Double Copy from Homotopy Algebras
Leron Borsten, Branislav Jurco, Hyungrok Kim, Tommaso Macrelli, Christian Saemann, Martin Wolf
DDMUS–MP–21/04EMPG–21–03
Double Copy from Homotopy Algebras
Leron Borsten a , Hyungrok Kim a , Branislav Jurčo b , Tommaso Macrelli c ,Christian Saemann a , and Martin Wolf c ∗a Maxwell Institute for Mathematical SciencesDepartment of Mathematics, Heriot–Watt UniversityEdinburgh EH14 4AS, United Kingdom b Charles University PragueFaculty of Mathematics and Physics, Mathematical InstitutePrague 186 75, Czech Republic c Department of Mathematics, University of SurreyGuildford GU2 7XH, United Kingdom
Abstract
We show that the BRST Lagrangian double copy construction of N “ su-pergravity as the ‘square’ of Yang–Mills theory finds a natural interpretationin terms of homotopy algebras. We significantly expand on our previous workarguing the validity of the double copy at the loop level, and we give a de-tailed derivation of the double copied Lagrangian and BRST operator. Ourconstructions are very general and can be applied to a vast set of examples.22nd February 2021 ∗ E-mail addresses: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] , [email protected] a r X i v : . [ h e p - t h ] F e b ontents
1. Introduction and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1. Double copy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2. Homotopy algebras and quantum field theory . . . . . . . . . . . . . . . . 91.3. Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.5. Reading guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222. Double copy basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.1. Scattering amplitude generalities . . . . . . . . . . . . . . . . . . . . . . . 232.2. Colour–kinematics duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3. Double copy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4. Manifestly colour–kinematics-dual action . . . . . . . . . . . . . . . . . . . 313. Non-linear sigma model and special galileons . . . . . . . . . . . . . . . . . . . . 343.1. Review of the essentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2. Flavour–kinematics duality and double copy . . . . . . . . . . . . . . . . . 363.3. Formulation in terms of homotopy algebras . . . . . . . . . . . . . . . . . . 414. Field theories, Batalin–Vilkovisky complexes, and homotopy algebras . . . . . . 444.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2. Batalin–Vilkovisky formalism and L -algebras . . . . . . . . . . . . . . . . 494.3. Scattering amplitudes and L -algebras . . . . . . . . . . . . . . . . . . . . 565. Examples of homotopy algebras of field theories . . . . . . . . . . . . . . . . . . 625.1. Biadjoint scalar field theory . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2. Yang–Mills theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.3. Free Kalb–Ramond two-form . . . . . . . . . . . . . . . . . . . . . . . . . . 665.4. Einstein–Hilbert gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.5. N “ supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756. Factorisation of homotopy algebras and colour ordering . . . . . . . . . . . . . . 756.1. Tensor products of homotopy algebras . . . . . . . . . . . . . . . . . . . . 756.2. Colour-stripping in Yang–Mills theory . . . . . . . . . . . . . . . . . . . . . 786.3. Twisted tensor products of strict homotopy algebras . . . . . . . . . . . . . 817. Factorisation of free field theories and free double copy . . . . . . . . . . . . . . 837.1. Factorisation of the chain complex of biadjoint scalar field theory . . . . . 847.2. Factorisation of the chain complex of Yang–Mills theory . . . . . . . . . . 857.3. Canonical transformation for the free Kalb–Ramond two-form . . . . . . . 917.4. Canonical transformation for Einstein–Hilbert gravity with dilaton . . . . 937.5. Factorisation of the chain complex of N “ supergravity . . . . . . . . . . 968. Quantum field theoretic preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 1008.1. BRST-extended Hilbert space and Ward identities . . . . . . . . . . . . . . 1018.2. Quantum equivalence, correlation functions, and field redefinitions . . . . . 1068.3. Strictification of Yang–Mills theory . . . . . . . . . . . . . . . . . . . . . . 1118.4. Colour–kinematics duality for unphysical states . . . . . . . . . . . . . . . 1169. Double copy from factorisation of homotopy algebras . . . . . . . . . . . . . . . 1181.1. Biadjoint scalar field theory . . . . . . . . . . . . . . . . . . . . . . . . . . 1199.2. Strictified Yang–Mills theory . . . . . . . . . . . . . . . . . . . . . . . . . . 1209.3. BRST Lagrangian double copy . . . . . . . . . . . . . . . . . . . . . . . . . 1259.4. BRST Lagrangian double copy of Yang–Mills theory . . . . . . . . . . . . 1319.5. Equivalence of the double copied action and N “ supergravity . . . . . . 134Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138A. Definitions and conventions for homotopy algebras . . . . . . . . . . . . . . . . . 138A.1. A -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138A.2. C -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140A.3. L -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140A.4. Structure theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147B. Inverses of wave operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Data Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
1. Introduction and results
Recent years have witnessed a resurgence of the idea that gravity can, in some sense, beviewed as the product of two gauge theories,‘gravity “ gauge b gauge’ . (1.1)This notion goes back at least to the 1960’s [1, 2] and was realised concretely at the levelof tree-level scattering amplitudes via the Kawai–Lewellen–Tye (KLT) relations of stringtheory [3]: closed string tree amplitudes can be written as sums over products of openstring tree amplitudes. The field theory limit of the KLT relations implies a relationshipbetween tree-level Yang–Mills amplitudes and those of N “ supergravity, the universalmassless sector of closed string theories consisting of Einstein–Hilbert gravity coupled to aKalb–Ramond two-form and dilaton.This paradigm was dramatically advanced with the advent of the Bern–Carrasco–Johansson (BCJ) colour–kinematics duality and the double copy prescription [4–6]. Firstly,it was conjectured [4] that gluon amplitudes can be recast so as to manifest a duality bet-ween their colour and kinematical data. This was quickly established at the tree level [7, 8],however it remains conjectural at the loop level. Then, given a gluon amplitude in colour–kinematics-dual form, it can be ‘double-copied’ to yield a bona fide amplitude of N “ supergravity. 2n our previous work [9], we have shown that the double copy can be realised at thelevel of ‘off-mass-shell’ perturbative quantum field theories. Specifically,(i) the Yang–Mills Becchi–Rouet–Stora–Tyutin (BRST) Lagrangian can be made tomanifest tree-level colour–kinematics duality for the full BRST-extended Fock spaceand(ii) the Yang–Mills BRST Lagrangian itself double-copies to yield the perturbative N “ supergravity BRST Lagrangian.An immediate corollary is that the double copy of gluon amplitudes yields the amplitudesof N “ supergravity to all orders in perturbation theory, at trees and loops. Goals of this work.
In this work, we shall(i) give a detailed exposition of the central ideas contained in [9]. As a warm-up we applyour methodology to the Lagrangian double copy of the non-linear sigma model, whichgives a special galileon theory to all orders in perturbation theory;(ii) show that colour–kinematics duality and the BRST Lagrangian double copy can beelegantly articulated in terms of homotopy algebras. In particular, the Batalin–Vilkovisky (BV) L -algebra of Yang–Mills theory admits a twisted factorisation, andthis makes the double copy construction manifest;(iii) address some of the implications of this perspective for generalisations beyond Yang–Mills theory, colour–kinematics duality, the double-copy, and scattering amplitudes.Since one of our aims is to cater to both the scattering amplitudes and the homotopyalgebra communities, our discussion will be self-contained to a high level. We also provideseparate introductory sections for both the double copy (Section 1.1) and homotopy algebras(Section 1.2) in the following. The reader familiar with both of these areas may want toskip directly to the results presented in Section 1.3, the outlook given in Section 1.4, or thereading guide provided in Section 1.5. Heuristically, by (1.1) we mean that one can regard the tensorproduct of two gauge potentials as the field content of a gravitational theory summarisedby ‘ A µ b ¯ A ν “ g µν ‘ B µν ‘ ϕ ’ . (1.2)Here, A µ and ¯ A ν are the gauge potentials of two distinct Yang–Mills theories with twocolour or gauge Lie algebras g and ¯ g . After stripping off the colour component, the tensor3roduct of A µ and ¯ A µ yields a metric g µν , the Kalb–Ramond Abelian two-form gaugepotential B µν , and a scalar field ϕ called the dilaton. The latter form the field contentof N “ supergravity, the common Neveu–Schwarz-Neveu–Schwarz sector of the α Ñ limit of closed string theories with classical action S N “ : “ ż d d x ?´ g ! ´ κ R ´ d ´ B µ ϕ B µ ϕ ´ e ´ κd ´ ϕ H µνκ H µνκ ) , (1.3)where κ “ πG p d q N is Einstein’s gravitational constant, R the scalar curvature, and H µνκ the curvature of B µν .We can refine our interpretation of equation (1.2) to on-shell states of scattering amp-litudes, if we regard it as the tensor product of the corresponding space-time little grouprepresentations, R d ´ b R d ´ – Ä R d ´ ‘ Ź R d ´ ‘ R ,A i b ¯ A j – g p ij q ‘ B r ij s ‘ ϕ , (1.4)where R d ´ is the vector representation of SO p d ´ q , Ä R d ´ denotes the trace-freesymmetric product, and i, j “ , . . . , d ´ . In the context of scattering amplitudes, thisamounts to tensor product of transverse gluon polarisation tensors into those of the graviton(transverse-traceless), Kalb–Ramond two-form (transverse) and dilaton, ε µ ¯ ε ν “ ` ε p µ ¯ ε ν q ´ d η µν ε ρ ¯ ε ρ ˘ ` ε r µ ¯ ε ν s ` d η µν ε ρ ¯ ε ρ . (1.5)Given this identification of on-shell states, it is natural to wonder about a correspondingidentification of scattering amplitudes of N “ supergravity. The latter relation, however,has to be subtle. In particular, given that the Weinberg–Witten theorem [10] forbidscomposite gravitons under the assumption that there exists a Lorentz covariant conservedenergy–momentum tensor, how should one make sense of such a proposal? Moreover, whathappens to the gauge groups and from where would the diffeomorphism invariance of (1.3)arise?The earliest non-trivial concrete realisation of (1.2) came from string theory in theguise of the aforementioned KLT relations [3]: the tree-level scattering amplitudes of closedstrings are sums of products of open string amplitudes. The intuition is clear as closedstring spectra are given by the tensor product of left and right moving open string spectra.The low energy effective field theory limits of closed (open) strings are given by gravity(Yang–Mills) theories, so the graviton, Kalb–Ramond two-form and dilaton states arise asthe tensor product of the gluon states and we should expect precisely the couplings of (1.3). This expression is meant to be schematic. In particular, the trace piece corresponding to the dilatonmust be supplemented by additional terms to render it left- and right-transverse. “ open b open’ picture was used to construct one-loop gravitonamplitudes [11], indicating that such relations may extend beyond the semi-classical regime.The key technical development in this regard was a shift to an ‘on-shell’ perspective thatdispensed with the familiar Lagrangian starting point and relied instead on gauge-invarianton-shell amplitude structures, such as recursion relations and unitarity cuts. For example,unitarity methods [12–18] were employed to build d “ , N “ supergravity two-loop amp-litudes from N “ supersymmetric Yang–Mills theory, by-passing the usual Lagrangianand Feynman diagram prescription entirely [19]. Such results motivated a search for ageneral all-loop amplitude factorisation. Colour–kinematics duality.
Importantly, the on-shell methodology uncovered proper-ties of amplitudes not readily visible in the underlying Lagrangians. This includes thecolour–kinematics duality of Yang–Mills theory [4, 5]. Let us briefly summarise this dualityhere; we will give the details in Section 2.2. For pedagogical introductions and furtherreferences see [20–24].Firstly, one can write any gluon amplitude entirely in terms of trivalent graphs by‘blowing’ up the four-point contact terms, see Section 2 for details and notation. Theresulting trivalent diagrams are thus not the Feynman diagrams of the original theory, butof an equivalent theory. Having done so, the colour–kinematics duality conjecture statesthat there exists a rewriting of the amplitude such that(i) for any triple of graphs, p i, j, k q with colour factors c i , c j , and c k , which are builtentirely from the structure constants f abc of the gauge Lie algebra, obeying a Jacobiidentity c i ` c j ` c k “ , the corresponding kinematic factors, n i , n j , and n k , whichare built from the momenta and polarisation tensors, also obey the same Jacobi-typeidentity n i ` n j ` n k “ and(ii) for any diagram, i , such that c i Ñ ´ c i under the interchange of two legs then n i Ñ ´ n i .Kinematic factors n i satisfying the colour–kinematics duality conditions are referred to asthe BCJ numerators.Whilst the colour factors satisfy Jacobi identities by definition, it is not at all obviousthat the kinematic factors should obey the same rules; it is certainly not evident form theYang–Mills Lagrangian. A reorganisation admitting this surprising relationship betweencolour and kinematic data exists for all n -point tree-level amplitudes, as has been demon-strated from a number of perspectives [25–29]. Although there is as yet no proof that the5olour–kinematics duality will continue to hold for general loop amplitudes, there are manyhighly non-trivial examples providing supportive evidence [30–37].The kinematical Jacobi identities have important implications for the structure of thescattering amplitudes themselves, such as the existence of BCJ relations amongst colour-ordered partial amplitudes, reducing the number of independent n -point partial amplitudesdown from p n ´ q ! to p n ´ q ! [4]. The perhaps most important implication is the doublecopy of tree-level scattering amplitudes. Double copy.
