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REVIEW 2 major objections 4 minor 63 references

Full S-matrices and Witten diagrams with (relative) L-infinity algebras

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2412.16106 v1 pith:NAV6SO5G submitted 2024-12-20 hep-th math-phmath.MP

Luigi Alfonsi , Leron Borsten , Hyungrok Kim , Martin Wolf , Charles Alastair Stephen Young This is my paper · ORCID
classification hep-thmath-phmath.MP MSC 81T1881T4081T1317B70
keywords L-infinityalgebrasS-matrixWittendiagramsAdS/CFThomologicalperturbationlemmaBV-BFVformalismChern-SimonstheoryYang-Mills
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

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.

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 (2)
  1. [Section 3.1.2, Eq. (6.1) and surrounding text] 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.
  2. [Section 3.1.2, Eq. (5.9) and the paragraph after (5.8c)] 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.
minor comments (4)
  1. [Section 3.1.2, Eqs. (4.7) and (5.7)] 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.
  2. [Section 2.2, Definition 4] 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.
  3. [Section 2.4, Eqs. (3.4a)-(3.4g)] 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.
  4. [Title and Abstract] 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.

Circularity Check

2 steps flagged · score 6.0 of 10

Two-point 'trivial' S-matrix and CFT two-point functions are engineered through the boundary pairing, π1, and fitted normalization κ; higher-point Witten diagram expansion remains an independent homotopy-transfer result.

  1. fitted input called prediction [Section 3.1.2, 'Relative L-infinity algebra', around eqs. (5.8c)-(5.9); result in (7.1)]
    "where κ is a constant; it will eventually be fixed by requiring that the coefficient for the quadratic term in (6.9) or (7.1) below is correctly normalised (when compared to the trivial part of the S-matrix or the CFT two-point function)."

    The free constant κ in the boundary contribution to the relative homotopy Maurer-Cartan action (5.9) is fitted so that the quadratic coefficient in the AdS minimal-model action (7.1) equals the known CFT two-point normalization. The functional form |x-x'|^{-2Δ+} is itself put in by the choice of boundary pairing (5.6) and by π1(φ)=φ_pw, which extracts the asymptotic expansion. Thus the two-point term in (7.1) is not derived from the relative L-infinity structure; it is the normalization condition used to fix an input. Presenting this as reproduction of the CFT two-point function is a fitted input called a prediction.

  2. self definitional [Section 2.4, Definition 9 and note after eq. (3.7); cf. Section 3.1.2 after eq. (7.2)]
    "Note also that, for the two-point scattering amplitude, whilst the first term in (3.7) vanishes since ◦l1 = 0 by construction, the second term will, in general, not be zero, so that we recover the trivial piece of the S-matrix."

    Definition 9 defines the generalized n-point amplitude to include the boundary terms Σ⟨˜◦π_i, ˜◦π_j⟩. With ◦l1=0 in the bulk minimal model, the two-point amplitude reduces by definition to ⟨π1(φ1), π1(φ2)⟩. The paper later confirms this: 'The CFT two-point correlator is not given by the usual homotopy transfer but is instead given by ⟨π1(−), π1(−)⟩.' Hence the 'recovery' of the identity part of the S-matrix and of the CFT two-point function is the chosen boundary data, not a consequence of homological perturbation theory or of the minimal-model construction.

full rationale

The higher-point machinery is substantially independent. Existence of minimal models is cited to Markl's coloured-operad minimal model, and the homological perturbation lemma to Crainic, Loday-Vallette, and Berglund, none of which are the authors' own work. Given the retract data (i,p,h), the recursion (3.4) is a genuine diagrammatic identity, and its identification with tree-level Witten diagrams is a real mathematical correspondence. The analytic gaps in the non-compact function spaces (convergence of (4.8)/(5.6), the unproved nature of 'a suitable left inverse of i' in (6.1)) are correctness risks, not circularity. However, the paper's advertised inclusion of the 'trivial' part—the identity component of the S-matrix and the CFT two-point function—is built in rather than derived: the boundary L-infinity algebra is chosen so that the boundary minimal model is the input boundary theory; π1 is taken to be the asymptotic-value map; the boundary pairing (5.6) is defined with the shift 1+bd-α so that it evaluates to the known two-point kernel; and the free constant κ is explicitly fixed by requiring the known normalization of (6.9)/(7.1). Consequently the two-point 'prediction' is equivalent by construction to the inputs. This is partial circularity: the higher-point connected Witten diagrams retain independent content, so the score is 6 rather than 8-10.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The construction rests on standard L-infinity and homotopy transfer results, plus two paper-specific modeling assumptions: the doubled boundary field space with pi equal to pullback and normal derivative, and the analytic well-definedness of the function spaces used for flat and AdS spaces. The only explicit free parameter is the constant kappa, which is fitted to the known normalization of the two-point function. No new physical entities are introduced; the relative L-infinity algebra is a mathematical structure.

