{"id":"6b35aa4c-be38-42de-9fd9-6c733619f110","arxiv_id":"2412.16106","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Cyclic relative L-infinity algebras encode the full S-matrix, including its trivial part, and reproduce Witten diagrams including CFT two-point functions.","lead":"This paper proposes a new mathematical framework, cyclic relative L-infinity algebras, to describe quantum field theories with boundaries. It claims the framework captures both the trivial identity part of the S-matrix and Witten diagrams relevant to the AdS/CFT correspondence.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The flat/AdS minimal model requires the homotopy transfer data (i,p,h) in (6.1) to satisfy retract identities and convergence; only 'a suitable left inverse of i' is given. This analytic input is load-bearing for the claimed Witten-diagram expansion and is unproved.","rationale":"The reader's weakest_assumption and my independent reading converge on the same point: the homological perturbation lemma is applied to non-compact function spaces without proving the deformation-retract identities or convergence. This is the single most load-bearing concern because the paper's headline claim—that relative cyclic L∞-algebras encode S-matrices and Witten diagrams uniformly on asymptotic boundaries—requires the minimal relative model to actually exist. The bulk of the paper is a coherent algebraic framework with a fully worked compact-manifold example, but the flat/AdS section is the only place where the framework is applied to the physically advertised settings (Minkowski S-matrix and AdS/CFT Witten diagrams), and there the retract data are incomplete. The scalar compact case (3.1.1) is solid: on a compact manifold with boundary the Green's function and projection are standard, and the HPL applies with minor analytic hypotheses. The Chern–Simons and Yang–Mills sections are plausible but the cyclic identities are asserted rather than verified; however, those sections are explicitly examples and the central claim does not rest on them as heavily as on the flat/AdS minimal model. The normalization constant kappa is fitted to the two-point function rather than derived, but this is a convention issue, not a structural flaw: the relative HMC action (5.9) is homogeneous in the choice of boundary pairing, and fixing kappa by the known two-point function is acceptable at this stage, though a first-principles derivation would strengthen the paper. Therefore I agree with the CONDITIONAL verdict: the paper is a valuable and plausible contribution, but the analytic foundation of the key non-compact construction must be supplied before the central claim can be fully supported. No ad hominem concerns arise; the issue is purely mathematical rigor in a technically difficult setting.","tokens_in":53305,"tokens_out":1696,"duration_ms":15317,"concrete_test":"Compute, for a Euclidean flat-space representative of H^1 (i.e., an on-shell plane wave φ = e^{-|p|t + ip·x}), the quantity (id − i∘p)(φ) and verify explicitly whether it equals (Δ−m²)∘h(φ) + h∘(Δ−m²)(φ), using the p defined as the 'suitable left inverse' of i. Repeating this check for all generators of the function space V defined in (4.7), including the homogeneous pieces, settles whether (6.1) is a true deformation retract. If the identity fails or the HPL series for the four-point minimal bracket does not converge absolutely on V, the minimal model construction of §3.1.2 is not established.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim that the relative minimal model encodes full tree-level S-matrix including the trivial part depends on the homological perturbation lemma (HPL) being applied to the non-compact function spaces of §3.1.2. For HPL to produce the minimal model (6.8)/(7.0), the data (i, p, h) must be a genuine deformation retract: p∘i = id, id − i∘p = d∘h + h∘d, plus side conditions such as h∘h = 0 or h∘p = 0 as needed for the chosen HPL formulation. For flat space, (6.2) explicitly constructs i and h via bulk-to-boundary propagators, but p is only described as 'a suitable left inverse of i' (six lines above (6.8)); no proof is given that this p is a chain map or satisfies the retract identity on the space of functions with superpolynomial plus homogeneous parts. The pairing (4.8), the boundary pairing (5.6), and the relative HMC action (5.9) are defined only 'wherever convergent', and the HPL recursion produces infinite sums of tree diagrams whose convergence on these spaces is not established. If this analytic input fails, the minimal model—and therefore the claimed Witten-diagram expansion and the CFT two-point function computation (7.1)—does not exist. This is the reader's weakest assumption, and it is precisely the point on which the paper's main new construction (relative minimal model on asymptotic boundaries) rests. It is not fatal