Consider the BCJ double copy prescription [5,6]. Concretely, take the two n -point L -loop Yang–Mills amplitudes, both written in trivalent form with respective colourand kinematic factors p c i , n i q and p ˜ c i , ˜ n i q , at least one of which has been successfully cast ina colour–kinematics-duality respecting form, say p c i , n i q . We can construct a correspondinggravitational theory amplitude by simply replacing each colour factor in p c i , ˜ n i q with thecorresponding kinematic factor of p c i , n i q , that is, p c i , ˜ c i q Ñ p n i , ˜ n i q . We have removed allreference to the gauge group and ‘doubled’ the kinematic terms. This addresses the first ofour earlier questions: the gauge Lie algebra is replaced by a ‘kinematic algebra’. The secondquestion concerning diffeomorphisms is more subtle, but also rests on colour–kinematicsduality. For example, assuming colour–kinematics duality, the residual gauge invarianceof the Yang–Mills amplitudes implies the invariance of the double copy amplitude underresidual diffeomorphisms [38]. For two Yang–Mills theories, again with possibly unrelatedgauge Lie algebras, this double copy procedure generates all possible tree amplitudes of N “ supergravity, giving precise meaning to the heuristic equation (1.2), at least at thesemi-classical level.This prescription generalises to supersymmetric Yang–Mills theory with both unrelatedsupersymmetry and gauge algebras. For example, we could take the p c i , n i q from N “ supersymmetric Yang–Mills amplitudes and the p ˜ c i , ˜ n i q from purely bosonic Yang–Millstheory and double-copy them to the amplitudes of N “ supergravity [39]. Alternatively,if both factors are N “ supersymmetric Yang–Mills theories, we produce the amplitudes of N “ supergravity [5]. This can be thought of as the (low energy limit of the) dimensionalreduction on a six-dimensional torus of the ‘type II “ type I b type I’ relation of d “ ` superstring theory. By varying the left and right factors over all colour–kinematics dualitycompatible gauge theories, we generate all double-copy constructible gravitational theories.Whilst concrete constructions are complicated, there is nonetheless a rapidly multiplyingzoology of double copy constructible gravity theories [30, 5, 6, 40–45, 42, 46–59, 38, 60–62].The double copy is clearly conceptually provocative, suggesting a deep relationshipbetween perturbative quantum Yang–Mills theory and gravity. It is also computationally6xpedient, bringing seemingly intractable calculations within reach. This has advanced ourunderstanding of perturbative quantum gravity [30, 63, 41, 34, 64–66, 31, 67, 68], revealinga number of unexpected features and calling into question hitherto accepted argumentsregarding divergences.For instance, the early expectations [69, 70] regarding the onset of divergences werefalse in the case of the four-point graviton amplitude of N “ supergravity, which wasshown to be finite to four loops in [30]. This four-loop cancellation can be accounted forby supersymmetry and E p q U-duality [71–76]. However, at seven loops any cancellationscould not be ‘consequences of supersymmetry in any conventional sense’ [72] and would bedue to ‘enhanced cancellations’, where the terminology reflects the fact that they cannotbe explained by any standard symmetry argument .The seven loop case has not yet been verified, but there is evidence for enhanced cancel-lations from theories with less supersymmetry and, correspondingly, less protection againstdivergences. For example, the four-point amplitude of d “ , N “ supergravity hasbeen shown to be finite to four loops, contrary to expectations from standard symmetryarguments [31]. This casts serious doubt on the conclusion that N “ supergravity willdiverge at seven loops.Currently, the cutting edge is the N “ four-point five-loop amplitude, which wasperformed using generalised colour–kinematics duality and the double copy [78, 68]. Itwas found to be finite, but the degree of finiteness was in agreement with the standardsymmetry arguments, a disappointing outcome for anyone looking for enhanced cancellationthat might make for a seven-loop miracle. However, this conclusion was reached in d “ where the five-loop amplitude first diverges and there is a B R counter-term.Altogether, without a complete understanding of the amplitudes, including hidden fea-tures such as the double copy construction and enhanced cancellations, such questions aboutdivergences of amplitudes remain open in the absence of explicit calculations. Why do colour–kinematics duality and the double copy work.
Given these re-markable results, we are compelled to ask why the colour–kinematics duality and the doublecopy prescription work and whether they remain valid in the full quantum perturbation the-ory. Although colour–kinematics duality and the double copy were arrived at through anon-shell lens, it may prove instructive to step back to an off-shell, or Lagrangian, point ofview. In our previous work [9], we took the middle road, incorporating elements of the on-shell picture to facilitate a fully off-shell BRST Lagrangian double copy. This construction See [77] for possible explanations at three loops that nonetheless fail at four loops. See also [79] for recent work on a plain Lagrangian double copy. internal –kinematics duality.Here ‘internal’ stands for the (possibly trivial) Lie algebra of any internal symmetry,such as colour or flavour symmetries. For example, we shall discuss the flavour –kinematics duality of the non-linear sigma model in Section 3. Even Maxwell theoryhas a trivial U p q –kinematics duality.(ii) The tree-level colour–kinematics duality manifesting action can itself be double-copied [6]. By construction, the tree-level amplitudes of the double copy Lag-rangian will match exactly those obtained by the double copy of the tree-level colour–kinematics-dual amplitudes themselves. In the Lagrangian double copy the (polyno-mials of) colour structure constants are replaced by a second copy of the correspondingdifferential operators, which can be regarded as ‘kinematic structure constants’. Forself-dual Yang–Mills theory it has been shown that there is a corresponding ‘kin-ematic algebra’ of area preserving diffeomorphisms [81–83] with further generalisa-tions given in [84, 85]. Note that complementary to the Lagrangian double copy isthe idea that gravitational actions can be written in a form that factorises order-by-order [86, 6, 87–90].(iii) There is an off-shell field theory ‘product’ of BRST quantised gauge theories, in-cluding the ghost fields, that generates the BRST complex of the double copy the-ory [91–93]. Applying the Lagrangian double copy and truncating out the dilaton andKalb–Ramond sector, it has been explicitly shown to give Einstein–Hilbert gravityto cubic order (where colour–kinematics duality is trivially satisfied) [94]. The ne-cessity of the inclusion of BRST ghosts in the context of ‘closed “ open b open’ instring theory was stressed some time ago by Siegel [95, 96]. For our purposes, the keyobservation is that the linear BRST transformations of the resulting gravity theoryfollow from those of the gauge theory factors [92, 93]. A related perspective on thedouble copy of symmetries has recently been used to derive all-order diffeomorph-isms and extended Bondi–Metzner–Sachs symmetries in the self-dual sector [97] (seee.g. [98–104] for the construction of all hidden symmetries in self-dual Yang–Mills Specifically those gravity theories that derive from the double copy.
Other aspects of the double copy.
Let us also mention some of the other generalisa-tions and applications of the double copy. At the classical level, one can apply a double-copy-type construction for classical solutions to generate non-perturbative Kerr–Schild solutionsin theories of gravity, such as black holes, or bi-adjoint scalar solutions from gauge theorysolutions [111–123]. This classical double-copy can be used to relate other features of gaugeand gravity theories [124–126] and may also be implemented perturbatively [127]. Thereis an elegant formulation of this idea connecting Yang–Mills field strengths to the Weyltensor, which has expanded the space of amenable solutions [128–132]. The field theoryproduct [91] can also be used to elucidate the classical solution double-copy [133] and toconstruct, for example, supersymmetric (single/multi-centre) black hole solutions in N “ supergravity [134, 135], in the weak-field limit.Alternatively, one can bend amplitudes and the double copy to the problem of classicalblack hole scattering, strongly motivated by the advent gravity-wave astronomy [136–149].Another interesting approach is to seek a geometric and/or world-sheet understanding ofthese relations through string theory [7, 8, 25, 150–153] or ambitwistor strings and the scat-tering equations [154–164]. In physics, we describe infinitesimal symmetries by Lie algebras and their action on field con-figurations by Lie algebroids. In the case of gauge symmetry, the latter are more familiar intheir dual realisation, known as the Chevalley–Eilenberg picture. The Chevalley–Eilenbergdifferential encoding the Lie algebra of gauge symmetries as well as its action is called theBRST operator. 9 -algebras and the BV formalism. The classical observables of a field theory aregiven by field configurations that satisfy the equations of motion. To incorporate theequations of motion into the differential graded algebra picture, we can — and sometimes must — extend the BRST operator to the Batalin–Vilkovisky (BV) operator [165–170].This Chevalley–Eilenberg differential, however, no longer describes a mere Lie algebra, buta homotopy generalisation thereof, known as L -algebra or strong homotopy Lie algebra.These algebras first emerged in string field theory [171] taking inspiration from the definitionof A -algebras [172, 173], and were further developed in [174–176]. Note that in the samesense as L -algebras generalise Lie algebras, A -algebras generalise associative algebras.To be somewhat more explicit, an L -algebra is a generalisation of a differential gradedLie algebra in which the Jacobi identity, as well as its nested forms, is satisfied only up tohomotopies. This means that besides the differential µ and the binary products µ , thereare also products of higher arity µ i such that, for example, µ p (cid:96) , µ p (cid:96) , (cid:96) qq ˘ µ p (cid:96) , µ p (cid:96) , (cid:96) qq ˘ µ p (cid:96) , µ p (cid:96) , (cid:96) qq ““ µ p µ p (cid:96) , (cid:96) , (cid:96) qq ˘ µ p µ p (cid:96) q , (cid:96) , (cid:96) q ˘ µ p (cid:96) , µ p (cid:96) q , (cid:96) q ˘ µ p (cid:96) , (cid:96) , µ p (cid:96) qq , (1.6)where the signs depend on the precise Z -grading of the arguments. The right-hand side isthe Jacobiator, measuring the failure of the Jacobi identity to hold and importantly, it isgiven by a homotopy. For µ i “ for i ě , we recover a differential graded Lie algebraand a (graded) Lie algebra if also µ “ . We collected more details on L -algebras inAppendix A.3.Classically, any BV quantisable field theory is fully described by an L -algebra [177,178], see [179–193] for earlier and partial accounts. The Maurer–Cartan theory of L -algebras, which in itself is a vast generalisation of Chern–Simons theory for Lie algebras,encompasses the action, the field equations, and all symmetries of general observables, suchas gauge and Noether symmetries. Most interestingly, Maurer–Cartan theory also describesthe (tree-level) scattering amplitudes of the field theory in question.In this sense, the L -framework provides a very natural and unifying description ofLagrangians and scattering amplitudes of a field theory, resolving the question about whatshould be regarded as fundamental:(i) The mathematically appropriate notion of equivalence between L -algebras is givenby quasi-isomorphisms, and classically equivalent field theories correspond to L -algebras which are quasi-isomorphic; see Section 4.3 and Appendix A.3 for details. For example, in the case of open BRST complexes such as the ones arising in (unadjusted) highergauge theories. L -algebra is quasi-isomorphic to an L -algebra in which the differential µ vanishes identically [194, 195], and such L -algebras are known as minimal models.A minimal model and its higher products describe precisely the tree-level scatteringamplitudes of the corresponding field theory [196,195,177,178], and we explain this inSection 4.3. Notice that minimal models are related to a Feynman diagram expansionin general, following earlier suggestions [197], and this was used in [198] to deriveWick’s theorem and Feynman rules for finite-dimensional integrals.(iii) The notion of an L -algebra can be generalised to that of a quantum L -algebra [171,199,200], which corresponds to a solution to the quantum master equationin the BV formalism, see Section 4.2. Ultimately, such an quantum homotopy algebraencapsulates the quantum aspects of the corresponding field theory. In particular,quantum L -algebras also come with a (quantum) minimal model [200], and theirhigher products describe precisely the full scattering amplitudes of the correspondingfield theory [201, 202], as we shall review this in Section 4.3. For aspects regardingrenormalisation in this context see e.g. [203, 204] and in particular [205, 206].(iv) Since both classical and quantum minimal models can be computed recursively bythe homological perturbation lemma [207, 208], see Section 4.3 for details, we obtainBerends–Giele-type recursion relations for amplitudes in any BV quantisable fieldtheory both at the tree and loop levels [209, 201, 210, 202]. See also [211] for relateddiscussions of the S-matrix in the L -language, [212,213] for the tree-level perturbinerexpansion, [214] for an L -interpretation of tree-level on-shell recursion relations,and [215] for the construction of a homotopy BV algebra description of the BCJrelations and BCJ colour–kinematics duality at the tree level.Because homotopy algebras are the key algebraic structure underlying string field theory,it is perhaps not very surprising that they also play an important role in analysing string andfield theories. They are vital in approaches to non-perturbatively completing string theoryto M-theory, and we refer the interested reader to the recent review [217] (and referencestherein) which gives a condensed overview about the various applications of homotopyalgebras in physics as well as a basic introduction into higher structures. Homotopy algebras and factorisations.
Besides the strong homotopy Lie algebras,or L -algebras, there are other homotopy algebras that are important for our purposes. Inparticular, we will make use of strong homotopy associative algebras, or A -algebras and See e.g. [216] for the definition of homotopy BV algebras. C -algebras. A good example to demonstratetheir use is colour-stripping of Yang–Mills theory.The L -algebra of Yang–Mills theory L YM can be obtained as the anti-symmetrisationof an underlying A -algebra A YM [201]. As explained there, this A -algebra allows for aninteresting factorisation, A YM “ A col b A kin , (1.7)where A col is a gauge matrix algebra, regarded as an A -algebra concentrated in degreezero which encodes the colour structure, and A kin is concentrated in degrees , . . . , andencodes the colour-stripped interactions.As we shall show in Section 6.2, this interpretation of colour-stripping can be improvedby factorising the L -algebra of Yang–Mills theory L YM as L YM “ g b C YM , (1.8)where g is indeed the colour or gauge Lie algebra and C YM is the (unique) C -algebrawhich fully describes the colour interactions. A related application of C -algebras wasgiven in [189].At first glance, the latter factorisation seems suitable for the description of the doublecopy. On closer inspection, however, we observe that the factorisation of the Yang–Millsscattering amplitudes is really a factorisation into three parts: the colour part, the formkinematics part and an underlying scalar field theory with cubic interactions, which actsas a ‘skeleton’ for the Feynman diagram expansion.Since homotopy algebras underlie the string field theory actions and because the doublecopy prescription linking gauge theory and gravity amplitudes is motivated by ‘closed =open b open’ string duality, it is not surprising that homotopy algebras provide a goodframework for understanding this duality. In this paper, we provide an explicit account of the BRST Lagrangian double copy [9] andits articulation in terms of homotopy algebras. Here we summarise the key results andfeatures of the BRST Lagrangian double copy, its implications for scattering amplitudes Note that colour-stripping is not automatically possible, even if all fields take values in the adjointrepresentation: it requires that the colour coefficients in the interaction terms consist exclusively of (con-tractions of) the Lie algebra structure constants f abc . For example, the non-Abelian Dirac–Born–Infeldaction fails this criterion, even though all fields are adjoint, since its interactions also involve the coefficient d abc : “ tr pt e a , e b u e c q . BRST Lagrangian double copy.