free parameters (1)
  • kappa = not given; fixed by matching known normalization
    Introduced in eq. (5.8c) as a constant in the boundary term of the relative homotopy Maurer-Cartan action for flat and AdS spaces. It is fixed by requiring the quadratic term in eqs. (6.9) or (7.1) to match the known coefficient of the trivial part of the S-matrix or the CFT two-point function. The functional form is derived, but the overall normalization is fitted to a known result.
assumptions (5)
  • standard math Cyclic L-infinity algebras correspond to classical BV actions, and minimal models of L-infinity algebras encode the non-trivial connected S-matrix.
    Background assumption from Section 1, Table 1, citing references [2-10,14-18]. This is the standard dictionary the paper extends.
  • standard math Relative L-infinity algebras are the minimal model of the two-coloured operad of relative Lie algebras via Koszul duality, and homotopy transfer applies to them.
    Used in Definition 3 and Section 2.4, citing Markl and homological perturbation lemma references. This provides the existence of minimal models.
  • ad hoc to paper The physical action with boundary terms is captured by the relative homotopy Maurer-Cartan action with a doubled boundary space and pi equal to the pullback and normal derivative.
    Introduced in Section 2.1, eqs. (1.11)-(1.14). It is motivated by the BV-BFV formalism but is a specific modeling choice, not derived from a more fundamental principle.
  • domain assumption The function spaces V and V_partial for flat and AdS spaces, together with the deformation retract (6.1), are well-defined and the homological perturbation lemma series converge.
    Assumed in Section 3.1.2. The pairings are declared to hold 'wherever convergent', and the projection p is only described as a suitable left inverse of i, without explicit verification of the retract identities.
  • domain assumption The cyclic identities of Definition 4 hold for the Chern-Simons and Yang-Mills relative L-infinity algebras.
    Asserted without proof in Section 3.2. These identities are needed for the relative homotopy Maurer-Cartan action to be well-defined and for crossing symmetry of the amplitudes.

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Pith. "Pith review of Full S-matrices and Witten diagrams with (relative) L-infinity algebras." pith.science (2026). https://pith.science/paper/NAV6SO5G

@misc{pith2026241216106,
  author       = {Pith},
  title        = {Pith review of: Full S-matrices and Witten diagrams with (relative) L-infinity algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NAV6SO5G}},
  note         = {Machine review of arXiv:2412.16106}
}
abstract

The $L_\infty$-algebra approach to scattering amplitudes elegantly describes the nontrivial part of the $S$-matrix but fails to take into account the trivial part. We argue that the trivial contribution to the $S$-matrix should be accounted for by another, complementary $L_\infty$-algebra, such that a perturbative field theory is described by a cyclic relative $L_\infty$-algebra. We further demonstrate that this construction reproduces Witten diagrams that arise in AdS/CFT including, in particular, the trivial Witten diagrams corresponding to CFT two-point functions. We also discuss Chern-Simons theory and Yang-Mills theory on manifolds with boundaries using this approach.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

63 extracted references · 16 canonical work pages

  1. [1]

    Damgaard, Vittorio Del Duca, Lance Jenkins Dixon, Daniele Dorigoni, Claude Duhr, Yvonne Geyer, Michael Boris Green, Enrico Herrmann, Paul Heslop, Henrik Johansson, Gregory P

    Gabriele Travaglini, Andreas Brandhuber, Patrick Dorey, Tristan McLoughlin, Samuel Abreu, Zvi Bern, Niels Emil Jannik Bjerrum-Bohr, Johannes Blümlein, Ruth Britto, John Joseph Murphy Carrasco, Dmitry Chicherin, Marco Chiodaroli, Poul H. Damgaard, Vittorio Del Duca, Lance Jenkins Dixon, Daniele Dorigoni, Claude Duhr, Yvonne Geyer, Michael Boris Green, Enri...