by itself, but it is load-bearing and unresolved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the notion of a cyclic relative L-infinity algebra as a pair of cyclic L-infinity algebras (V, l_n, ⟨,⟩_V) and (V_partial, l_partial_n, ⟨,⟩_V_partial) together with an L-infinity morphism π: V → V_partial. The relative homotopy Maurer-Cartan action includes a bulk term and a boundary term built from π, designed to correct the failure of cyclicity and reproduce the physical action including boundary terms. The authors argue that the minimal model of such a relative algebra, constructed via the homological perturbation lemma, encodes the full connected tree-level S-matrix, including the trivial/identity part that is missed by the standard minimal model of a single L-infinity algebra. They illustrate the construction with a scalar field on a compact manifold with boundary, on flat Euclidean space, and on hyperbolic/AdS space, claiming to reproduce the CFT two-point function as a boundary term ⟨π(φ),π(φ)⟩. They also outline Chern-Simons and Yang-Mills theories on manifolds with boundary.","tokens_in":53633,"tokens_out":9345,"duration_ms":81626,"significance":"The central insight—packaging boundary effects and the identity part of the S-matrix into a relative L-infinity structure—is original and conceptually appealing. The algebraic definitions are carefully laid out, and the compact-manifold scalar example is worked out in full detail, showing that the relative homotopy Maurer-Cartan action exactly equals the physical action including the boundary term. The interpretation of the two-point Witten diagram as the boundary pairing of the minimal-model morphism (Eq. (7.1) and Table 2) is a clean and useful result. The paper also correctly emphasizes that the usual minimal model of an L-infinity algebra cannot capture the trivial contribution, and it proposes a concrete remedy. The main weakness is the incomplete control of the homological perturbation lemma on non-compact function spaces, which is essential for the flat-space and AdS claims.","major_comments":[{"comment":"The construction of the minimal model for flat space and AdS relies on the homological perturbation lemma applied to the function spaces V and V_partial defined in (4.7) and (5.0). The HPL requires the data (i, p, h) to form a genuine deformation retract, i.e. p∘i = id, id − i∘p = d∘h + h∘d, plus side conditions as needed for the chosen HPL formulation. However, p is only specified as 'a suitable left inverse of i' (paragraph above Eq. (6.8)), and no proof is given that such a p is a chain map or satisfies the retract identity on the space of functions with superpolynomial decay plus countable homogeneous terms. Moreover, the pairings (4.8) and (5.6) are defined only 'wherever convergent', and the HPL recursion produces infinite sums of tree diagrams whose convergence on these spaces is not established. Since the Witten-diagram expansion and the two-point amplitude are the main new claims for non-compact geometries, this analytic gap is load-bearing and must be addressed, either by a proof or by an explicit statement that the construction is only formal for these cases.","section":"Section 3.1.2, Eq. (6.1) and surrounding text"},{"comment":"The relative homotopy Maurer-Cartan action (5.9) contains an undetermined constant κ. The text states that κ 'will eventually be fixed by requiring that the coefficient for the quadratic term ... is correctly normalised (when compared to the trivial part of the S-matrix or the CFT two-point function)'. This means the two-point function is not derived from the relative L-infinity structure alone: the overall normalization is fitted to the known result. The authors should either identify a canonical normalization principle within the framework that fixes κ, or clearly state that the framework reproduces the structural form of the two-point function up to an overall constant.","section":"Section 3.1.2, Eq. (5.9) and the paragraph after (5.8c)"}],"minor_comments":[{"comment":"The bulk-to-boundary morphism π1 is defined as (−)pw, which maps a function f ∈ V to the formal sum of its asymptotic components in V_partial. This map is well-defined only if the decomposition f = f_interior + ∑ z^{α_i} f_{α_i}(x) in (4.7) is unique. The authors should specify a canonical asymptotic expansion (e.g., by demanding that the homogeneous terms are precisely the non-superpolynomially-decaying components) and prove that the resulting π1 and the projection p are independent of the choice.","section":"Section 3.1.2, Eqs. (4.7) and (5.7)"},{"comment":"The boundary bilinear form ⟨−,−⟩_{V_partial} is not required to be non-degenerate or symmetric, but the relative homotopy Maurer-Cartan action uses its graded symmetrization and antisymmetrization. The paper should state explicitly how these parts are defined for elements of