation 'S algebra' is not defined in the text; the reader would benefit from a one-line statement that this is the celestial symmetry algebra of pure Yang-Mills theory introduced in refs. [9,10].","section":"§6.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically rich and the final results are very likely correct in substance, but the current presentation leaves two central limiting steps in a gray zone: the generalized distributional identity (44) and the residue renormalization (70). Both are acknowledged by the authors as not fully rigorous, and both sit exactly between the CFT3 input and the announced celestial CFT output. I would be willing to accept the paper after the authors either justify these steps or clearly re-frame them as conjectural with the burden placed on the matching to independent results. The comparison with refs. [9,44,45] is a genuine check and not a fit, which is a strength."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline: this paper gives the first explicit derivation of the towers of conformally soft photon/gluon and graviton theorems, and the S and w1+∞ algebras, from the Carrollian limit of three-point OPE blocks in CFT3. If it holds up, it shows these celestial structures are not mysterious 2d accidents but follow from the decomposition of unitary so(3,2) representations (2,1) and (3,2) into sl(2,C) towers. That is a genuine conceptual step.\n\nWhat is new is not the individual ingredients — the sl(2,C) decomposition of so(3,2) blocks and the celestial algebras already exist — but the fact that they all drop out of a single CFT3 input: the three-point function of scalars, currents and stress tensors, via weight-shifting operators. The authors are careful about normalizations, check against known celestial OPE blocks from [9,45], and do not fit any parameters. The integral representation trick to access the magnetic Carroll sector before taking Mellin moments is clever and seems to be the key enabling step.\n\nThe soft spots are localized but real. The biggest is eq. (70): the derived OPE coefficients have poles at exactly the negative-integer conformal dimensions where the soft modes live, and the authors remove them by redefining the 2d operators via residues. They state in Section 5.1 that they do not fully understand why the dimensionally reduced operators do not coincide with the celestial ones. That is an honest admission, but it means the central derivation is not yet complete — the final agreement with [45] depends on a prescription that is currently an extra input, not a consequence of the construction. The stress-test note sharpens this correctly: the poles could be artifacts of taking c→0 before the u-integrals, and the order-of-limits issue is acknowledged in Section 4.\n\nTwo smaller issues: the distributional identity (44) is used for real λ although it strictly holds for imaginary λ; the authors defer to [70], which is a reasonable stopgap. And they only treat the positive-helicity transverse sector, discarding electric-sector distributional terms without a fully systematic justification. Neither is fatal if the renormalization point is clarified.\n\nWho this is for: anyone working on celestial or Carrollian holography, or on flat-space limits of AdS/CFT. It deserves a serious referee. My recommendation: send it to review, but flag eq. (70) as load-bearing and ask the authors to either derive the residue prescription from a well-defined operator definition or show that the pole-structure is scheme-independent.\n\nBest,","headline":"A mostly solid derivation of celestial soft algebras from CFT3 Carrollian limits, with one load-bearing renormalization step the authors themselves do not yet understand; worth refereeing, but that step needs to be settled before the central claim is fully established.","tokens_in":36078,"tokens_out":2733,"would_cite":true,"duration_ms":25801,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that the Carrollian limit of CFT3 OPE blocks decomposes them into towers of sl(2,C) blocks whose current and stress tensor sectors reproduce the conformally soft photon and graviton theorems and the S and w1+∞ algebras…","keywords":["Carrollian limit","celestial CFT","OPE blocks","conformal soft theorems","S algebra","w1+∞ algebra","weight-shifting operators","AdS/CFT"],"falsifier":"Compute the unrenormalized current-current OPE block (eq. (89) with coefficient (90)) for a concrete CFT3, such as the free fermion or free boson with a non-abelian current, without applying the residue prescription (70); if the resulting 2d correlator