    arXiv 2022

  2. [2]

    Amplitudes in YM and GR as a minimal model and recursive characterization

    Andrea N \"u tzi and Michael Reiterer. Amplitudes in YM and GR as a minimal model and recursive characterization . Communications in Mathematical Physics , 392:427--482, June 2022. https://arxiv.org/abs/1812.06454 arXiv:1812.06454 , https://doi.org/10.1007/s00220-022-04339-4 doi:10.1007/s00220-022-04339-4

    arXiv 2022

  3. [3]

    Scattering amplitude recursion relations in Batalin--Vilkovisky quantizable theories

    Tommaso Macrelli, Christian Sämann, and Martin Wolf. Scattering amplitude recursion relations in Batalin--Vilkovisky quantizable theories . Physical Review D , 100(4):045017, August 2019. https://arxiv.org/abs/1903.05713 arXiv:1903.05713 , https://doi.org/10.1103/PhysRevD.100.045017 doi:10.1103/PhysRevD.100.045017

    arXiv 2019

  4. [4]

    The L _ -algebra of the \( S \)-matrix

    Alexandros Spyridion Arvanitakis. The L _ -algebra of the \( S \)-matrix. Journal of High Energy Physics , 2019(07):115, 2019. https://arxiv.org/abs/1903.05643 arXiv:1903.05643 , https://doi.org/10.1007/JHEP07(2019)115 doi:10.1007/JHEP07(2019)115

    arXiv 2019

  5. [5]

    Loop Amplitudes and Quantum Homotopy Algebras

    Branislav Jur c o, Tommaso Macrelli, Christian Sämann, and Martin Wolf. Loop amplitudes and quantum homotopy algebras. Journal of High Energy Physics , 2007:3, July 2020. https://arxiv.org/abs/1912.06695 arXiv:1912.06695 , https://doi.org/10.1007/JHEP07(2020)003 doi:10.1007/JHEP07(2020)003

    work page Pith review arXiv 2020

  6. [6]

    L_ -algebras and the perturbiner expansion

    Cristhiam Lopez-Arcos and Alexander Quintero V \'e lez. L_ -algebras and the perturbiner expansion . Journal of High Energy Physics , 2019(11):010, 2019. https://arxiv.org/abs/1907.12154 arXiv:1907.12154 , https://doi.org/10.1007/JHEP11(2019)010 doi:10.1007/JHEP11(2019)010

    arXiv 2019

  7. [7]

    Perturbative Quantum Field Theory and Homotopy Algebras

    Branislav Jur c o, Hyungrok Kim, Tommaso Macrelli, Christian Saemann, and Martin Wolf. Perturbative quantum field theory and homotopy algebras. Proceedings of Science , 376:199, 2020. https://arxiv.org/abs/2002.11168 arXiv:2002.11168 , https://doi.org/10.22323/1.376.0199 doi:10.22323/1.376.0199

    work page Pith review arXiv 2020

  8. [8]

    Symmetry Factors of Feynman Diagrams and the Homological Perturbation Lemma

    Christian Saemann and Emmanouil Sfinarolakis. Symmetry factors of F eynman diagrams and the homological perturbation lemma. Journal of High Energy Physics , 2012(12):88, December 2020. https://arxiv.org/abs/2009.12616 arXiv:2009.12616 , https://doi.org/10.1007/JHEP12(2020)088 doi:10.1007/JHEP12(2020)088

    work page Pith review arXiv 2020

Show all 63 references
  1. [9]

    The L _ structure of gauge theories with matter

    Humberto Gomez, Renann Lipinski Jusinskas, Cristhiam Lopez-Arcos, and Alexander Quintero V\'elez. The L _ structure of gauge theories with matter. Journal of High Energy Physics , 2021(02):093, 2021. https://arxiv.org/abs/2011.09528 arXiv:2011.09528 , https://doi.org/10.1007/J...