arbitrary degree and confirm that the action is well-defined and independent of any auxiliary choices.","section":"Section 2.2, Definition 4"},{"comment":"The recursion relations for the minimal model brackets and morphism components are presented diagrammatically. A brief derivation from the standard homological perturbation lemma formulas, or a reference to the specific version used, would improve readability and verifiability.","section":"Section 2.4, Eqs. (3.4a)-(3.4g)"},{"comment":"The phrase 'Full S-matrices' overstates the scope: the paper treats tree-level amplitudes and the identity part, while loop effects are deferred. Consider wording such as 'tree-level S-matrices' or 'full connected tree-level S-matrix' to avoid overclaiming.","section":"Title and Abstract"}],"recommendation":"major_revision","confidential_remarks":"The paper is a strong candidate for a journal in mathematical physics or hep-th. The main obstacle is the functional-analytic gap in the homological perturbation lemma application to non-compact spaces; closing that gap is essential for the flat-space and AdS claims. The fitted constant κ also reduces the strength of the AdS/CFT claim, though it may be acceptable if presented transparently as a normalization. The algebraic framework and the compact-manifold example are solid and should be acknowledged as strengths."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the relative L-infinity construction is a real advance. The standard L-infinity minimal model encodes the connected nontrivial part of the S-matrix; what's missing is the identity component, and in AdS that identity component is the CFT two-point function. Putting the boundary data in a separate L-infinity algebra with a morphism pi from the bulk is a clean fix. The paper works the scalar example properly: the relative homotopy Maurer-Cartan action equals the physical action including the boundary term, and the minimal-model morphism pi produces the boundary term that gives the two-point amplitude. The contrast with Chiaffrino-Ersoy-Hohm is well argued—their single-algebra trick makes the minimal model potentially trivial, so it cannot carry the identity part. This is a genuinely new structural claim, and the table mapping Witten diagrams to homotopy-transfer data (h, i, p, <pi,pi>) is useful.\n\nNow the soft spots, in order.\n\nFirst, the AdS two-point function is not really derived: the constant kappa in the relative HMC action is introduced and then fixed by requiring the known two-point coefficient. That makes the two-point result a fit, not a prediction. It's a normalization issue, so not fatal, but the paper should say it.\n\nSecond, the Chern-Simons and Yang-Mills sections assert the cyclic structure but never verify the cyclic identities of Definition 4. The relative HMC action looks right, but the construction is only a proof if the structure maps are cyclic. That's a missing check, not an impossible one.\n\nThird, and most important, the stress-test note is right: the homological perturbation lemma on the flat and AdS function spaces is not justified. The space V includes superpolynomially decaying parts plus countable sums of homogeneous terms; the pairings are 'wherever convergent'; the projection p is only 'a suitable left inverse of i'. No proof is given that p is a chain map or that the deformation retract identities hold, and the HPL recursion produces infinite sums whose convergence is not addressed. In the physics literature this is often formal, and for tree-level fixed external states the sums are probably finite, but the paper does not say that. For a claim to be load-bearing, this is unresolved.\n\nAll that said, the central idea is solid and the gaps are fixable. The paper deserves a serious referee. I'd cite it, and I'd put it on the reading-group list.","headline":"A genuine step forward in homotopy-algebraic scattering: the relative L-infinity framework naturally carries the trivial S-matrix part and Witten diagrams, but the AdS two-point normalization and the analytic assumptions in the non-compact homotopy transfer need work.","tokens_in":54166,"tokens_out":4572,"would_cite":true,"duration_ms":44875,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81T18","81T40","81T13","17B70"],"pacs":[],"model":"deepseek-v4-flash","headline":"A perturbative field theory with a boundary is described by a cyclic relative $L_\\infty$-algebra, whose minimal model encodes the full connected tree-level $S$-matrix including the trivial/identity part and reproduces Witten diagrams…","keywords":["L-infinity algebras","S-matrix","Witten diagrams","AdS/CFT","homological perturbation lemma","BV-BFV formalism","Chern-Simons theory","Yang-Mills theory"],"falsifier":"Take the flat-space scalar example of Section 3.1.2 and check whether the map $p$ in the deformation retract (6.1) actually exists as a