Our central result is that the Yang–Mills BRST Lag-rangian double-copies to give the perturbative N “ supergravity BRST Lagrangian toall orders. The logic of the underlying argument, and the key sub-results entering into it,are summarised here (cf. Figure 1.1):(i) The tree-level Yang–Mills scattering amplitudes with external states from the ex-tended BRST Hilbert space including the physical transverse gluons, the unphysical forward/backward polarised gluons, and (anti)ghost states, can be made to satisfycolour–kinematics duality. See Section 8.4.(ii) This extended tree-level BRST colour–kinematics duality can be made manifest inthe Yang–Mills BRST Lagrangian. Unlike the colour–kinematics duality for physicalgluons, this requires the addition of non-vanishing vertices to the Yang–Mills BRSTLagrangian. However, they may be introduced exclusively through the gauge-fixingfermion and so preserve perturbative quantum equivalence. See Section 8.4.(iii) The extended tree-level BRST colour–kinematics duality manifesting Yang–MillsBRST Lagrangian can be ‘strictified’ to possess purely cubic interactions in an ex-tended colour–kinematics duality preserving manner through the introduction of aninfinite tower of auxiliary fields. See Section 8.3.(iv) The strict (i.e. cubic) Yang–Mills Lagrangian which manifests tree-level BRST colour–kinematics duality can be double copied to give a putative perturbative N “ super-gravity BRST Lagrangian. Similarly, the Yang–Mills BRST operator is double copiedto give a putative N “ supergravity BRST operator. See Section 9.4.(v) By construction, the physical tree-level N “ supergravity amplitudes of the doublecopy Lagrangian match those of N “ supergravity.(vi) The double copy BRST charge is valid on-shell due to tree-level BRST colour–kinematics duality and the linear double copy BRST charge implies that the doublecopy amplitudes satisfy the BRST Ward identities. This implies perturbative quantumequivalence to N “ supergravity. See Section 9.5.Some comments are in order here. Firstly, we work perturbatively. This implies that, asin [6, 80], the BRST colour–kinematics duality manifesting action of [9] requires an infinitetower of vertices and hence the strictified action contains an infinite tower of auxiliary fields.The intuition is clear: perturbative gravity has all order interactions and these are generated13 V action S YMBV of Yang–Mills theory ga u g e fi x i n ga nd fi e l d r e d e fi n i t i o n s BV action S N “ of N “ supergravity ga u g e fi x i n ga nd fi e l d r e d e fi n i t i o n s gauge-fixed action ˜ S YMBV o n - s h e ll c o l o u r – k i n e m a t i c s - du a l s tr i c t i fi c a t i o n gauge-fixed action ˜ S N “ o n - s h e ll c o l o u r – k i n e m a t i c s - du a l s tr i c t i fi c a t i o n double copy at the levelof chain complexes,i.e. the kinematic terms agreestrictified action ˜ S YM , stBV strictified action ˜ S N “ , stBV double copy at the levelof L -algebras,i.e. full equivalence Figure 1.1: Diagrammatic description of the double copy.by the double copy of the vertices enforcing BRST colour–kinematics duality. The n ą point interactions of gravity ensure diffeomorphism invariance, and in the BRST frameworkthis follows from the BRST colour–kinematics duality. Note, however, that perturbatively,i.e. at any finite n -point, L -loop order we only require a finite number of auxiliary fieldsand terms in the actions. Scattering amplitudes and Bern–Carrasco–Johansson numerators.
An immedi-ate corollary of this argument is that the Yang–Mills scattering amplitudes double copy intoamplitudes of N “ supergravity to all orders, tree and loop. The former are computeddirectly from the tree-level BRST colour–kinematics duality manifesting Lagrangian, whichcan be used to construct ‘almost BCJ numerators’ that double-copy correctly. Note that toany finite n -point, L -loop order, deriving this Yang–Mills Lagrangian is a purely algebraicexercise, i.e. there is no need to solve for functional colour–kinematics duality relations.To be more precise: 14i) At n points and L loops, one constructs the tree-level BRST colour–kinematics dualitymanifesting Yang–Mills Lagrangian up to the necessary finite order in auxiliary fields.Being exclusively tree-level, this is a purely algebraic operation. Nonetheless, thenumber of required auxiliary fields grows quickly, as one needs the largest trivalenttree that can be glued into a cubic n -point and L -loop diagram, i.e. at one loop and n points one needs n ` point vertices. Already at four points and two loops one needsup to eight points, which requires about auxiliary fields. We should stress, this isthe worst case scenario. It is likely that one can do better by incorporating on-shellmethods, in particular generalised unitarity. Also, the process can be automated usingcomputer algebra programmes.(ii) Equipped with such an action, the ‘almost BCJ numerators’ are given by the sumsthe numerators of all Feynman diagrams with the same topology (i.e. one ignores thedistinction amongst the different fields that can sit on the internal lines), which bydefinition have the same colour numerators and propagators.(iii) The ‘almost’ qualifier indicates that the numerators so constructed will not necessarilysatisfy perfect colour–kinematics duality at the loop level. It might be that there aresome hidden miracles and they do satisfy perfect colour–kinematics duality, but ourarguments do not ensure this and we have not encountered any reason to think thatthis should happen generically.(iv) Nonetheless, these ‘almost BCJ numerators’ will double copy to yield the correspond-ing N “ supergravity numerators. This gives a bona fide N “ supergravity n -point and L -loop amplitude integrand.(v) From a pragmatic point of view, perfect colour–kinematics duality at the loop levelis (very probably, i.e. barring the miracles mentioned above) unnecessarily strong.It would be of practical importance to turn this statement into a precise set of loopintegrand ‘almost colour–kinematics duality’ conditions independent of the underlyingBRST Lagrangian argument. We intend to address this in future work. The mostpowerful ‘almost colour–kinematics duality’ conjecture motivated by our construction(i.e. the conjecture with the weakest condition for loop-level double copy) is thatloop integrands with enough internal lines cut to be tree must satisfy perfect colour–kinematics duality. Double copy from homotopy algebras.
Our central result is that the L -algebraof the strict Yang–Mills Lagrangian which manifests tree-level BRST colour–kinematicsduality factorises into a colour factor, a kinematic vector space, and a scalar theory factor.15chematically, L YM “ g b Kin b τ Scal , (1.9)where g is the colour part, Kin the kinematic part,
Scal the scalar part. The tensor productbetween the kinematic and scalar factors is twisted with twist datum τ ; see Section 6.3 fordetails. One can think of this twisting as a form of semi-direct tensor product and itgenerates a kinematic algebra acting on the scalar factor.Given the factorisation, to double copy is to replace the colour factor with another copyof the twisted kinematic factor. This yields the L -algebra L N “ of perturbatively BRSTquantised N “ supergravity, up to a quasi-isomorphism compatible with quantisation.The action, scattering amplitudes and all other features are encoded in the L -algebra. Perturbatively, it is the complete quantum gravity theory up to the point of renormalisation.Alternatively, one can replace kinematics with colour to give the cubic biadjoint scalar fieldtheory. The scalar factor is common to all three theories.Schematically,Biadjoint scalar field theory ÐÝ Yang–Mills theory ÝÑ N “ supergravity g b g b Scal g b Kin b τ Scal Kin b τ Kin b τ Scal (1.10)Let us expand on the key elements entering this picture. Our starting point is the obser-vation that BV quantised Yang–Mills theory corresponds to an L -algebra, denoted L YMBV ,which upon gauge-fixing yields a BRST L -algebra, denoted L YMBRST . The BV operator Q YMBV is uniquely determined by the higher products of L YMBV , i.e. it is the dual Chevalley–Eilenberg differential, cf. Section 4.2. The BRST operator Q YMBRST then follows from itsgauge-fixing.As we saw above, a crucial step in the double copy is the reformulation of scatteringamplitudes in terms of cubic interaction vertices. This is particularly natural from the pointof view of homotopy algebras, where this is a well-known process known as strictification or rectification . The statement of the strictification theorem for homotopy algebras is that anyhomotopy algebra is quasi-isomorphic to a strict homotopy algebra with higher productsthat have either one or two inputs and one output, see Appendix A.4. Field theories withcubic interaction vertices then simply correspond to strict homotopy algebras. Moreover,quasi-isomorphisms are the proper homotopy algebraic articulation of physical equivalence,since they are isomorphisms on the cohomology and thus preserve the space of physicalstates while allowing for field redefinitions and for the integrating in and out of auxiliary Strictly speaking, we also have to provide a path integral measure for the loop amplitudes; we alwayswork with the evident one for our field content. L -algebra ˜ L YM , stBRST is quasi-isomorphic to L YMBRST . The non-trivial observation making the double copy manifest is thefactorisation of ˜ L YM , stBRST : ˜ L YM , stBRST “ g b p Kin st b τ Scal q . (1.11)Let us provide some further details on the identification (1.11):(i) g is the familiar colour Lie algebra, i.e. an L -algebra with only µ , the Lie bracket,being non-trivial.(ii) Kin st is the kinematic algebra. It is a graded vector space of the Poincaré represent-ations carried by all the fields of the theory, including the strictification auxiliaries.Restricting to the familiar BRST fields A , c , ¯ c , and b , it is given by Kin st : “ g R r s loomoon “ : Kin ´ ‘ ` v µ M d ‘ n R ˘loooomoooon “ : Kin ‘ a R r´ s loomoon “ : Kin ‘ ¨ ¨ ¨ , (1.12)where we have labelled the basis vectors of the ghost c , the gauge potential A , theNakanishi–Lautrup b , and anti-ghost ¯ c Poincaré modules suggestively by g , v µ , n , and a . The ellipses denote the Poincaré modules of all the auxiliary fields required for thestrictification of the BRST colour–kinematics duality manifesting Lagrangian.(iii) Scal is the L -algebra of a cubic scalar field theory. Since it is cubic, this L -algebrahas only two higher products, the unary and the binary ones µ and µ . Explicitly, µ is simply the wave operator, the unique Lorentz invariant possibility, and µ encodesa cubic scalar interaction, the skeleton of the strictified Yang–Mills interactions.(iv) The map τ is the twist datum of the tensor product, Kin st b τ Scal . Physically, itencodes the kinematic (differential operator) factors of the Yang–Mills interactions.It is fully determined by the BRST colour–kinematics-dual form of the colour-strippedaction of Yang–Mills theory, and it induces the correct tensor product for factorisingthe L -algebra of N “ supergravity.For this factorisation of the L -algebra to be sensible, the expected tensor products ofhomotopy algebras need to exist first. One of our collateral results is that, as mentionedabove, colour-stripping in Yang–Mills theory can be regarded as a factorisation of the L -algebra of Yang–Mills theory into a colour or gauge Lie algebra and a kinematical C -algebra. Here, we have ˜ L YM , stBRST “ g b C YM , stBRST , (1.13)17here C YM , stBRST describes the colour-stripped part of manifestly colour–kinematic-dual, stric-tified Yang–Mills theory.The double copy strongly suggests the further factorisation C YM , stBRST “ Kin st b τ Scal , (1.14)and our notion of twisted tensor product of homotopy algebras is essentially constructedsuch that this factorisation is possible, see Section 6.1. For example, the action of thedifferential m τ : Kin st b τ Scal Ñ Kin st b τ Scal on v µ b ϕ p x q P Kin st b τ Scal looksschematically like m τ p v µ b ϕ p x qq „ v µ b l ϕ p x q ` n b B µ ϕ p x q . (1.15)Given the full factorisation of the Yang–Mills L -algebra, the double copy prescriptionbecomes manifest. Replace the colour g with another copy of the kinematics and twist p Kin st , τ q : ˜ L YM , stBRST “ g b p Kin st b τ Scal q double copy ÝÝÝÝÝÝÝÑ
Kin st b τ p Kin st b τ Scal q “ ˜ L N “ , stBRST , (1.16)where ˜ L N “ , stBRST fully determines the double copy theory. The space of fields in this theoryis determined by the tensor product of graded vector spaces, which is an extension of thetensor product (1.2) that includes ghosts, anti-ghosts, Nakanishi–Lautrup fields and thefurther auxiliary fields arising in the strictification. Since it is constructed from the higherproducts, the (gauge-fixed) BV differential Q YMBV also factorises and double copies into the(gauge-fixed) BV differential Q N “ .We note that one can also replace kinematics with colour to produce the cubic biadjointscalar theory, sometimes referred to as the zeroth copy, ˜ L YM , stBRST “ g b p Kin st b τ Scal q zeroth copy ÝÝÝÝÝÝÝÑ g b ¯ g b Scal “ ˜ L biadjBRST . (1.17)In this case, the homotopy algebraic discussion becomes straightforward.To summarise, the homotopy algebraic structure underlying the double copy is thefactorisation of the L -algebra L of a field theory as L : “ V b ¯ V b Scal , (1.18)where V and ¯ V are two (graded) vector spaces, with the most prominent examples being V ¯ V Biadjoint scalar field theory g ¯ g Yang–Mills theory g Kin N “ supergravity Kin Kin
18f the factorisation is suitable, which means that it is compatible with colour–kinematicsduality, then the double copy is a mapping between L -algebras of classical field theoriesobtained from substitutions of the factors V and ¯ V . The advantage of this homotopydouble copy is that it is fully off-shell and goes beyond on-shell amplitudes. Furthermore,it suggests a lift to homotopy algebraic structures in string field theory.Let us also list a few secondary results which we obtained collecting the necessary toolsfor our homotopy algebraic discussion of the double copy.(i) We demonstrate in Section 6.2 that the familiar colour-stripping of Yang–Mills scatter-ing amplitudes corresponds to a factorisation of the L -algebra of Yang–Mills theoryinto a colour or gauge Lie algebra factor and a kinematical C -algebra. This factor-isation extends to the level of actions.(ii) A mathematical argument that we were not able to find in the literature is that therather evident tensor product between certain strict homotopy algebras guarantees theexistence of a tensor product between the corresponding general homotopy algebrasby homotopy transfer via the homological perturbation lemma. The full argument isgiven in Section 6.1.(iii) Finally, we show in Appendix A.3 that in the homotopy algebraic picture, finite gaugetransformations can be regarded as curved morphisms of L -algebra. In this work, we focus on the case of Yang–Mills theory and N “ supergravity. However,when making the replacement g b p Kin b τ Scal q ÝÑ ¯ Kin b ¯ τ p Kin b τ Scal q (1.19)there is no reason to restrict to p ¯ Kin , ¯ τ q – p Kin , τ q . We could have taken p ¯ Kin , ¯ τ q from anyBV Lagrangian field theory admitting a factorisation ¯ Kin b ¯ τ Scal and enjoying a generalisednotion of tree-level colour–kinematics duality. For example, taking the p ¯ Kin , ¯ τ q of N “ super Yang–Mills theory, ¯ Kin b ¯ τ p Kin b τ Scal q would be the L -algebra of pure N “ supergravity. In this example, both theories have vanilla colour–kinematics duality, but thisalso need not be the case. For instance, the flavour–kinematics duality of the non-linearsigma model is an example of (rather trivially) generalised colour–kinematics duality. Onecan even consider theories that have no ‘colour’ factor at all, such as Maxwell theory. This is not a morphism of L -algebras, which would imply a map between elements of the L -algebras.A simple analogy is the mapping of vector spaces V ÞÑ V b W for some fixed vector space W . L Max “ Kin b τ Scal , can be enhanced by introducing graded antisymmetric higher products in
Scal that satisfy kinematic Jacobi identities. Since
Kin b τ Scal is a C -algebra, these higherproducts do not contribute to the Maxwell action, but if we then tensor with a colour Liealgebra g we recover BRST colour–kinematics-dual Yang–Mills theory.With these ingredients, there are numerous immediate generalisations. We only require(i) tree-level colour–kinematics duality — there is growing zoology of such theories;(ii) an underlying L -algebra — this includes all BV quantisable Lagrangian field theoriesand so is very general;(iii) that the L -algebra factorises in an appropriate manner — this is essentially therequirement that the gauge and space-time symmetries do not mix.The last condition restricts the apparent vast generality a bit. There are examples, suchas the non-Abelian Dirac–Born–Infeld theory, where colour structure constants arise whosecompatibility with a factorisation is not apparent. In such cases further work is required,before proceeding directly on the homotopy double copy. Up to this issue, our machinery ispowerful enough to derive all order validity of the double copy from the validity at the treelevel. Let us summarise here some of the possibilities, indicating the outstanding questionsthat must be addressed to realise this claim. Supergravity.