    2021 arXiv

  2. [10]

    Tree-level scattering amplitudes via homotopy transfer

    Roberto Bonezzi, Christoph Chiaffrino, Felipe Díaz-Jaramillo, and Olaf Hohm. Tree-level scattering amplitudes via homotopy transfer. December 2023. https://arxiv.org/abs/2312.09306 arXiv:2312.09306 , https://doi.org/10.48550/arXiv.2312.09306 doi:10.48550/arXiv.2312.09306

    2023 doi arXiv

  3. [11]

    Anton M. Zeitlin. B atalin-- V ilkovisky Y ang-- M ills theory as a homotopy C hern-- S imons theory via string field theory. International Journal of Modern Physics A , 24(07):1309--1331, 2009. https://arxiv.org/abs/0709.1411 arXiv:0709.1411 , https://doi.org/10.1142/S0217751...

    2009 arXiv

  4. [12]

    Anton M. Zeitlin. Conformal field theory and algebraic structure of gauge theory. Journal of High Energy Physics , 2008(1003):056, 2008. https://arxiv.org/abs/0812.1840 arXiv:0812.1840 , https://doi.org/10.1007/JHEP03(2010)056 doi:10.1007/JHEP03(2010)056

    2010 arXiv

  5. [13]

    L_ algebras and field theory

    Olaf Hohm and Barton Zwiebach Cantor. L_ algebras and field theory . Fortschritte der Physik , 65(3-4):1700014, 2017. https://arxiv.org/abs/1701.08824 arXiv:1701.08824 , https://doi.org/10.1002/prop.201700014 doi:10.1002/prop.201700014

    2017 arXiv

  6. [14]

    L_ -algebras of classical field theories and the B atalin-- V ilkovisky formalism

    Branislav Jur c o, Lorenzo Raspollini, Christian Saemann, and Martin Wolf. L_ -algebras of classical field theories and the B atalin-- V ilkovisky formalism . Fortschritte der Physik , 67(7):1900025, July 2019. https://arxiv.org/abs/1809.09899 arXiv:1809.09899 , https://doi.or...

    2019 arXiv

  7. [15]

    L_ -algebras, the BV formalism, and classical fields

    Branislav Jur c o, Tommaso Macrelli, Lorenzo Raspollini, Christian Saemann, and Martin Wolf. L_ -algebras, the BV formalism, and classical fields . Fortschritte der Physik , 67(8--9):1910025, 2019. https://arxiv.org/abs/1903.02887 arXiv:1903.02887 , https://doi.org/10.1002/pro...

    2019 arXiv

  8. [16]

    Homotopy transfer and effective field theory I : Tree-level

    Alexandros Spyridion Arvanitakis, Olaf Hohm, Christopher Michael Hull, and Victor Lekeu. Homotopy transfer and effective field theory I : Tree-level. Fortschritte der Physik , 70(2--3):2200003, March 2022. https://arxiv.org/abs/2007.07942 arXiv:2007.07942 , https://doi.org/10....

    2022 arXiv

  9. [17]

    Homotopy transfer and effective field theory II : Strings and double field theory

    Alexandros Spyridion Arvanitakis, Olaf Hohm, Christopher Michael Hull, and Victor Lekeu. Homotopy transfer and effective field theory II : Strings and double field theory. Fortschritte der Physik , 70(2--3):2200004, March 2022. https://arxiv.org/abs/2106.08343 arXiv:2106.08343...

    2022 arXiv

  10. [18]

    Out-of-time-order asymptotic observables are quasi-isomorphic to time-ordered amplitudes

    Leron Borsten, David Simon Henrik Jonsson, and Hyungrok Kim. Out-of-time-order asymptotic observables are quasi-isomorphic to time-ordered amplitudes. Journal of High Energy Physics , 2024(08):074, August 2024. https://arxiv.org/abs/2405.11110 arXiv:2405.11110 , https://doi.or...

    2024 arXiv

  11. [19]

    S-matrix and homological perturbation lemma

    Ján Pulmann. S-matrix and homological perturbation lemma . Diplomová práce, Univerzita Karlova, Matematicko-fyzikální fakulta, Prague, June 2016. URL: http://hdl.handle.net/20.500.11956/75884

    2016

  12. [20]

    Quantum L_ algebras and the homological perturbation lemma

    Martin Doubek, Branislav Jur c o, and J \' a n Pulmann. Quantum L_ algebras and the homological perturbation lemma . Communications in Mathematical Physics , 367:215--240, April 2019. https://arxiv.org/abs/1712.02696 arXiv:1712.02696 , https://doi.org/10.1007/s00220-019-03375-...