fails to satisfy the sl(2,C) block decomposition with the exchanged dimension δ3=1−s1−s2, or if the residue prescription cannot be reproduced by a well-defined contour deformation, the central claim collapses. Alternatively, test the first subleading correction in z12 in eq. (89) against the celestial OPE of [45]: the paper's blocks are derived to leading order, and the next term should also match if the identification is correct.","tokens_in":34999,"feed_emoji":"🌌","tokens_out":7068,"duration_ms":60236,"temperature":0.7,"pith_summary":"This paper claims that the Carrollian limit of three-dimensional conformal field theory—setting t=cu and sending c→0—is not just a way to extract scattering data, but a mechanism that generates the infinite 2d symmetry algebras of celestial holography. Starting from the OPE blocks (the universal pieces of an operator product expansion that encode the exchange of a given operator) of a scalar, a conserved current, and the stress tensor in CFT3, the authors show that each so(3,2) primary decomposes into a tower of sl(2,C) primaries labeled by negative integer dimensions. The current-scalar-scalar and stress tensor-scalar-scalar blocks reproduce the infinite towers of conformally soft photon and graviton theorems, while the current-current and stress tensor-stress tensor blocks reproduce the S and w_{1+∞} algebras of celestial CFT. If correct, these 2d algebras are not additional structures that must be put in by hand; they are forced by the representation theory of the 3d conformal group and the existence of a conserved current or stress tensor.","feed_headline":"Carrollian limit of CFT3 yields celestial S and w1+∞ algebras","feed_subtitle":"Soft gluon and graviton theorems in 4d flat space emerge as sl(2,C) OPE blocks of a 3d CFT's current and stress tensor.","key_machinery":"The central object is the CFT3 OPE block written in an integral representation (eq. (33)) in which the three-point function is expressed as a Gaussian integral over an auxiliary point; taking the Carrollian limit t_i=c u_i inside this representation before evaluating the u_i moments projects the correlator onto the magnetic Carroll sector and yields standard 2d CFT three-point functions of sl(2,C) primaries of dimensions δ_i=Δ_i−s_i−1. For the spinning operators, the three-point functions are re-expressed as weight-shifting operators acting on scalar correlators, so the scalar result carries over to currents and the stress tensor. A final residue renormalization (eq. (70)) replaces the naive 2d operators by residues that remove unwanted integer poles and produces the celestial OPE coefficients.","core_discovery":"In the Carrollian limit of CFT3, the OPE block of two scalar primaries of dimensions Δ1 and Δ2 reduces to a collection of sl(2,C) OPE blocks whose exchanged operator has dimension δ1+δ2−2, with δi=Δi−si−1 for integers si (eq. (52)). Applying the same dimensional reduction to the conserved current (Δ=2, ℓ=1) and the stress tensor (Δ=3, ℓ=2) OPE blocks, the authors find that current modes of dimension 1−s and stress tensor modes of dimension 2−s, with s∈N, satisfy the sl(2,C) OPE blocks (eqs. (89) and (97)) that, after a residue renormalization of the operators, exactly reproduce the celestial OPE blocks of gluons and gravitons from which the S and w_{1+∞} algebras are extracted. The same blocks, expanded in the OPE limit, reproduce the towers of tree-level soft photon and graviton theorems in a conformal primary basis. The paper's claim is that these infinite 2d symmetry algebras are a consequence of nothing more than the decomposition of the unitary so(3,2) representations (2,1) and (3,2) into integer-dimension sl(2,C) representations.","pith_inferences":["Because the derivation only uses kinematically determined three-point functions, the same towers should appear in any CFT3 with a conserved current or stress tensor, including free-field and perturbative fixed points; computing the free-field current block would test universality directly.","The residue renormalization (70) may be equivalent to summing over the two global-time-slice expansions of the Lorentzian cylinder, which the authors note differ by phases; if so, the 'unwanted poles' would cancel once both in/out configurations are included.","The graviton OPE block is obtained from the current block by squaring the weight-shifting operator (eqs. (88) vs (96)), suggesting that the w_{1+∞} structure constants are a kind of double copy of the S algebra at the level of OPE data—an interpretation the paper notes via the AdS double copy but does not