The first obvious generalisation is the inclusion of supersymmetry. For ir-reducible super Yang–Mills multiplets, colour–kinematics duality for gluons ensures colour–kinematics duality for the entire multiplet. This can be shown using a supersymmetric Wardidentity argument, entirely analogous to the BRST Ward identity argument for BRSTcolour–kinematics duality given in Section 8.1. The factorisation requirement is obviouslysatisfied, so in principle, there is no obstruction. For gauge theory factors of N “ supersymmetric Yang–Mills theory and N “ Yang–Mills theory, which yields N “ su-pergravity minimally coupled to a single chiral multiplet, it is particularly straightforward,since there is a convenient superfield formalism as described in [91].It is more subtle and interesting when both factors are supersymmetric and there isa Ramond–Ramond sector. Since the gauginos have no linear gauge (BRST) transforma-tion, their product must be identified with field strengths that are to be regarded as thefundamental fields. The intuition from string theory is clear — the Ramond–Ramond sec-tor couples to the string world-sheet only through the field strengths and never the barepotentials. Indeed, the type II supergravity Lagrangians can both be written without anybare Ramond–Ramond potentials and the Lagrangian double copy in the Ramond–Ramond20ector does indeed generate a Lagrangian that is formulated purely in terms of fundamentalfield strengths. Of course, it is perturbatively equivalent to the familiar formulation in termof field strengths of potentials. Interestingly, this is automatically achieved via Sen’s mech-anism for writing Lagrangians for self-dual field strengths [218] , but without necessarilyimposing self-duality. We shall spell out the details in forthcoming work.With these basic ingredients accounted for, the door is then open to the plethora ofdouble-copy constructible theories, such as (almost, cf. [59]) all N ě ungauged super-gravity theories, (super) Einstein–Yang–Mills–scalar theories [49] and gauged supergravity(with Poincaré background) [60]. Each comes with interesting features that must still beaddressed in the BRST Lagrangian double copy formalism, e.g. spontaneous symmetrybreaking, but none that present an obvious obstruction. Abelian Dirac–Born–Infeld theory.
The Abelian Dirac–Born–Infeld (DBI) scatteringamplitudes have been double copy constructed [158,220,164]. They follow from the productof the non-linear sigma model and (super) Yang–Mills amplitudes. Given that we alreadyhave both the BRST manifestly colour–kinematics-dual formulations of the non-linear sigmamodel, Yang–Mills theory and their L -algebras, we can immediately apply the homotopydouble copy to obtain the perturbative DBI BRST Lagrangian by replacing the colourfactor of Yang–Mills with the kinematics of the non-linear sigma model. Massive gravity.
There has been recent work [221] suggesting that massive gravity canbe double-copy constructed. Here, the principal impediment is that tree-level colour–kinematics duality is not known to hold beyond four points. If we assume that its goesthrough, then it should prove doable using a BRST-quantised Stückelberg formulation ofmassive Yang–Mills theory.
Conformal gravity.
More ambitiously, one can approach conformal gravity by includ-ing higher-dimensional operators in the Yang–Mills theories [57, 222]. Again, if colour–kinematics duality holds to all points, then it may be possible to turn the homotopy doublecopy handle, although the higher dimensional operators will have to be treated carefully.
Closed string theory.
Given the KLT origins of the double copy, it is natural to seekan ‘open b open “ closed’ stringy extension. Moreover, both open and closed string fieldtheory are built on homotopy algebras [171, 195], and so the homotopy double copy is a See also [219] where it is shown that this mechanism arises very directly from a, again, homotopyalgebraic perspective. a priori automatic. All choices will be quasi-isomorphic, but thereseems to be no reason to suppose that every partition of the moduli space gives rise to astrictification compatible with colour–kinematics duality. In this case, one would have asituation similar to Yang–Mills theory, where only certain strictifications are compatiblewith colour–kinematics duality. Alternatively, it could be that all quasi-isomorphic choicescan be double copied and only on taking the field theory limit is this structure broken.Ambitiously, one could consider the more general (or other) formulations of the open/closedstring duality.
As stated before, it is our intention to be highly self-contained in our presentation, inorder to make the homotopy algebraic perspective on the double copy accessible for readersunfamiliar with either homotopy algebras or the double copy (or both!). This reading guidemay provide some further help.For readers unfamiliar with the double copy, good stating points are Section 2, with aconcise review of the basics, and Section 3, which spells out the details in the case of therelated but much simpler double copy of the amplitudes of the non-linear sigma model tothose for the special galileon. The explicit details for the factorisation of homotopy algebrasinvolved in the gauge–gravity double copy are then spelled out in Section 7 at the level offree field theories and Section 9 at the level of the full actions. The latter section alsocontains the proof that translates the double copy of amplitudes from tree to full quantum(i.e. loop) level.Readers unfamiliar with the BV formalism and the BV formulation of standard fieldtheories will benefit from the general discussion in Section 4 as well as the concrete examplespresented in Section 5. Some quantum field theoretic preliminaries that are crucial toextending the double copy of amplitudes to loop level are reviewed or developed in Section 8.For readers unfamiliar with homotopy algebras, we have collected the basic definitionsand results in Appendix A. The link to field theories is gently introduced in Section 3.3,using the example of the non-linear sigma model. The general picture and the link to theBV formalism are then developed in Section 4; the correspondence between actions and L -algebras is explained in Section 4.2, while the link between scattering amplitudes and L -algebras is presented in Section 4.3. Concrete examples of L -algebras for a number of22eld theories relevant in the gauge–gravity double copy are then given in Section 5.
2. Double copy basics
We start with a brief review of gluon scattering amplitudes to set some notation and to beself-contained.
Gluon scattering amplitudes.
Consider Yang–Mills theory with a semi-simple compactmatrix Lie algebra g as gauge algebra. Because the Lie bracket in g describes naturally acubic interaction vertex r´ , ´s : g ˆ g Ñ g , the possibility of relating colour to kinematicsrelies on writing the amplitude in terms of trivalent diagrams only, A n,L “ p´ i q n ´ ` L g n ´ ` L ÿ i ż L ź l “ d d p l p π q d S i c i n i d i . (2.1)Here, A n,L is the n -point L -loop gluon scattering amplitude, and g the Yang–Mills couplingconstant. The sum is over all n -point L -loop diagrams, labelled i , with only trivalent vertices( not the Feynman diagrams of the original theory). The colour numerator or colour factor c i associated to a diagram i is composed of gauge algebra structure constants and can beread off directly from the trivalent diagram. The kinematic numerator or kinematic factor n i associated to diagram i is a polynomial of Lorentz-invariant contractions of polarisationvectors and momenta. The denominator d i associated to a diagram i is the product of theFeynman–’t Hooft propagators, i.e. the product of the squared momenta of all internal linesof the diagram i . Finally, S i P N is the symmetry factor associated to a diagram i , definedin the same way as for Feynman diagrams, accounting for any over-counting due to thediagram symmetries. At the tree level, i.e. for L “ , (2.1) simplifies to A n, “ p´ i q n ´ g n ´ p n ´ q !! ÿ i “ c i n i d i , (2.2)since there are p n ´ q !! trivalent tree diagrams at n points.This trivalent form exists because the four-point contact terms can always be ‘blown-up’and absorbed into corresponding three-point diagrams: ÝÑ s ` t ` u (2.3)23 our-point tree-level gluon scattering amplitude from trivalent diagrams. Con-sider the simplest example of the four-point tree-level scattering amplitude, A , “ s ` t ` u ` . (2.4)Explicitly, with all momenta incoming, s “ ´ i g f abe f ecd n s s “ : ´ i g c s n s s , t “ ´ i g f aed f ebc n t t “ : ´ i g c t n t t , u “ ´ i g f aec f edb n u u “ : ´ i g c u n u u , “ ´ i g ´ c s n p q s ´ c t n p q t ´ c u n p q u ¯ . (2.5a)Here, we have made use of the standard Mandelstam variables s : “ p p ` p q , t : “ p p ` p q ,and u : “ p p ` p q in the trivalent s -, t -, and u -channel diagrams, respectively. Furthermore,upon setting p ij : “ p i ´ p j where i, j, . . . “ , . . . , label the different gluons, the kinematic24umerators are given by n s : “ “ p ε ¨ p q ε ´ p ε ¨ p q ε ` p ε ¨ ε q p ‰ ¨¨ “ p ε ¨ p q ε ´ p ε ¨ p q ε ` p ε ¨ ε q p ‰ , n t : “ ´ “ p ε ¨ p q ε ´ p ε ¨ p q ε ` p ε ¨ ε q p ‰ ¨¨ “ p ε ¨ p q ε ´ p ε ¨ p q ε ` p ε ¨ ε q p ‰ , n u : “ “ p ε ¨ p q ε ´ p ε ¨ p q ε ` p ε ¨ ε q p ‰ ¨¨ “ p ε ¨ p q ε ´ p ε ¨ p q ε ` p ε ¨ ε q p ‰ , (2.5b)where ε i is the polarisation vector of the i -th gluon and x ¨ y : “ η µν x µ x ν . We also havesuggestively labelled the kinematic numerators appearing in the four-point contact termin (2.5a) by n p q s , n p q t , and n p q u . They are given by n p q s : “ p ε ¨ ε qp ε ¨ ε q ´ p ε ¨ ε qp ε ¨ ε q , n p q t : “ ´p ε ¨ ε qp ε ¨ ε q ` p ε ¨ ε qp ε ¨ ε q , n p q u : “ ´p ε ¨ ε qp ε ¨ ε q ` p ε ¨ ε qp ε ¨ ε q . (2.5c)Upon summing up (2.5a), the four-point tree-level scattering amplitude is a sum overthe three trivalent diagrams, A , “ ´ i g ˆ c s n s s ` c t n t t ` c u n u u ˙ (2.6a)with n s : “ n s ` s n p q s , n t : “ n t ´ t n p q t , n u : “ n u ´ u n p q u . (2.6b)Note that any n -point L -loop diagram with a four-point contact term is accompanied bythree diagrams that are identical except that the four-point contact term is replaced by the s -, t -, and u -channel trivalent diagrams and the above argument can be applied. Of course,this can be realised at the Lagrangian level by introducing an auxiliary field strictifying theaction to be cubic [6]. Remark 2.1.
We may introduce the colour-stripped vertex ˜ F µνρ in momentum space, in-tentionally written as ‘structure constants’ analogous to f abc , ˜ F µ µ µ p p , p , p q : “ p p p : “ p µ η µ µ ` p µ η µ µ ` p µ η µ µ , (2.7)25 o that n s “ ε µ ε µ ˜ F µ µ ρ ˜ F ρµ µ ε µ ε µ , n t “ ε µ ε µ ˜ F µ ρµ ˜ F ρµ µ ε µ ε µ , n u “ ε µ ε µ ˜ F µ ρµ ˜ F ρµ µ ε µ ε µ . (2.8) This observation will become important in Section 2.3.
Generalised gauge transformations.
We note that the three colour numerators satisfythe Jacobi identity, c s ´ c t ´ c u “ f ea r b f ecd s “ , (2.9)so that a shift of the kinematic numerators by an arbitrary function α , n s ÞÑ n s ´ sα , n t ÞÑ n t ` tα , n u ÞÑ n u ` uα , (2.10)leaves the amplitude (2.6a) invariant. These shifts, corresponding to an additional freedomin the choice of the kinematic numerators, were referred to as generalised gauge trans-formations in [4]. Of course, this applies to any triple of trivalent diagrams p i, j, k q thatonly differ in a common four-point subdiagram with colour numerators satisfying a Jacobiidentity of the form c i ` c j ` c k “ , where the generalised gauge transformation acting onthe corresponding kinematic numerators is given by n i ÞÑ n i ` s i α , n j ÞÑ n j ` s j α , n k ÞÑ n k ` s k α , (2.11)and s i , s j , and s k are the Mandelstam variables of the common four-point subdiagram inwhich the three diagrams differ. The Bern–Carrasco–Johansson (BCJ) colour–kinematics duality conjecture now states the following.
Conjecture 2.2. (Bern–Carrasco–Johansson, [4, 5]) There exists a choice of kinematicnumerators of the trivalent diagrams entering the scattering amplitude A n,L such that(i) whenever a triple of trivalent diagrams p i, j, k q has colour numerators obeying theJacobi identity c i ` c j ` c k “ , (2.12a) then the corresponding kinematic numerators obey precisely the same identity n i ` n j ` n k “ (2.12b) Here, square brackets indicate anti-symmetrisation of the enclosed indices. ii) if in any individual diagram, c i ÞÑ ´ c i under the interchange of two legs, then n i ÞÑ ´ n i at the same time. Three- and four-point tree-level gluon scattering amplitudes.