    2019 arXiv

  13. [21]

    A homotopy BV algebra for Y ang-- M ills and color--kinematics, December 2019

    Michael Reiterer. A homotopy BV algebra for Y ang-- M ills and color--kinematics, December 2019. https://arxiv.org/abs/1912.03110 arXiv:1912.03110 , https://doi.org/10.48550/arXiv.1912.03110 doi:10.48550/arXiv.1912.03110

    2019 doi arXiv

  14. [22]

    Double copy from homotopy algebras

    Leron Borsten, Branislav Jur c o, Hyungrok Kim, Tommaso Macrelli, Christian Saemann, and Martin Wolf. Double copy from homotopy algebras . Fortschritte der Physik , 69(8--9):2100075, September 2021. https://arxiv.org/abs/2102.11390 arXiv:2102.11390 , https://doi.org/10.1002/pr...

    2021 arXiv

  15. [23]

    Kinematic lie algebras from twistor spaces

    Leron Borsten, Branislav Jur c o, Hyungrok Kim, Tommaso Macrelli, Christian Saemann, and Martin Wolf. Kinematic lie algebras from twistor spaces. Physical Review Letters , 131:041603, 2023. https://arxiv.org/abs/2211.13261 arXiv:2211.13261 , https://doi.org/10.1103/PhysRevLett...

    2023 arXiv

  16. [24]

    Colour--kinematics duality, double copy, and homotopy algebras

    Leron Borsten, Hyungrok Kim, Branislav Jur c o, Tommaso Macrelli, Christian Saemann, and Martin Wolf. Colour--kinematics duality, double copy, and homotopy algebras. Proceedings of Science , ICHEP2022:426, June 2023. https://arxiv.org/abs/2211.16405 arXiv:2211.16405 , https://...

    2023 arXiv

  17. [25]

    Double copy from tensor products of metric BV ^ gray -algebras

    Leron Borsten, Branislav Jur c o, Hyungrok Kim, Tommaso Macrelli, Christian Saemann, and Martin Wolf. Double copy from tensor products of metric BV ^ gray -algebras. Fortschritte der Physik , 2024. https://arxiv.org/abs/2307.02563 arXiv:2307.02563 , https://doi.org/10.1002/pro...

    2024 arXiv

  18. [26]

    Tree-level color--kinematics duality from pure spinor actions

    Leron Borsten, Branislav Jur c o, Hyungrok Kim, Tommaso Macrelli, Christian Saemann, and Martin Wolf. Tree-level color--kinematics duality from pure spinor actions. Physical Revew D , 108:126012, December 2023. https://arxiv.org/abs/2303.13596 arXiv:2303.13596 , https://doi.or...

    2023 arXiv

  19. [27]

    Double-copying self-dual Yang-Mills theory to self-dual gravity on twistor space

    Leron Borsten, Branislav Jur c o, Hyungrok Kim, Tommaso Macrelli, Christian Saemann, and Martin Wolf. Double-copying self-dual Yang-Mills theory to self-dual gravity on twistor space . Journal of High Energy Physics , 2023(11):172, November 2023. https://arxiv.org/abs/2307.103...

    2023 arXiv

  20. [28]

    Gauge independent kinematic algebra of self-dual Y ang- M ills theory

    Roberto Bonezzi, Felipe Díaz-Jaramillo, and Silvia Nagy. Gauge independent kinematic algebra of self-dual Y ang- M ills theory. Physical Review D , 108(9):065007, September 2023. https://arxiv.org/abs/2306.08558 arXiv:2306.08558 , https://doi.org/10.1103/PhysRevD.108.065007 do...

    2023 arXiv

  21. [29]

    Double copy of 3D Chern-Simons theory and 6D Kodaira-Spencer gravity

    Roberto Bonezzi, Felipe Díaz-Jaramillo, and Olaf Hohm. Double copy of 3D Chern-Simons theory and 6D Kodaira-Spencer gravity . Physical Review D , 110(4):045024, August 2024. https://arxiv.org/abs/2404.16830 arXiv:2404.16830 , https://doi.org/10.1103/PhysRevD.110.045024 doi:10....

    2024 arXiv

  22. [30]

    White, and Sam Wikeley

    Kymani Armstrong-Williams, Silvia Nagy, Chris D. White, and Sam Wikeley. What can abelian gauge theories teach us about kinematic algebras? Journal of High Energy Physics , 2024(08):169, 2024. https://arxiv.org/abs/2401.10750 arXiv:2401.10750 , https://doi.org/10.1007/JHEP08(2...