develop for the 2d algebras.","The restriction to leading order in the Carrollian limit means the 2d OPE coefficients capture only the collinear/soft part; higher-point CFT3 correlators should produce corrections to the celestial OPE that are currently invisible to the celestial bootstrap."],"forward_implications":["The S algebra and the w_{1+∞} algebra are universal consequences of the existence of a conserved spin-1 or spin-2 operator in a 3d CFT, not of a priori assumptions about a Carrollian or celestial field theory.","The towers of conformally soft photon and graviton theorems in 4d flat space arise from the current-scalar-scalar and stress tensor-scalar-scalar OPE blocks of CFT3, to leading order in the Carrollian limit.","The Δ=2 conformally soft graviton, which plays a special role in celestial holography, appears as the s=0 mode in the tower of CFT3 stress tensor modes.","Subleading corrections in the Carrollian limit are expected to deform w_{1+∞} by terms proportional to the cosmological constant, matching known AdS4 deformations."],"supporting_citations":[{"why":"Supplies the spinning three-point structures used to express current and stress tensor blocks as weight-shifting operators acting on scalar correlators.","marker":"[38]"},{"why":"Provides the celestial symmetry algebras S and w_{1+∞} whose OPE blocks the paper reproduces through dimensional reduction.","marker":"[9]"},{"why":"Gives the celestial OPE blocks for all spins used to compare eqs. (68), (78), (89), and (97).","marker":"[45]"},{"why":"Derives the towers of conformally soft photon and graviton theorems in a conformal primary basis that the CFT3 blocks reproduce.","marker":"[11]"},{"why":"Establishes the discrete basis of sl(2,C) primaries with positive and negative integer dimensions used for the mode decomposition.","marker":"[37]"},{"why":"Previous work establishing the electric and magnetic Carroll sectors and the order of limits, justifying the projection onto the magnetic sector.","marker":"[32]"},{"why":"Provides AdS4 bulk-to-boundary propagators and OPE coefficients for spinning Witten diagrams that fix the normalizations used here.","marker":"[59]"}],"fun_headline_variants":["Carrollian CFT3 limit spawns infinite symmetry towers","From 3d Carrollian blocks to celestial S and w1+∞ algebras","Carrollian limit of CFT3 yields celestial soft theorems","Infinite sl(2,C) towers from Carrollian CFT3","Celestial S and w1+∞ algebras from Carrollian CFT3"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The calculation matches celestial OPE blocks only after the 2d operators are renormalized by taking residues at integer dimensions, a step the authors state they do not fully understand; if that renormalization is not justified, the OPE coefficients carry extra poles and the claimed agreement with the celestial OPE of [45] fails.","fun_headline_variants_meta":{"raw":{"variants":["Carrollian CFT3 limit spawns infinite symmetry towers","From 3d Carrollian blocks to celestial S and w1+∞ algebras","Carrollian limit of CFT3 yields celestial soft theorems","Infinite sl(2,C) towers from Carrollian CFT3","Celestial S and w1+∞ algebras from Carrollian CFT3"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00064,"raw_usage":{"total_tokens":2994,"prompt_tokens":1039,"completion_tokens":1955,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":655,"completion_tokens_details":{"reasoning_tokens":1860}},"tokens_in":655,"tokens_out":1955,"duration_ms":13319,"temperature":1.0,"reasoning_tokens":1860,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:48:47.421315+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the unrenormalized current-current OPE block (eq. (89) with coefficient (90)) for a concrete CFT3, such as the free fermion or free boson with a non-abelian current, without applying the residue prescription (70); if the resulting 2d correlator fails to satisfy the sl(2,C) block decomposition with the exchanged dimension δ3=1−s1−s2, or if the residue prescription cannot be reproduced by a well-defined contour deformation, the central claim collapses. Alternatively, test the first subleading correction in z12 in eq. (89) against the celestial OPE of [45]: the paper's blocks are derived to leading order, and the next term should also match if the identification is correct.","supporting_citations":[{"cited_title":"Spinning ads propagators,","cited_arxiv_id":null,"evidence_quote":"Provides AdS4 bulk-to-boundary propagators and OPE coefficients for spinning Witten diagrams that fix the normalizations used here."}],"review_version":1}