Evidently, thetree-level three-point scattering amplitude (allowing complex momenta), which consistsof a single diagram, trivially satisfies the duality since under interchange of any twoedges c “ f abc ÞÑ ´ c , since f abc is totally anti-symmetric, and the same is true for n “ ε µ ε µ ε µ F µ µ µ , since F µ µ µ is totally anti-symmetric.The tree-level four-point scattering amplitude was known to satisfy colour–kinematics-duality before the notion of this duality had been articulated [224, 225]. Indeed, usingmomentum conservation ř i p i “ and transversality ε i ¨ p i “ , (2.5b), (2.5c), and (2.6b)immediately imply that the kinematic numerators satisfy n s ´ n t ´ n u “ . (2.13)This agrees with the colour Jacobi identity c s ´ c t ´ c u “ . Note that this would havefailed without the additional contributions from the four-point contact term (2.5c). Athigher points, one would also need the on-shell conditions p i “ .The fact that the kinematic identity holds without any intervention besides blowing upthe four-point contact term is due to the special kinematics of the four-point amplitude.At higher points, not all possible choices of n i will satisfy the required kinematic identities.Already at five points it is non-trivial [4], although there is a particularly nice representationof the colour–kinematics-dual amplitude in this case [226]. General tree-level gluon scattering amplitudes.
Thinking of the p n ´ q !! colour c i and kinematic numerators n i as column vectors, denoted by c and n , we can triviallyrewrite the n -point tree amplitude as A n, “ c T Dn with D ij : “ δ ij d j . (2.14)The number of linearly independent (under the Jacobi identities) colour numerators c i is p n ´ q ! , which, using the multi-peripheral colour decomposition of [227], is seen to be thesame as the number of linearly independent partial colour-stripped scattering amplitudes A i n : “ A n, p σ i p ¨ ¨ ¨ n qq under the Kleiss–Kuijf relations [228], where t σ i u n ´ i “ “ S n ´ and A n, p ¨ ¨ ¨ n q is the colour-ordered n -gluon tree amplitude.Thus, we can choose a subset consisting of p n ´ q ! linearly independent colour numer-ators, which we shall refer to as the primaries , and put them into a p n ´ q ! -component27olumn vector c m . The rest are generated by the ř t p n ´ q u k “ k C n ´ k C kk p n ´ q ! linearlyindependent Jacobi identities, c “ J c m , (2.15)where J is a p n ´ q !! ˆ p n ´ q ! matrix encoding these relations. For example, at fourpoints, in the conventions of (2.5a), we can choose c t , c u as our primary colour numerators,and then J “ ¨˚˚˝ ˛‹‹‚ . (2.16)In this form, colour–kinematics duality requires the existence of kinematic numerators sat-isfying n “ J n m . (2.17)We also have the p n ´ q ! ostensibly linearly independent (prior to applying the BCJ rela-tions) n -point partial amplitudes A i : “ A i n which may be written as A “ P n , (2.18)where P is an p n ´ q ! ˆ p n ´ q !! matrix of propagators with signs determined by thepermutations defining the components of A relative to the colour order of the correspondinggraphs.If (2.16) can be realised then A “ P J n m . (2.19)Note that, while this relation looks as if it immediately identifies the colour–kinematics du-ality respecting n in terms of the partial scattering amplitudes, P J is necessarily singular.However, the required relations are purely algebraic, and we can solve for p n ´ q ! elements of n m in terms of p n ´ q ! partial amplitudes and the remaining p n ´ q ! ´p n ´ q ! “p n ´ q ! p n ´ q elements of n m . On substituting this solution back into (2.18), we encountera surprise: the dependence of the remaining p n ´ q ! p n ´ q partial amplitudes on n m dropsout entirely and we are left with a new set of relations amongst the p n ´ q ! partial scatteringamplitudes. These are known as the BCJ relations and were introduced in [4], where theywere shown to hold explicitly up to eight points. Assuming the colour–kinematics-duality,the p n ´ q ! partial amplitudes are in fact an overcomplete basis, which is reduced to p n ´ q ! linearly independent partial amplitudes by the implied BCJ relations.Conversely, given the BCJ relations it is possible to explicitly construct a representationof the total amplitude such that colour–kinematics duality holds [25,229]. The n -point BCJ28elations were shown to hold in [7,8] by considering the α Ñ limit of string theory mono-dromy relations, confirming the colour–kinematics duality conjecture at the tree level. TheBCJ relations may also be deduced from pure spinor cohomology [230]. There are a numberof powerful stringy perspectives on the BCJ relations, see for example [231, 232, 150, 27], in-cluding α deformations respecting colour–kinematics duality [233]. A purely field theoreticderivation was given in [234] using only Britto–Cachazo–Feng–Witten recursion [235]. Theywere also established [236] in N “ super Yang–Mills, which contains the Yang–Mills case,using the Roiban–Spradlin–Volovich–Witten connected formalism. Recently, it has beenshown [29], via a residue theorem, that the tree-amplitudes written in terms of intersectionnumbers [237, 238] automatically satisfy the colour–kinematics duality.Our discussion so far has been restricted to tree level. The statement of the duality forloops is not affected, up to some minor subtleties. In particular, the kinematic numeratorsare functions of the loop momenta. At the loop level, the kinematic Jacobi-type identit-ies are functional identities. Hence, one cannot straightforwardly solve for the kinematicnumerators via a pseudo-inverse as in the tree-level case.The four-point one-loop example in N “ supersymmetric Yang–Mills theory is par-ticularly simple, due to the simple structure of one-loop amplitudes [239, 240]. See forexample [20,21,23]. For Yang–Mills theory at one and two loops, see [241]. For detailed ex-amples at three loops, see for example [5,242]. These simple cases make it clear that colour–kinematics duality can work at the loop level. However, the proof of colour–kinematicsduality at the tree level given in [229, 25] relied on the Kawai–Lewellen–Tye relations andtherefore does not extend to loop level. At the time of writing there is no proof that colour–kinematics duality will hold to all loops, despite an impressive number of highly non-trivialconcrete examples [5, 39, 32, 243, 244, 63, 41, 33–36, 65, 37, 241, 64, 66, 31, 245, 56]. Expanding the Einstein–Hilbert action perturbat-ively around the Minkowski background g µν “ η µν ` κh µν , we can construct graviton scat-tering amplitudes as pioneered by DeWitt [246–248]. The Feynman diagrams for gravitonsinclude n -point vertices for all n , and schematically we have S EH “ ´ κ ż d d x ?´ g R „ ż d d x ÿ n “ κ n BB h n ` , (2.20)where κ “ πG p d q N is Einstein’s gravitational constant. However, just as for the four-point vertex in Yang–Mills theory, these can all be absorbed into the kinematic numerators See Section 5.4 for more details on the perturbative analysis of the Einstein–Hilbert action.
29f the purely trivalent diagrams. For example, consider a purely trivalent diagram i , con-tributing N i d i , where N i is the kinematic numerator, to the amplitude integrand, and anotherdiagram i p q , contributing N i p q d i p q , which is identical except that one trivalent four-point sub-diagram with propagator s has been contracted to a four-point vertex. Then, sd i p q “ d i and so, N i d i ` N i p q d i p q “ N i ` sN i p q d i “ N i d i . (2.21)This argument is not affected by the inclusion of the Kalb–Ramond and dilaton sectors.Consequently, the N “ supergravity scattering amplitudes structurally resemble closelythe gluon scattering amplitudes (2.1), H n,L “ p´ i q n ´ ` L ´ κ ¯ n ´ ` L ÿ i ż L ź l “ d d p l p π q d S i N i d i . (2.22) Double copy.
What is less immediately apparent is that the BCJ double copy prescrip-tion implies that, given colour–kinematics duality, the N “ kinematic numerators, forfactorisable external states, can always be written as a product: N i “ n i ˜ n i . More precisely,let us state the BCJ double copy prescription for Yang–Mills theory [5, 6]: given any two n -point L -loop gluon scattering amplitudes, A n,L “ p´ i q n ´ ` L g n ´ ` L ÿ i ż L ź l “ d d p l p π q d S i c i n i d i , ˜ A n,L “ p´ i q n ´ ` L g n ´ ` L ÿ i ż L ź l “ d d p l p π q d S i c i ˜ n i d i , (2.23)at least one of which respects colour–kinematics duality , let us assume it is A n,L , we may‘double copy’ by replacing the colour numerators by kinematic numerators which respectcolour–kinematics duality, while sending g ÞÑ κ , to generate a new scattering amplitude, H n,L “ p´ i q n ´ ` L ´ κ ¯ n ´ ` L ÿ i ż L ź l “ d d p l p π q d S i n i ˜ n i d i , (2.24)which is guaranteed to be a bona-fide scattering amplitude of N “ supergravity. Thisremarkable fact was conjectured in [4, 5] and shown to be true in [6]. Note that this is anall-loop-order statement: if colour–kinematics duality holds to all loop orders, then N “ supergravity is the double copy of Yang–Mills theory to all orders. The open question, inthis context, is whether or not colour–kinematics duality holds to arbitrary loop order. The c i should not be explicitly evaluated under the integral (i.e. internal indices should not be summedwhen the corresponding momentum is undetermined) in case they accidentally vanish before being replacedby the loop-momenta dependent kinematic numerators.
30e have discussed only pure Yang–Mills theory and N “ supergravity. However, asmentioned in Section 1.4, there is an ever-growing zoo of colour–kinematics duality respect-ing and double copy constructible theories. There are also various counterexamples, themost obvious being Yang–Mills theory coupled to adjoint fermions in a non-supersymmetricway [47]. Such a coupling is incompatible with the assumption of colour–kinematics duality,which implies that the fermion couplings obey a Fierz identity that ensures supersymmetry.The latter is also evident from the double copy perspective: the product of gluon and fer-mion states yields gravitino states, which must couple supersymmetrically. The decomposition of scattering amplitudes into diagrams with trivalent vertices raises thequestion if there is an action principle for which these diagrams are the genuine Feynmandiagrams. The homotopy algebraic perspective which we adopt in the rest of the paperyields the general statement that for any BV quantisable field theory there is a physicallyequivalent action with only trivalent vertices, cf. Section 8.3. We call the latter theory a strictification of the former, because the homotopy algebras underlying a field theory withtrivalent vertices is called strict . An example of a strictification of four-dimensional Yang–Mills theory is the first-order formulation with an additional self-dual two-form [249]. Inthe case of the double copy, however, manifest colour–kinematics duality requires a differentstrictification, which we review below. Reorganisation of general tree-level scattering amplitudes.
For each tree Γ con-taining a higher-than-trivalent vertex with n external vertices, consider its contribution A Γ to a total scattering amplitude A . We split A Γ into a sum over trivalent trees with thesame number of external vertices, A Γ “ ÿ Γ P Tree ,n A p Γ q Γ , (2.25)where Tree ,n is the set of trivalent trees with n external vertices. The value of eachsummand is to be determined later by the colour–kinematics condition. Doing this forevery tree involving a vertex which is higher than trivalent (the case where Γ is alreadytrivalent is trivial), we have reorganised the tree-level amplitudes of our theory into a sumover trivalent diagrams: ÿ Γ P Tree n A Γ “ ÿ Γ P Tree ,n A Γ , (2.26) Mathematicians would also use the term rectification . A Γ “ ÿ Γ P Tree n A p Γ q Γ . (2.27)We require that our reorganisation be local : the way each n -ary vertex is resolved intoa trivalent tree subdiagram is independent of the rest of the diagram. It then suffices tospecify, for each n -ary vertex with n ě , how to split it into a sum over trivalent trees.That is, for the tree diagram T n with n external legs and one internal n -ary vertex, i.e. the n -point contact contribution with a single n -ary vertex, we specify its decomposition A T n “ ÿ Γ P Tree ,n A p T n q Γ “ ÿ Γ P Tree ,n N p Γ q D p Γ q , (2.28)where we have chosen a particular ansatz in order to satisfy the colour–kinematics identities:(i) D p Γ q is a differential operator of degree p n ´ q which is the product of the inversepropagators l corresponding to the n ´ internal edges in Γ , i.e. the kinematicdenominator. In particular, D “ l ´ for n “ and D “ for n “ .(ii) N p Γ q is a differential operator, corresponding in momentum space to a polynomialof the external momenta p , . . . , p n and the polarisation vectors ε , . . . , ε n .Because only vertices of degree at most n contribute to the n -point amplitude, we cansolve for these decompositions recursively. Concretely, supposing that one knows the de-composition for all vertices of degrees at most n , then one simply writes an equation in asmany unknowns as there are for all possible ways to decompose the p n ` q -ary vertex intotrees, and solves for the colour–kinematics duality equations (a system of linear equations)at n ` points. The initial case of the iteration is n “ , where the decomposition oftrivalent vertices is trivial, and one must verify that the colour–kinematics identities holdfor tree-level three-point functions.A priori, at each level of the iteration, i.e. for any given n , there may be infinitely manysolutions or no solutions; tree-level colour–kinematics duality of the amplitudes of a fieldtheory then amounts to the assertion that the latter is never the case. Colour–kinematics-dual Yang–Mills action.