    2024 arXiv

  23. [31]

    Vertex operators for the kinematic algebra of Y ang- M ills theory

    Roberto Bonezzi, Christoph Chiaffrino, and Olaf Hohm. Vertex operators for the kinematic algebra of Y ang- M ills theory. August 2024. https://arxiv.org/abs/2408.17341 arXiv:2408.17341 , https://doi.org/10.48550/arXiv.2408.17341 doi:10.48550/arXiv.2408.17341

    2024 doi arXiv

  24. [32]

    Holography as homotopy

    Christoph Chiaffrino, Talha Ersoy, and Olaf Hohm. Holography as homotopy. Journal of High Energy Physics , 2024(09):161, 2024. https://arxiv.org/abs/2307.08094 arXiv:2307.08094 , https://doi.org/10.1007/JHEP09(2024)161 doi:10.1007/JHEP09(2024)161

    2024 arXiv

  25. [33]

    Classical BV theories on manifolds with boundary

    Alberto Sergio Cattaneo, Pavel Nikolaevich Mnëv, and Nicolai Yuryevich Reshetikhin. Classical BV theories on manifolds with boundary. Communications in Mathematical Physics , 332(2):535–603, August 2014. https://arxiv.org/abs/1201.0290 arXiv:1201.0290 , https://doi.org/10.1007...

    2014 arXiv

  26. [34]

    B \'e rard

    Pierre H. B \'e rard. Spectral Geometry: Direct and Inverse Problems , volume 1207 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, Heidelberg, 1986. https://doi.org/10.1007/BFb0076330 doi:10.1007/BFb0076330

    1986 doi

  27. [35]

    Towards holography in the BV-BFV setting

    Pavel Nikolaevich Mnëv, Michele Schiavina, and Konstantin Wernli. Towards holography in the BV-BFV setting. Annales Henri Poincaré , 21(3):993–1044, October 2019. https://arxiv.org/abs/1905.00952 arXiv:1905.00952 , https://doi.org/10.1007/s00023-019-00862-8 doi:10.1007/s00023-...

    2019 arXiv

  28. [36]

    Asymptotic symmetries in the BV-BFV formalism

    Katarzyna Anna Rejzner and Michele Schiavina. Asymptotic symmetries in the BV-BFV formalism. Communications in Mathematical Physics , 385(2):1083–1132, April 2021. https://arxiv.org/abs/2002.09957 arXiv:2002.09957 , https://doi.org/10.1007/s00220-021-04061-7 doi:10.1007/s00220...

    2021 arXiv

  29. [37]

    J\'an Pulmann, Pavol S evera, and Donald R. Youmans. Renormalization group flow of C hern-- S imons boundary conditions and generalized R icci tensor. Journal of High Energy Physics , 2020(10):096, 2020. https://arxiv.org/abs/2009.00509 arXiv:2009.00509 , https://doi.org/10.10...

    2020 arXiv

  30. [38]

    Twisted supergravity and K oszul duality: A case study in \( AdS _3\)

    Kevin Joseph Costello and Natalie Marie Paquette. Twisted supergravity and K oszul duality: A case study in \( AdS _3\) . Communications in Mathematical Physics , 384(1):279–339, April 2021. https://arxiv.org/abs/2001.02177 arXiv:2001.02177 , https://doi.org/10.1007/s00220-021...

    2021 arXiv

  31. [39]

    Williams

    Natalie Marie Paquette and Brian R. Williams. Koszul duality in quantum field theory. Confluentes Mathematici , 14(2):87--138, 2022. https://arxiv.org/abs/2110.10257 arXiv:2110.10257 , https://doi.org/10.5802/cml.88 doi:10.5802/cml.88

    2022 arXiv

  32. [40]

    Homotopy algebras are homotopy algebras

    Martin Markl. Homotopy algebras are homotopy algebras. Forum Mathematicum , 16(1):129--160, 2004. https://arxiv.org/abs/math/9907138 arXiv:math/9907138 , https://doi.org/10.1515/form.2004.002 doi:10.1515/form.2004.002

    2004 arXiv

  33. [41]

    Homotopy diagrams of algebras

    Martin Markl. Homotopy diagrams of algebras. In Jan Slovák and Martin Čadek, editors, Proceedings of the 21st Winter School ``Geometry and Physics'' , pages 161--180, Palermo, 2002. Circolo Matematico di Palermo. URL: https://hdl.handle.net/10338.dmlcz/701787, https://arxiv.or...