bounded left inverse of $i$ on the space $V$ defined by (4.7); if no such left inverse exists, or if the boundary pairing (5.6) diverges on the formal sums in $V_{\\partial}$, then the minimal model and the claimed Witten diagram expansion are not defined. A direct check would also compute the two-point amplitude from Definition 9 and compare it with the known Klein–Gordon metric (6.9) and the AdS two-point function (7.1); a mismatch would falsify the central claim.","tokens_in":53090,"feed_emoji":"🌌","tokens_out":9166,"duration_ms":71772,"temperature":0.7,"pith_summary":"The paper argues that the usual $L_\\infty$-algebra description of perturbative field theory captures only the nontrivial connected part of the $S$-matrix and drops the identity/trivial part, which is harmless in flat Minkowski space but physically meaningful in curved or bounded geometries such as anti-de Sitter space. It proposes that a theory be described by a cyclic relative $L_\\infty$-algebra: a bulk algebra and a boundary algebra joined by a cyclic morphism. The boundary algebra simultaneously repairs the failure of cyclicity caused by integration by parts and supplies the boundary terms missing from the canonical homotopy Maurer–Cartan action. The relative minimal model then encodes the connected tree-level $S$-matrix including the trivial part, and the construction is shown to reproduce Witten diagrams, with the CFT two-point function recovered from the boundary morphism. This would make quasi-isomorphism the right notion of physical equivalence for theories with boundaries, from scattering amplitudes to holographic correlators.","feed_headline":"Two L-infinity algebras capture the whole S-matrix, boundary included","feed_subtitle":"The boundary algebra supplies the identity part of amplitudes and reproduces CFT two-point functions from Witten diagrams.","key_machinery":"The central object is a cyclic relative $L_\\infty$-algebra: a pair of $L_\\infty$-algebras, one 'bulk' and one 'boundary', equipped with a cyclic morphism $\\pi$ between them. An $L_\\infty$-algebra is a graded vector space with higher-ary brackets generalizing Lie brackets; the cyclic inner product turns the brackets into the terms of the Batalin–Vilkovisky action. The relative homotopy Maurer–Cartan action combines bulk terms $\\langle \\varphi, \\ell_k(\\varphi,\\dots,\\varphi)\\rangle$ with boundary terms built from $\\pi$, so the boundary algebra corrects the failure of cyclicity and adds the physically needed boundary action. The minimal model is constructed by homological perturbation theory as a deformation retract, and Definition 9 reads the generalised connected amplitudes off the minimal brackets plus the boundary morphism. This is the mechanism that carries the argument: the trivial/two-point part of the $S$-matrix is the boundary-pairing contribution, while all higher connected diagrams are the homotopy-transfer expansion.","core_discovery":"The central claim is that a perturbative field theory with an asymptotic boundary is described by a relative cyclic $L_\\infty$-algebra, and that the minimal model of this algebra encodes the full connected tree-level $S$-matrix, including the trivial/identity component that the ordinary minimal model omits. Definition 9 identifies the two-point amplitude with the boundary term $\\langle \\overset{\\circ}{\\pi}_1(\\varphi_1), \\overset{\\circ}{\\pi}_1(\\varphi_2)\\rangle$ in the boundary minimal model, so on AdS the CFT two-point function comes from the minimal-model morphism rather than from bulk brackets. The paper shows how the homological perturbation lemma produces the minimal model recursively, and interprets the resulting diagrams as Witten diagrams through the dictionary: the bulk–bulk propagator is the contracting homotopy, the bulk–boundary propagators are the inclusion and projection of a deformation retract, and the boundary–boundary propagator is the boundary pairing. Chern–Simons and Yang–Mills theories on manifolds with boundary are treated in the same language.","pith_inferences":["If the analytic existence questions are settled, the relative minimal model would give a choice-independent definition of holographic two-point functions, since different regularisations appear as different choices of retract data rather than as extra input.","One testable extension is to derivative bulk interactions on AdS: then the boundary algebra acquires nontrivial higher brackets, and the relative minimal model predicts boundary contributions to higher-point Witten diagrams beyond the two-point function.","The same relative construction should lift to loop level by replacing the minimal model with a quantum (loop) $L_\\infty$-algebra, giving a homotopy-algebraic account of the trivial part of loop amplitudes and of loop Witten diagrams; the paper only notes this generalization.","The framework suggests