We now specialise to the case of Yang–Mills theory with the physical field being the gauge field. For simplicity, we shall work inFeynman gauge. The Yang–Mills action only contains terms up to quartic order in thefield, but this does not mean that we cannot split the vanishing quintic and higher-order in fact, pseudo-differential for n “ For a general linear Lorentz-covariant gauge, the kinematic numerator n will instead be a rationalfunction containing terms such as p µ p ν p . A T n “ ÿ Γ P Tree ,n A p T n q Γ “ ÿ Γ P Tree ,n c p Γ q n p Γ q D p Γ q , (2.29)where the denominator D p Γ q is as before, but the numerator has been split into the colournumerator c p Γ q and the kinematic numerator n p Γ q . More explicitly,(i) c p Γ q is a group theory factor, corresponding to contractions of n ´ copies of the colourgroup structure constants corresponding to the vertices of the trivalent diagram Γ .(ii) n p Γ q is a differential operator of degree n ´ , corresponding to the kinematic nu-merator; equivalently, in momentum space, a polynomial expression of the externalmomenta and polarisation vectors, whose degree is homogeneously n ´ in terms ofthe momenta and k in terms of the polarisation vectors.The exact form of this splitting was computed in [80], with the first few terms containedin [6]. In the former paper, the authors present their result in terms of a (non-local) actionwith the Lagrangian L YM “ L YM2 ` L YM3 ` ¨ ¨ ¨ , (2.30)in which the n -th order term is L YM n “ ÿ Γ P Tree ,n O µ ¨¨¨ µ n n, Γ tr ! r A µ σ p q , A µ σ p q s r . . . r A µ σ p q , A µ σ p q s . . . , A µ σ p n q s ) l j n, Γ , ¨ ¨ ¨ l j n, Γ ,n ´ , (2.31)where the permutation σ is determined by the tree-level diagram Γ and where O µ ¨¨¨ µ n n, Γ isa sum of polynomials in the inverse Minkowski metric η µν and n ´ partial differentialoperators B µ acting on one of the n occurrences of the field A in the numerator.Note that the expressions for the splitting of the higher-arity vertices are simply substi-tuted into the ordinary Yang–Mills action, even though the resulting expression is simplyalgebraically equal to the original Yang–Mills action (in particular, the higher-order ver-tices vanish due to colour Jacobi identities), giving an impression reminiscent of the ‘ghostsof departed quantities’ of Newtonian calculus [250]. However, there is nothing mysteriousabout this action; it simply expresses how higher-order vertices, most of which are zero, aresplit apart and distributed into trivalent trees.The order-by-order calculation of the splitting of the higher-degree Feynman vertices is,in principle, a straightforward exercise, and there is nothing specific to Yang–Mills theory33apart from perhaps the ansatz of the numerator), provided that the tree-level colour–kinematics identities in fact hold. In particular, one can readily compute such a splittingfor gravity, except that there the ansatz is of a different form, A T n “ ÿ Γ P Tree ,n A p T n q Γ “ ÿ Γ P Tree ,n n p Γ q D p Γ q , (2.32)where the colour numerator is replaced by another copy of the kinematic numerator.In dealing with scattering amplitudes, we can freely use the on-shell condition p i “ andthe transversality condition ε i ¨ p i “ for the external momenta p i and polarisation vectors ε i when performing the above manipulations. Thus, the action is colour–kinematics-dualonly on shell.
3. Non-linear sigma model and special galileons
Before delving into the details of the double copy of Yang–Mills theory to N “ supergrav-ity, we first consider the simpler example of the double copy of the non-linear sigma modelon a Lie group to the special galileon [220, 251, 28, 54, 89]. This example is considerablysimpler because we can ignore the technicalities due to gauge symmetry and Becchi–Rouet–Stora–Tyutin (BRST) quantisation.In this simpler example, the non-linear sigma model enjoys a flavour–kinematics du-ality , the analogue of colour–kinematics duality in Yang–Mills theory. The roles of thesetwo dualities differ slightly: whereas in Yang–Mills theory the colour–kinematics dualityensures the existence of a BRST operator in the double copy, the flavour–kinematics dualityin the non-linear sigma model ensures (amongst other things) avoidance of the Ostrogradskyinstability. This instability generically arises for Lagrangians involving derivatives of higherorder than two, in which the Hamiltonian is unbounded from below, cf. e.g. [252]. Let us first recall some of the background material.
Non-linear sigma model.
Consider d -dimensional Minkowski space M d : “ R ,d ´ withmetric p η µν q “ diag p´ , , . . . , q with µ, ν, . . . “ , , . . . , d ´ and local coordinates x µ together with a semi-simple compact matrix Lie group G . To define the non-linear sigmamodel action, we are interested in maps g : M d Ñ G , or rather their flat current j µ : “ g ´ B µ g , (3.1)34hich takes values in the Lie algebra g of G . We take e a as a basis of g with a, b, . . . “ , , . . . , dim p g q , r e a , e b s “ f abc e c with r´ , ´s the Lie bracket on g , and x e a , e b y : “ ´ tr p e a e b q “ δ ab with ‘ tr ’ the matrix trace. The action without a potentialterm is then given by S NLSM : “ ż d d x tr t j µ j µ u “ ´ ż d d x p g ´ B µ g q a p g ´ B µ g q a , (3.2)and this special case is also called the principal chiral model .Upon setting g : “ e φ and ad φ p´q : “ r φ, ´s for φ : M d Ñ g , and using the formula j µ “ e ´ φ B µ e φ “ ´ e ´ ad φ ad φ pB µ φ q “ ÿ n “ p´ q n p n ` q ! ad nφ pB µ φ q , (3.3)which follows from the Baker–Campbell–Hausdorff formula, we may rewrite (3.2) as S NLSM “ ż d d x ÿ n “ p´ q n pp n ` q ! q tr (cid:32) B µ φ ad nφ pB µ φ q ( . (3.4)Because of the symmetry φ ÞÑ ´ φ , corresponding to the symmetry g ÞÑ g ´ of (3.2), thereare only Feynman vertices of even degree, each of which contains exactly two derivatives. Galileons.
As mentioned above, a generic Lagrangian involving derivatives of higher orderthan two runs into the Ostrogradsky instability. We can avoid this if we carefully select anansatz such that even though the action contains higher-derivative terms, the correspondingequations of motion are at most of second order in the derivatives. For a scalar field theory,the most general such ansatz is that of the galileon [254–258], see also [259] for a review.The galileon theory is a theory of a scalar field φ which is invariant under the Galilean-type symmetry φ p x q ÞÑ φ p x q ` c ` b µ x µ , (3.5)where c is a constant and b µ is a constant vector on M d . In d space-time dimensions, thereare d ` possible terms that satisfy the Galilean-type symmetry. Specifically, the generalaction is of the form [258] S Gal : “ ż d d x d ` ÿ n “ α n L Gal n with L Gal n : “ φ ε µ ¨¨¨ µ d ε ν ¨¨¨ ν d ˜ n ´ ź i “ B µ i B ν i φ ¸ ˜ d ź i “ n η µ i ν i ¸ , (3.6) We use here the exponential parametrisation g “ e φ . There are other possible parametrisations usedin this context in the literature, such as the Cayley parametrisation (see e.g. [253]). Our treatment below,however, does not depend on the choice of parametrisation as long as this parametrisation is defined by anequivariant map g Ñ G , where both g and G are equipped with adjoint actions. ε µ ¨¨¨ µ d is the usual Levi–Civita symbol. This action is parametrised by the d ` coefficients α i , among which α , corresponding to the tadpole L Gal1 φ , should be set tozero, and α , corresponding to the kinetic term L Gal2 φ l φ where l : “ B µ B µ , should becanonically normalised. Thus, one obtains a p d ´ q -dimensional moduli space of possiblegalileon theories in d space-time dimensions.There is a special point in this moduli space called the special galileon [260, 261, 158], α n : “ n ˆ d n ´ ˙ M n ´ and α n ` : “ , (3.7)where M is a mass scale. At this point, the scattering amplitudes become extremal ina specific sense. In particular, all amplitudes with an odd number of external particlesvanish, which is not the case for the generic galileon theory [158]. Operationally, it may bedefined as the galileon theory obtained by double-copying the scattering amplitudes of thenon-linear sigma model. We will transform the non-linear sigma model action (3.4) into that of the special galileonvia the following steps:(i) Put the non-linear sigma model action into a manifestly flavour–kinematics-dual form.(ii) Introduce infinitely many auxiliary fields to render the action cubic (i.e. strictify) ina manner compatible with flavour–kinematics duality.(iii) We square the coefficients of the cubic action to obtain a raw double-copied action.(iv) Upon integrating out the infinite tower of auxiliary fields of the double copy and asuitable field redefinition, we are guaranteed to recover the special galileon action.Unlike the case of Yang–Mills theory, we will not need a detailed argument to ensure theexistence of a BRST operator.
Manifestly flavour–kinematics-dual action.
As explained in Section 2.4, once it isknown that flavour–kinematics duality (and hence double copy) holds at the tree level, it isautomatic that one can write down a manifestly flavour–kinematics-dual form of the action.In particular, we may organise the infinitely many terms or the Lagrangian L NLSM : “ ř n “ L NLSM2 n of the non-linear sigma model Lagrangian (3.4) into a manifestly flavour–kinematics-dual form as L NLSM2 n “ ÿ Γ P Tree , n O n, Γ tr ! r φ, φ s r . . . r φ, φ s . . . , φ s ) l j n, Γ , ¨ ¨ ¨ l j n, Γ , n ´ , (3.8)36nd this expression needs to be read as follows. Firstly, O n, Γ is a sum of polynomials inthe inverse Minkowski metric η µν and p n ´ q partial differential operators B µ acting onone of the n occurrences of the field φ in the numerator (and rendering the commutat-ors non-trivial). Secondly, the subscripts on the inverse wave operators similarly indicatewhich product of fields they act on. Finally, the sum ranges over trivalent trees Γ with n external legs, in which the flavour contraction is determined by the topology of thetree Γ . The Lagrangian (3.8) is the analogue of the manifestly flavour–kinematics-dualLagrangian (2.31) for Yang–Mills theory.In particular, for n “ , L NLSM2 “ tr t φ l φ u (3.9)is the canonical kinetic term.This action seems perhaps less strange than the manifestly colour–kinematics-dual ac-tion for Yang–Mills theory (2.31), because unlike Yang–Mills theory, the non-linear sigmamodel already has an infinite number of terms to begin with. The method of constructionof the manifestly dual action, however, is exactly the same in both cases. Strictification.
The flavour–kinematics-dual action (3.8) has two defects: it is neitherlocal nor cubic , and thus does not produce exclusively cubic Feynman vertices. We canremedy both defects by introducing an infinite tower of auxiliary fields. We note that to afixed order in perturbation theory (with a bounded number of external legs and loops), thenumber of auxiliary fields that enter is always finite, just as in ordinary Yang–Mills theory,cf. Observation 8.10 below.In the case of the non-linear sigma model, we require two scalar auxiliary fields C and C , both in the adjoint representation, at the quartic order; quintic and higher orders willeach require multiple auxiliary fields, all of which are in the adjoint and Lorentz tensors,but with varying numbers of vector indices. The strictified action takes the form S NLSM , st “ ż d d x tr (cid:32) φ l φ ` C µ l C ,µ ` αC µ r φ, B µ φ s ` β p l C ,µ qr φ, B µ φ s ` ¨ ¨ ¨ ( , (3.10)where the dimensionful coefficients α and β are tuned so as to give the correct four-pointamplitude and to manifest flavour–kinematics duality. Notice that the strictification is notarbitrary, but mostly determined by the form of the manifestly flavour–kinematics-dualform of the action (3.8). corresponding to an internal propagator in the Feynman diagrams or strict, in homotopy algebraic language; cf. Appendix A.4 ouble copy. The next step is to engineer an action which reproduces the double copyof the flavour–kinematics-dual amplitudes on the nose, and it is not hard to see that thisconsists essentially of the following:(i) We take the tensor square of the field content p φ, C , C , . . . q , so as to obtain aninfinite quadrant of fields, ˜ φ : “ φ b φ ˜ C ,µ : “ C µ b φ ˜ C ,µ : “ C µ b φ ¨ ¨ ¨ ˜ C ,µ : “ φ b C µ ˜ C ,µν : “ C µ b C ν ˜ C ,µν : “ C µ b C ν ¨ ¨ ¨ ˜ C ,µ : “ φ b C µ ˜ C ,µν : “ C µ b C ν ˜ C ,µν : “ C µ b C ν ¨ ¨ ¨ ... ... ... . . . (3.11)all of which except for φ b φ are regarded as auxiliary.(ii) The kinematical terms are given by (off-diagonal) wave operators.(iii) The interaction vertices are simply the products of two of the interaction vertices ofthe non-linear sigma model.The double-copied action is then ˜ S DC : “ ż d d x !` ˜ φ l ˜ φ ` ˜ C µ l ˜ C ,µ L ` ˜ C ,µ l ˜ C ,µ R ` ˜ C µν l ˜ C ,µν LR ` ¨ ¨ ¨ ˘ `` Λ d ´ ` α ˜ C ,µν pB µ B ν ˜ φ q ˜ φ ` α ˜ C ,µν LR pB µ ˜ φ qB ν ˜ φ `` α pB ν ˜ C ,µ L qpB µ ˜ C ,ν R q ˜ φ ` α pB ν ˜ C ,µ L qpB µ ˜ φ q ˜ C ,ν R `` α ˜ C ,µ L pB µ B ν ˜ φ q ˜ C ,ν R ` α ˜ C ,µ L pB µ ˜ C ,ν R qB ν ˜ φ ` ¨ ¨ ¨ ˘) , (3.12)where in the interests of brevity we have only shown the terms corresponding to the doublecopy of φ and C . Note that we have introduced a new mass scale Λ in the double copy, inthe same way that the Planck scale enters during the double copy of Yang–Mills theory toEinstein gravity.The six interaction terms shown above, corresponding to the strictification of a quarticgalileon interaction, illustrate the six possible ways of double copying the flavour-strippedinteraction vertex C µ B µ φφ namely, schematically, ˆ C B φ φ b C B φ φ ˙ , ˆ C B φ φ b C φ B φ ˙ , ˆ C B φ φ bB φ C φ ˙ , ˆ C B φ φ bB φ φ C ˙ , ˆ C B φ φ b φ B φ C ˙ , ˆ C B φ φ b φ C B φ ˙ . (3.13)38y construction, the action (3.12) reproduces the double copy of the amplitudes ofthe non-linear sigma model and its tree-level amplitudes coincide with those of the specialgalileon. Physical equivalence.