    2002 doi arXiv

  34. [42]

    Alberto Sergio Cattaneo and Carlo A. Rossi. Higher-dimensional \( BF \) theories in the B atalin-- V ilkovisky formalism: The BV action and generalized W ilson loops. Communications in Mathematical Physics , 221:591--657, 2001. https://arxiv.org/abs/math.QA/0010172 arXiv:math....

    2001

  35. [43]

    Noncommutative homotopy algebras associated with open strings

    Hiroshige Kajiura. Noncommutative homotopy algebras associated with open strings. Reviews in Mathematical Physics , 19(1):1--99, 2007. https://arxiv.org/abs/math.QA/0306332 arXiv:math.QA/0306332 , https://doi.org/10.1142/S0129055X07002912 doi:10.1142/S0129055X07002912

    2007

  36. [44]

    Introduction to supergeometry

    Alberto Sergio Cattaneo and Florian Schaetz. Introduction to supergeometry . Reviews in Mathematical Physics , 23:669--690, 2011. https://arxiv.org/abs/1011.3401 arXiv:1011.3401 , https://doi.org/10.1142/S0129055X11004400 doi:10.1142/S0129055X11004400

    2011 arXiv

  37. [45]

    Global theory of graded manifolds

    Jan Vysoký. Global theory of graded manifolds. Reviews in Mathematical Physics , 34(10), August 2022. https://arxiv.org/abs/2105.02534 arXiv:2105.02534 , https://doi.org/10.1142/s0129055x22500350 doi:10.1142/s0129055x22500350

    2022 arXiv

  38. [46]

    Unimodular homotopy algebras and C hern-- S imons theory

    Christopher Braun and Andrey Yurievich Lazarev. Unimodular homotopy algebras and C hern-- S imons theory. Journal of Pure and Applied Algebra , 219(11):5158--5194, November 2015. https://arxiv.org/abs/1309.3219 arXiv:1309.3219 , https://doi.org/10.1016/j.jpaa.2015.05.017 doi:1...

    2015 arXiv

  39. [47]

    Minimal models of quantum homotopy L ie algebras via the BV -formalism

    Christopher Braun and James Maunder. Minimal models of quantum homotopy L ie algebras via the BV -formalism. Journal of Mathematical Physics , 59:063512, June 2018. https://arxiv.org/abs/1703.00082 arXiv:1703.00082 , https://doi.org/10.1063/1.5022890 doi:10.1063/1.5022890

    2018 arXiv

  40. [48]

    On the perturbation lemma, and deformations, 2004

    Marius Crainic. On the perturbation lemma, and deformations, 2004. https://arxiv.org/abs/math.AT/0403266 arXiv:math.AT/0403266 , https://doi.org/10.48550/arXiv.math/0403266 doi:10.48550/arXiv.math/0403266

    2004 doi

  41. [49]

    Algebraic Operads

    Jean-Louis Loday and Bruno Vallette. Algebraic Operads . Springer-Verlag, Berlin, Heidelberg, 2012. https://doi.org/10.1007/978-3-642-30362-3 doi:10.1007/978-3-642-30362-3

    2012 doi

  42. [50]

    Homological perturbation theory for algebras over operads

    Alexander Frans Olof Berglund. Homological perturbation theory for algebras over operads. Algebraic & Geometric Topology , 14, 2014. https://arxiv.org/abs/0909.3485 arXiv:0909.3485 , https://doi.org/10.2140/agt.2014.14.2511 doi:10.2140/agt.2014.14.2511

    2014 arXiv

  43. [51]

    Seolhwa Kim, Per Kraus, Ruben Monten, and Richard M. Myers. S-matrix path integral approach to symmetries and soft theorems. Journal of High Energy Physics , 2023:36, October 2023. https://arxiv.org/abs/2307.12368 arXiv:2307.12368 , https://doi.org/10.1007/JHEP10(2023)036 doi:...