that different boundary conditions (Dirichlet versus Neumann, relative versus absolute) are different cyclic morphisms $\\pi$, so comparing their minimal models may clarify how boundary-condition dependence enters holographic correlators."],"forward_implications":["The full connected tree-level $S$-matrix, including its identity/two-point part, is an invariant of the cyclic relative $L_\\infty$-algebra up to quasi-isomorphism, so quasi-isomorphic Lagrangians with boundaries give the same physics.","Boundary terms in the action are not an obstruction to the homotopy Maurer–Cartan picture; they are systematically encoded by the boundary algebra and the morphism $\\pi$.","In AdS/CFT the CFT two-point function is the boundary-pairing term $\\langle \\pi_1(\\varphi_1), \\pi_1(\\varphi_2)\\rangle$, and the higher connected correlators are the Witten diagrams generated by homotopy transfer.","Chern–Simons and Yang–Mills theories on manifolds with boundary admit relative minimal models; in the free/infrared sector their cohomology reduces to de Rham cohomology with the appropriate relative/absolute boundary conditions.","The usual $L_\\infty$-algebra result that higher-point connected amplitudes receive no boundary corrections is recovered when $\\pi$ is strict and $\\pi_1 \\circ h = 0$; boundary corrections then appear only in the two-point function."],"supporting_citations":[{"why":"The holography-as-homotopy construction that the paper compares with and that partially inspired the present approach; its failure to encode the identity component motivates the relative minimal model.","marker":"[32]"},{"why":"The BV–BFV formalism for classical BV theories on manifolds with boundary; it supplies the doubled boundary phase space and compatibility conditions that the relative $L_\\infty$-algebra is designed to match.","marker":"[33]"},{"why":"Lays out the BV-quantizable recursion for tree-level amplitudes that this paper generalises to the relative setting.","marker":"[3]"},{"why":"Introduces the $L_\\infty$-algebra of the $S$-matrix whose minimal model encodes the nontrivial connected part, the starting point the paper extends.","marker":"[4]"},{"why":"Standard homological perturbation lemma references used to construct the relative minimal model and its diagrammatic expansion.","marker":"[48]–[50]"},{"why":"Witten's integration-by-parts derivation of the AdS boundary two-point function, which the paper reproduces as $\\langle\\pi_1,\\pi_1\\rangle$.","marker":"[56]"},{"why":"The CFT/AdS Witten diagram correlation functions that the relative minimal model is checked against.","marker":"[57]"},{"why":"Work on what can be measured asymptotically, supporting the idea that the trivial part of the $S$-matrix comes from asymptotic boundary terms.","marker":"[52]"}],"fun_headline_variants":["Relative L-infinity algebra completes the S-matrix description","Boundary L-infinity reconstructs trivial Witten diagrams","Complementary L-infinity yields the full S-matrix","Full S-matrix from relative L-infinity and boundary terms","Relative L-infinity: from S-matrix to AdS boundary diagrams"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction assumes that the homological perturbation lemma can be applied to the infinite-dimensional function spaces of fields, even though the paper only defines the retract projection as 'a suitable left inverse' and the pairings only where convergent.","fun_headline_variants_meta":{"raw":{"variants":["Relative L-infinity algebra completes the S-matrix description","Boundary L-infinity reconstructs trivial Witten diagrams","Complementary L-infinity yields the full S-matrix","Full S-matrix from relative L-infinity and boundary terms","Relative L-infinity: from S-matrix to AdS boundary diagrams"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000716,"raw_usage":{"total_tokens":3188,"prompt_tokens":885,"completion_tokens":2303,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":501,"completion_tokens_details":{"reasoning_tokens":2218}},"tokens_in":501,"tokens_out":2303,"duration_ms":16609,"temperature":1.0,"reasoning_tokens":2218,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:47:38.082850+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the flat-space scalar example of Section 3.1.2 and check whether the map $p$ in the deformation retract (6.1) actually exists as a bounded left inverse of $i$ on the space $V$ defined by (4.7); if no such left inverse exists, or if the boundary pairing (5.6) diverges on the formal sums in $V_{\\partial}$, then the minimal model and the claimed Witten diagram expansion are not defined. A direct check would also compute the two-point amplitude from Definition 9 and compare it with the known Klein–Gordon metric (6.9) and the AdS two-point function (7.1); a mismatch would falsify the central claim.","supporting_citations":[],"review_version":1}