To compare against the usual action of the special galileon S sGal ,we can either integrate out the auxiliaries of ˜ S DC or rewrite S sGal in manifestly flavour–kinematics-dual cubic terms, introducing a tower of auxiliary fields. We will give argumentsin both cases; the former involves a small suspension of disbelief, the latter does not havethis gap, but is more indirect.In the first version of the argument, we straightforwardly integrate out all auxiliaryfields, i.e. all fields except for ˜ φ . One may worry that the result is non-local. However,since the tree amplitudes are all local, the resulting terms will have to conspire to hidethis non-locality. Let us assume, and this is the slight gap in the argument, that this doesnot happen. That is, we obtain a local action ˜ S DC2 of a single scalar field ˜ φ whose treeamplitudes agree with those of the special galileon.The two actions ˜ S DC2 and S sGal certainly agree at the quadratic and the cubic levels(for which there are no vertices). They can differ at the quartic level, but only up to termsthat are not visible in the four-point tree-level amplitude, because the tree-level amplitudesagree. That is, the difference has to be of the form ż d d x tr ! Z l ˜ φ ` O p ˜ φ q ) , (3.14)where Z is some local cubic Lie algebra-valued functional of ˜ φ . In this case, we can performa field redefinition of either side of the form ˜ φ ÞÑ ˜ φ ` θZ (3.15)with the coefficient θ tuned such that ˜ S DC2 r ˜ φ ` θZ s agrees up to the quartic level with theaction of the special galileon.We can then iterate this argument at the quintic, sextic, etc. orders. The scepticalreader may still worry about whether this sequence of field redefinitions converges, but thisis irrelevant from the perspective of perturbative quantum field theory, since to any desiredorder in perturbation theory, only finitely many interaction vertices contribute.Note that for the computation of n -loop amplitudes at certain loop orders, the degreeof the appearing vertices is bounded from above, and we can conclude that from the per-spective of perturbative quantum field theory, the two actions agree up to a local fieldredefinition. 39otice that we started with an action with infinitely many terms which, after a suitablefield redefinition, reduces to only finitely many terms. This is reminiscent of how the in-finitely many terms of perturbative gravity reduce to a single term of the Einstein–Hilbertaction. Furthermore, a Galilean-type symmetry has appeared that avoids the Ostrogradskyinstability. Presumably, if one had started from a generic theory of adjoint scalars withoutflavour–kinematics duality, or if one started with a strictification of the non-linear sigmamodel that did not manifest flavour–kinematics duality, then our construction would haveyielded a galileon-like theory that, nevertheless, would not avoid the Ostrogradsky instabil-ity. The miracle of the Galilean-type symmetry is opaque in our formalism (unlike theBRST symmetry in the gauged case), but it nevertheless occurs.Upon path-integral quantisation, such field redefinitions produce a Jacobian in the formof local operators, which can be cancelled by local counterterms. Hence, the double copiedaction and the action of the special Galilean define equivalent perturbative quantum theoriesup to local counterterms.We now give the alternative, gap-free but less direct argument. To ensure localitythroughout, instead of integrating out auxiliary fields from ˜ S DC , we will introduce auxiliariesinto S sGal . In the strictified action (3.10), the auxiliary fields can be formally put on externallegs, with well defined, local tree-level scattering amplitudes. The latter are particularcollinear limits of ˜ φ -only amplitudes, i.e. tree-level scattering amplitudes whose externallegs are exclusively copies of the field ˜ φ . Similarly, the action of the special galileon canbe put into a manifestly flavour–kinematics-dual form, by which we mean the kinematicnumerators factorise into kinematic Jacobi identity respecting pieces. This form of theaction can then be strictified introducing auxiliary fields. In doing so, we must take care tointroduce the special galileon auxiliary fields in a manner compatible with that of (3.10):namely, the special galileon auxiliaries’ equations of motion are the double copies of thenon-linear sigma model auxiliaries. Then tree-level flavour–kinematics duality implies thatthe double copy relation holds not only between φ -only amplitudes of S NLSM and ˜ φ -onlyamplitudes of S sGal , but also between S NLSM amplitudes with external auxiliaries and S sGal amplitudes with external auxiliaries.Thus, by construction, all tree-level amplitudes agree between ˜ S DC and S sGal , includ-ing those with auxiliaries on external legs; both are local, cubic actions. Now, the field-redefinition argument above applies to this pair of actions as follows. If ˜ S DC and S sGal differ, then the difference must be of the form ˜ S DC r ˜ ϕ s ´ S sGal r ˜ ϕ s “ ÿ i Z i r ˜ ϕ s l ˜ ϕ i (3.16)for some local functionals Z i r ˜ ϕ s , where ˜ ϕ i stands for an arbitrary field, physical or auxiliary.40pon a field redefinition in ˜ S DC of the form ˜ ϕ i ÞÑ ˜ ϕ i ` θ i Z i r ˜ ϕ s , (3.17)(no summation implied) we can tune the coefficients θ i so that ˜ S DC and S sGal coincide at thecubic level. Of course, this comes at the cost of potentially introducing quartic terms into ˜ S DC . But the argument continues to work nevertheless: since the quartic terms possiblypresent in ˜ S DC are not visible at four-point tree amplitudes, they must be proportionalto l ˜ ϕ i , which in turn can be absorbed by a field redefinition, possibly producing quinticterms in ˜ S DC . Another field redefinition pushes the quintic terms to sextic ones; sextic toseptic; ad infinitum.Again, the convergence of the field redefinitions is not of interest to us: to any desiredorder in perturbation theory, we only need finitely many interaction vertices. Thus, theloop amplitudes of ˜ S DC and S sGal agree to any desired order in perturbation theory, upto local counterterms; they define equivalent perturbative quantum theories up to localcounterterms. Relation to prior work.
Let us briefly point out how our approach differs from therelated prior work of Cheung–Shen [89]. Inspired by a particular dimensional reductionof Yang–Mills theory in d ` space-time dimensions [262], the non-linear sigma model ishere effectively embedded as a subsector into a theory of two vector-like fields and a scalar.This action of this larger theory is already in flavour–kinematics duality manifesting form.The special galileon then appears as a particular subsector of the square of this theory.The way that Cheung–Shen construct the double copy is similar to ours, except thattheir action is already strictified with a finite number of fields. We have seen that if wedrop this restriction and allow ourselves an infinite tower of auxiliary fields, then it is notnecessary to add any new degrees of freedom (except auxiliary fields), and the double copybecomes exactly the special galileon upon integrating out the auxiliary fields. The previous construction can be elegantly formulated using the language of homotopyalgebras , and this reformulation serves again as a simpler example of our perspective onthe gauge-gravity double copy. Relevant definitions and results on homotopy algebras are collected in Section 4.2 and Appendix A,but we shall not need them yet in this section, which is a gentle motivation of some of these definitions. -algebra of the strictified non-linear sigma model. Feynman diagrams are con-structed using n -point vertices that, with some bias, can be regarded as taking n ´ inputfields and combining them into a new field. They also involve propagators, which are theinverses of differential operators mapping a single input field into a new field. Both of thesestructures can be regarded as ‘higher products’ µ : L Ñ L and µ : L ˆ L Ñ L (3.18)of an L -algebra L with underlying graded vector space L i “ À i P Z L i . Here, L is thevector space of all fields and L is a second copy of L shifted by one in degree and identifiedwith the space of BV anti-fields. This is the space in which the ‘right-hand side’ of theequations of motion takes values. Usually one has further nontrivial vector subspaces L i with i ă describing gauge symmetries and L i with i ą describing Noether symmetries;we will come to this later when discussing Yang–Mills theory.In the case of the strictified non-linear sigma model, we have no gauge symmetry, merelychiral fields and exclusively trivalent vertices. Therefore L NLSM , st : “ L NLSM , st1 ‘ L NLSM , st2 , (3.19)where L NLSM , st1 is the space of all possible field configurations (including the auxiliaryfields), L NLSM , st2 – L NLSM , st1 , and we have only two non-vanishing maps µ NLSM , st1 : L NLSM , st1 Ñ L NLSM , st2 ,µ NLSM , st2 : L NLSM , st1 ˆ L NLSM , st1 Ñ L NLSM , st2 , (3.20)and, for example, µ NLSM , st1 p φ q “ l φ P L NLSM , st2 ,φ ` , µ NLSM , st2 p φ, φ q “ l p φ B µ φ q looomooon P L NLSM , st2 ,C ` ` . . . , (3.21)where L NLSM , st2 ,φ ` and L NLSM , st2 ,C ` are the subspaces of L NLSM , st2 in which the anti-fields φ ` and C ` take their values. We note that µ NLSM , st2 is graded anti-symmetric (i.e. symmetricwhen forgetting about the degree of the fields), and we have the usual polarisation identity µ NLSM , st2 p φ , φ q “ ´ µ NLSM , st2 p φ ` φ , φ ` φ q´´ µ NLSM , st2 p φ , φ q ´ µ NLSM , st2 p φ , φ q ¯ . (3.22)We also have an inner product x´ , ´y on L NLSM , st of degree ´ , which pairs elements in L NLSM , st1 with elements in L NLSM , st2 . The action of the non-linear sigma model is then givenby the homotopy Maurer–Cartan action for L NLSM , st , S NLSM , st “ x a, µ NLSM , st1 p a qy ` x a, µ NLSM , st2 p a, a qy , (3.23)42here a is a generic element in L NLSM , st1 . Factorisation.
Since every field (including the auxiliary fields) carries an adjoint repres-entation of the flavour symmetry, we can flavour-strip the fields in the theory in a similarway that one can colour-strip the fields in the case of Yang–Mills theory. In homotopyalgebraic language, this corresponds to a factorisation L NLSM , st “ g b C NLSM , st “ g b C NLSM , st1 loooooomoooooon “ L NLSM , st1 ‘ g b C NLSM , st2 loooooomoooooon “ L NLSM , st2 , (3.24)where C NLSM , st1 – C NLSM , st2 can be interpreted as the field space of a theory of an unchargedscalar field, together with a tower of auxiliary fields. Importantly, also the two maps µ NLSM , st1 and µ NLSM , st2 factorise, µ NLSM , st1 “ id b m NLSM , st1 and µ NLSM , st2 “ r´ , ´s b m NLSM , st2 . (3.25)Because the map m NLSM , st2 is now graded symmetric (i.e. anti-symmetric after forgettingthe fields’ degrees), ´ C NLSM , st , m NLSM , st1 , m NLSM , st2 ¯ (3.26)is not an L -algebra but a C -algebra. In the double copy of Yang–Mills theory, there is a kinematical factor that is treatedon equal footing and interchanged with the colour factor. We therefore should strip off thisfactor as well. In the non-linear sigma model, this corresponds to factorising the fields intoa single scalar field times the vector space of the Lorentz tensors on M d that the auxiliaryfields form.The graded vector space underlying C NLSM , st factorises as C NLSM , st “ Kin b Scal “ Kin b Scal loooooomoooooon “ L scal1 ‘ Kin b Scal loooooomoooooon “ L scal2 , (3.27)where Scal – Scal is simply the space of a single scalar (anti)field, while Kin : “ R loomoon φ ‘ M d loomoon C ‘ M d loomoon C ‘ ¨ ¨ ¨ (3.28)The higher product m NLSM , st1 factorises trivially as m NLSM , st1 “ id b m NLSM , st1 , (3.29)but the factorisation of m NLSM , st2 is harder. Notice that this higher product does two things: Recall that the tensor product of a Lie algebra and a commutative algebra is a Lie algebra. We havejust encountered an example of the corresponding homotopy algebras. twisted tensor product , whose precisedefinition will be given later in Section 6.3; it suffices to say that it is custom-made toimplement precisely the above separation. In terms of this twisted tensor product, we canfactorise m NLSM , st2 “ m Kin b τ µ Scal , (3.30)where µ Scal : p φ , φ q ÞÑ φ φ (3.31)is simply the pointwise product of two scalar fields, and the twist datum τ encodes thedifferential operators that appear as numerators in flavour–kinematics duality. We furthernote that µ Scal is symmetric (graded-anti-symmetric) such that p Scal , µ
Scal , µ Scal q formsan L -algebra. Double copy.
Altogether, we have factorised the L -algebra of (the manifestly flavour–kinematics-dual formulation of) the non-linear sigma model as L NLSM , st “ g b L scal “ g b p Kin b τ Scal q . (3.32)In terms of this factorisation, the double copy prescription is straightforward to phrase.The field space and the action of the double-copied form of the special galileon is given bythe L -algebra L sGal1 “ Kin b τ p Kin b τ Scal q , (3.33)where the double appearance of the twist datum τ reflects the fact that the differentialoperators appearing in the numerator of the flavour–kinematics-dual representation havebeen squared.The validity of the factorisation (3.33) is not hard to see, in particular once one under-stands the link between higher products of L -algebras and tree-level scattering amplitudesof the encoded field theories that we shall explain later in detail.
4. Field theories, Batalin–Vilkovisky complexes, and homotopy algebras
In the following, we summarise how perturbative quantum field theory is naturally formu-lated in the language of homotopy algebras. The bridge between field theories and homotopy44lgebras is provided by the Batalin–Vilkovisky (BV) formalism [166, 170]. Our discussionfollows the treatment in [177, 178]; see also [210] for a pedagogical summary and [202] for adetailed discussion of Feynman diagrams. Basic definitions and results on homotopy algeb-ras and homotopy Maurer–Cartan theory are collected in Appendix A for convenience. Westart with the Becchi–Rouet–Stora–Tyutin (BRST) formalism for the archetypal exampleof Yang–Mills theory. This will also prepare our discussion in Section 5.