    2023 arXiv

  44. [52]

    Hannesdottir, and Sebastian Mizera

    Simon Caron-Huot, Mathieu Giroux, Holmfridur S. Hannesdottir, and Sebastian Mizera. What can be measured asymptotically? Journal of High Energy Physics , 2024:139, 2024. https://arxiv.org/abs/2308.02125 arXiv:2308.02125 , https://doi.org/10.1007/JHEP01(2024)139 doi:10.1007/JHE...

    2024 arXiv

  45. [53]

    Positive energy in anti-de S itter backgrounds and gauged extended supergravity

    Peter Breitenlohner and Daniel Zissel Freedman. Positive energy in anti-de S itter backgrounds and gauged extended supergravity. Physics Letters B , 115(3):197--201, 1982. https://doi.org/10.1016/0370-2693(82)90643-8 doi:10.1016/0370-2693(82)90643-8

    1982 doi

  46. [54]

    Stability in gauged extended supergravity

    Peter Breitenlohner and Daniel Zissel Freedman. Stability in gauged extended supergravity. Annals of Physics , 144(2):249--281, 1982. https://doi.org/10.1016/0003-4916(82)90116-6 doi:10.1016/0003-4916(82)90116-6

    1982 doi

  47. [55]

    Jacobson

    Theodore A. Jacobson. Introduction to quantum fields in curved space-time and the H awking effect. In Lectures on Quantum Gravity , Series of the Centro De Estudios Científicos, pages 39--89, 8 2003. https://arxiv.org/abs/gr-qc/0308048 arXiv:gr-qc/0308048 , https://doi.org/10....

    2003 arXiv

  48. [56]

    Anti de S itter space and holography

    Edward Witten. Anti de S itter space and holography. Advances in Theoretical and Mathematical Physics , 2(2):253--291, 1998. https://arxiv.org/abs/hep-th/9802150 arXiv:hep-th/9802150 , https://doi.org/10.4310/ATMP.1998.v2.n2.a2 doi:10.4310/ATMP.1998.v2.n2.a2

    1998 arXiv

  49. [57]

    Correlation functions in the CFT\(_d\)/AdS\(_ d+1 \) correspondence

    Daniel Zissel Freedman, Samir Dayal Mathur, Alec Matusis, and Leonardo Rastelli. Correlation functions in the CFT\(_d\)/AdS\(_ d+1 \) correspondence. Nuclear Physics B , 546(1--2):96--118, April 1999. https://arxiv.org/abs/hep-th/9804058 arXiv:hep-th/9804058 , https://doi.org/...

    1999 arXiv

  50. [58]

    On de R ham's theorem

    Hans Samelson. On de R ham's theorem. Topology , 6(4):427--432, November 1967. https://doi.org/10.1016/0040-9383(67)90002-X doi:10.1016/0040-9383(67)90002-X

    1967 doi

  51. [59]

    A variational method in the theory of harmonic integrals

    Charles Bradfield Morrey, Jr. A variational method in the theory of harmonic integrals. Communications on Pure and Applied Mathematics , 9(3):499--508, August 1956. https://doi.org/10.1002/cpa.3160090321 doi:10.1002/cpa.3160090321

    1956 doi

  52. [60]

    Peter B. Gilkey. Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem . Studies in Advanced Mathematics. CRC Press, Boca Raton, second edition, December 1994. https://doi.org/10.1201/9780203749791 doi:10.1201/9780203749791

    1994 doi

  53. [61]

    Harmonic fields in R iemannian manifolds (generalized potential theory)

    Kunihiko Kodaira. Harmonic fields in R iemannian manifolds (generalized potential theory). Annals of Mathematics , 50(3):587--665, July 1949. https://doi.org/10.2307/1969552 doi:10.2307/1969552

    1949 doi

  54. [62]

    Hodge Decomposition -- A Method for Solving Boundary Value Problems , volume 1607 of Lecture Notes in Mathematics

    G \"u nter Schwarz. Hodge Decomposition -- A Method for Solving Boundary Value Problems , volume 1607 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, Heidelberg, July 1995. https://doi.org/10.1007/BFb0095981 doi:10.1007/BFb0095981

    1995 doi

  55. [63]

    Chern--Simons perturbation theory

    Scott Axelrod and Isadore Manuel Singer. Chern--Simons perturbation theory. In Sultan Catto and Alvany Rocha, editors, Differential Geometric Methods in Theoretical Physics. XXth International Conference on Differential Geometric Methods in Theoretical Physics (XX DGM), New Yo...

    1991 arXiv

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