As before, we consider d -dimensional Minkowski space M d : “ R ,d ´ with metric p η µν q “ diag p´ , , . . . , q with µ, ν, . . . “ , , . . . , d ´ and localcoordinates x µ . Let g be a semi-simple compact matrix Lie algebra with basis e a with a, b, . . . “ , , . . . , dim p g q , r e a , e b s “ f abc e c with r´ , ´s the Lie bracket on g , and x e a , e b y : “ ´ tr p e a e b q “ δ ab with ‘ tr ’ the matrix trace.The action for Yang–Mills theory in R ξ -gauge for some real constant ξ in the BRSTformalism reads as S YMBRST : “ ż d d x ! ´ F aµν F aµν ´ ¯ c a B µ p ∇ µ c q a ` ξ b a b a ` b a B µ A aµ ) (4.1a)with F aµν : “ B µ A aν ´ B ν A aµ ` f bca gA bµ A cν and p ∇ µ c q a : “ B µ c a ` gf bca A bµ c c , (4.1b)where g is the Yang–Mills coupling constant, A aµ are the components of the g -valued one-form gauge potential on M d , and c a , b a , and ¯ c a are the components of g -valued functionscorresponding to the ghost, the Nakanishi–Lautrup field, and the anti-ghost field, respect-ively. Z -graded vector spaces. We note that the fields in the action (4.1a) are graded bytheir ghost number as detailed in Table 4.1. Therefore, we should view them as coordinatefunctions on a Z -graded vector space V “ À k P Z V k . Elements of V k are said to be homogen-eous of degree k , and we shall use the notation | (cid:96) | V to denote the degree of a homogeneouselement (cid:96) P V . field Φ I c a A aµ b a ¯ c a ghost number | Φ I | gh ´ Table 4.1: Ghost numbers of the fields in Yang–Mills theory.45he tensor product of two Z -graded vector spaces V and W is defined as V b W “ à k P Z p V b W q k with p V b W q k : “ à i ` j “ k V i b W j , (4.2)and the degree in V b W is thus the sum of the degrees in V and W .We shall denote the dual of a Z -graded vector space V by V ˚ , and we have V ˚ “ à k P Z p V ˚ q k with p V ˚ q k : “ p V ´ k q ˚ . (4.3)In particular, elements in V k have the opposite degree of elements in p V k q ˚ .Given a Z -graded vector space V , we can introduce the degree-shifted Z -graded vectorspace V r l s for l P Z by V r l s “ à k P Z p V r l sq k with p V r l sq k : “ V k ` l . (4.4)For an ordinary vector space V ” V , for instance, V r s consists of elements of degree ´ since only p V r sq ´ “ V is non-trivial. Note that p V b W qr l s “ V r l s b W “ V b W r l s and p V r l sq ˚ “ V ˚ r´ l s for all l P Z . For convenience, we introduce the notion of a shiftisomorphism σ : V Ñ V r s (4.5)which lowers the degree of every element of V , that is, σ : V k Ñ p V r sq k ´ .We note that the action (4.1a) is built of polynomial functions and their derivatives. Bythe algebra of polynomial functions on a Z -graded vector space V , we mean the Z -gradedsymmetric tensor algebra C p V q : “ Ä ‚ V ˚ . Basis elements of V ˚ can be regarded as thecoordinate functions on V . Explicitly, such a function looks like f “ f ` ξ α f α ` ξ α ξ β f αβ ` ¨ ¨ ¨ P C p V q , (4.6)where ξ α are basis elements of V ˚ and f, f α , f αβ , . . . are constants. We have ξ α ξ β “p´ q | ξ α | V ˚ | ξ β | V ˚ ξ β ξ α . Note that if V is a vector space of some suitably smooth functionsor, more generally, sections of some vector bundle, then the dual V ˚ , being the space ofdistributions, contains not only the ordinary dual coordinate functions but also all of theirderivatives. We will not discuss the analytical subtleties of this construction in the infinite-dimensional case, exceptto note that the dual spaces will be degree-wise topological duals. RST operator in Yang–Mills theory.
The reason for introducing ghosts in the firstplace is the gauge symmetry of Yang–Mills theory, which in the BRST and BV formalismsis captured in a dual formulation as a differential on a differential graded commutativealgebra that is called the
Chevalley–Eilenberg algebra . More specifically, this is the algebraof polynomial functions, and the differential is a nilquadratic vector field Q : C p V q Ñ C p V q of degree one, Q “ , known as the homological vector field . A Z -graded vectorspace with such a homological vector field is called a Q -vector space .The prime example of a Q -vector space is that of an ordinary vector space g withbasis e a for a, b, . . . “ , . . . , dim p g q , regarded as the Z -graded vector space g r s . On g r s , we have coordinates ξ a only in degree one and thus, the most general vector field Q : C p g r sq Ñ C p g r sq of degree one is of the form Q : “ ξ b ξ c f cba BB ξ a ñ Qξ a “ ξ b ξ c f cba (4.7)for some constants f abc “ ´ f bac . The condition Q “ is equivalent to the Jacobi identityfor the f abc so that Q induces a Lie bracket r e a , e b s “ f abc e c on g . The differential gradedalgebra p C p g r sq , Q q is the Chevalley–Eilenberg algebra of the Lie algebra p g , r´ , ´sq towhich we alluded above. In order to translate between Q and r´ , ´s , it is useful to definethe contracted coordinate functions a : “ ξ a b e a P p g r sq ˚ b g (4.8)of degree one in p g r sq ˚ b g . Consequently, Q a : “ p Qξ a q b e a “ ξ b ξ c f cba b e a “ ´ ξ b ξ c b f bca e a “ ´ ξ b ξ c b r e b , e c s“ : ´ r ξ b b e b , ξ c b e c s“ ´ r a , a s . (4.9)More general vector fields arise in the Chevalley–Eilenberg algebras of L -algebrasand L -algebroids, cf. e.g. [177] for further details. In the case of Yang–Mills theory, thehomological vector field Q YMBRST describing the gauge symmetry acts according to c a Q YMBRST
ÞÝÝÝÝÝÑ ´ gf bca c b c c , ¯ c a Q YMBRST
ÞÝÝÝÝÝÑ b a ,A aµ Q YMBRST
ÞÝÝÝÝÝÑ p ∇ µ c q a , b a Q YMBRST
ÞÝÝÝÝÝÑ . (4.10) These are often used in the string field theory literature, albeit shifted such that a is of degree zero. BRST transformations and Q YMBRST as the
BRSToperator . One readily verifies that p Q YMBRST q “ , that is, Q BRST is a differential. Inaddition, the action (4.1a) is Q YMBRST -closed, that is, Q YMBRST S YMBRST “ , which ensures gaugechoice independence and unitarity.We shall denote the minimal field space for gauge-fixed Yang–Mills theory by L YMBRST ,but the ghost number is the degree of coordinate functions on L YMBRST r s . Explicitly, L YMBRST “ L YMBRST , ‘ L YMBRST , ‘ L YMBRST , , L YMBRST , : “ C p M d q b g , L YMBRST , : “ p Ω p M d q ‘ C p M d qq b g , L YMBRST , : “ C p M d q b g (4.11)and c , A , b , and ¯ c are coordinate functions on L YMBRST , r s “ p L YMBRST r sq ´ , L YMBRST , r s “p L YMBRST r sq , L YMBRST , r s “ p L YMBRST r sq , and L YMBRST , r s “ p L YMBRST r sq and thus of degrees , , , and ´ , respectively. Moreover, the action (4.1a) is a polynomial function S YMBRST P C p L YMBRST r sq on L YMBRST r s of total ghost number zero, | S YMBRST | C p L YMBRST r sq “ . In thefollowing, we shall write | ´ | gh as a shorthand for both | ´ | p L YMBRST r sq ˚ and | ´ | C p L YMBRST r sq .The Q -vector space p L YMBRST r s , Q YMBRST q describes the Lie algebra of gauge transforma-tions as well as its action on the various fields, which together form an action Lie algebroid.This becomes clear when comparing (4.10) to (4.9); the latter is the evident generalisation,e.g. to the corresponding formulas for a differential graded Lie algebra.We note that gauge-invariant objects are Q YMBRST -closed and that gauge-trivial objectsare Q YMBRST -exact. Therefore, physical observables are in the cohomology of Q BRST . Thepair of fields p b, ¯ c q is known as a trivial pair , as Q YMBRST links the two fields by an identitymap. They vanish in the Q YMBRST -cohomology and thus are not observable.As in (4.8), it will turn out useful to define the contracted coordinates a : “ ż d d x ! c a p x q b p e a b s x q ` A aµ p x q b p e a b v µ b s x q `` b a p x q b p e a b s x q ` ¯ c a p x q b p e a b s x q ) , (4.12a)where e a , v µ , and s x are basis vectors on g , T ˚ x M d , and C p M d q , respectively (and thus, wehave an identification v µ ˆ “ d x µ ). It should be noted that a is an element of p L YMBRST r sq ˚ b L YMBRST of degree one, and it can be regarded as a superfield which contains all the fieldsof different ghost numbers. The component fields can be recovered by projecting onto the This graded vector space is, in fact, the space of sections of a graded vector bundle, and fields andtheir derivatives are sections of the corresponding jet bundle; but these details would not enlighten ourdiscussion any further so we suppress them. a “ Φ I b e I (4.12b)for DeWitt indices I, J, . . . , which contain Lorentz and gauge indices as well as space-timeposition. A contraction of DeWitt indices involves sums over all discrete indices and evidentintegrals over the continuous ones. L -algebras The above example of Yang–Mills theory has demonstrated how Z -graded vector spacesand homological vector fields enter into the description of a gauge field theory in the BRSTformalism. In particular, gauge-invariant observables were contained in the cohomology of Q BRST . To fully characterise classical observables, however, we also need to impose theequations of motion. This is the purpose of the more general Batalin–Vilkovisky (BV)formalism. As a byproduct, the BV formalism can cater for open gauge symmetries whichare gauge symmetries for which Q BRST is a differential only on-shell. The BV operator Q BV ,which generalises the BRST operator Q BRST , encodes the Chevalley–Eilenberg descriptionof a cyclic L -algebra (i.e. an L -algebra with a notion of inner product). The gauge-fixedform of this cyclic L -algebra will be crucial for our formulation of the double copy ofamplitudes. BV operator.
Let L BRST r s be a Z -graded vector space of fields of a general field theory.Then we have also a correspondence between the fields and the coordinate functions on thisspace. In order to encode the field equations for all the fields in the action of an operator Q BV , we ‘double’ this vector space such that we have for each field Φ I of ghost number | Φ I | gh an anti-field Φ ` I of ghost number | Φ ` I | gh : “ ´ ´ | Φ I | gh so that Q BV Φ ` I : “ p´ q | Φ I | δS BRST δ Φ I ` ¨ ¨ ¨ . (4.13)Here, the ellipsis denotes terms at least linear in the anti-fields. Formally, this doublingamounts to considering the cotangent space L BV r s : “ T ˚ r´ sp L BRST r sq ô L BV : “ T ˚ r´ s L BRST , (4.14)which yields a canonical symplectic form ω : “ δ Φ ` I ^ δ Φ I (4.15)49f ghost number ´ . This symplectic form ω , in turn, induces a Poisson bracket, alsoknown as the anti-bracket . It reads explicitly as t F, G u “ p´ q | Φ I | gh p| F | gh ` q δFδ Φ I δGδ Φ ` I ´ p´ q p| Φ I | gh ` qp| F | gh ` q δFδ Φ ` I δGδ Φ I , (4.16)and it is of ghost number one so that t F, G u “ ´p´ q p| F | gh ` qp| G | gh ` q t G, F u .The classical Batalin–Vilkovisky action is now a function S BV P C p L BV r sq of ghostnumber zero, which obeys the classical master equation t S BV , S BV u “ , (4.17a)which extends the original action S of the field theory (without ghosts or trivial pairs) S BV | Φ ` I “ “ S , (4.17b)and whose Hamiltonian vector field extends the BRST differential, p Q BV Φ I q| Φ ` J “ “ Q BRST Φ I (4.17c)with Q BV : “ t S BV , ´u . (4.18)We note that Q “ and (4.17a) are equivalent.The last two conditions fix the terms of S BV which are constant and linear in theanti-fields to read as S BV “ S ` p´ q | Φ I | gh Φ ` I Q BRST Φ I ` ¨ ¨ ¨ . (4.19)General theorems now state that for each action and compatible BRST operator, there isa corresponding BV action and a BV operator, cf. [263].In a general theory, we will usually have a physical field a of ghost number zero aswell as ghosts c together with higher ghosts c ´ k of each ghost number ´ k ` as coordin-ate functions on L BV r s . Higher ghosts are non-trivial only in theories with higher gaugeinvariance. All fields come with the corresponding anti-fields a ` , c ` , and c `´ k . To accom-modate gauge fixing, we will have to expand the field space further by trivial pairs andcorresponding anti-fields, as already encountered in the previous section. The signs arise as follows. Hamiltonian vector fields V F are given by V F (cid:32) ω “ δF for some function F . The Poisson bracket is then given by t F, G u : “ V F (cid:32) V G (cid:32) ω “ V F p G q from which the signs follow usingthe explicit form (4.15) of ω . The signs are often absorbed using left- and right-derivatives; however, forclarity we shall keep them explicitly. Here, | Φ ` I “ is the restriction to the subspace of BV field space where all anti-fields are zero. I in C p L BRST r sq , and the functions onthe solutions space are the quotient C p L BRST r sq{ I . Because of (4.18), Q BV Φ ` I “ p´ q | Φ I | δS BV δ Φ I , (4.20)and the gauge-invariant functions on the solutions space are described by the Q BV -cohomology. L -algebras. Following (4.12), we define again a superfield a : “ a I b e I “ Φ I b e I ` Φ ` I b e I (4.21)of degree one in p L BV r sq ˚ b L BV , where I runs over all fields, ghosts, ghosts for ghosts andthe corresponding anti-fields, as well as space-time and Lie algebra indices. As in (4.9), wemay extend the action of Q BV to elements in p L BV r sq ˚ b L BV by left action and write Q BV a “ t S BV , a u “ ´ f p a q with f p a q “ : ÿ i ě i ! µ i p a , . . . , a q . (4.22a)The µ i now encode i -ary graded anti-symmetric linear maps µ i : L BV ˆ ¨ ¨ ¨ ˆ L BV Ñ L BV ,which can be extracted by the formulas µ p a q : “ p´ q | a I | gh a I b µ p e I q ,µ i p a , . . . , a q : “ p´ q i ř ij “ | a I j | gh ` ř ij “ | a I j | gh ř j ´ k “ | e I k | L BV a I ¨ ¨ ¨ a I i b µ i p e I , . . . , e I i q , (4.22b)see [177] for a much more detailed exposition. The condition Q “ then amounts tothe homotopy Jacobi identities ÿ i ` i “ i ÿ σ P Sh p i ; i q p´ q i χ p σ ; (cid:96) , . . . , (cid:96) i q µ i ` p µ i p (cid:96) σ p q , . . . , (cid:96) σ p i q q , (cid:96) σ p i ` q , . . . , (cid:96) σ p i q q “ (4.23a)for all (cid:96) , . . . , (cid:96) i P L BV . The sum is over all p i ; i q unshuffles σ which consist of permutations σ of t , . . . , i u so that the first i and the last i ´ i images of σ are ordered. Moreover, χ p σ ; (cid:96) , . . . , (cid:96) i q is the Koszul sign given by (cid:96) ^ . . . ^ (cid:96) i “ χ p σ ; (cid:96) , . . . , (cid:96) i q (cid:96) σ p q ^ . . . ^ (cid:96) σ p i q . (4.23b)The pair p L BV , µ i q with products µ i subject to (4.23) is called an L -algebra , cf. Ap-pendix A.3. In our present setting, L BV is, in fact, a cyclic L -algebra because of the Note that the µ i define, in fact, an L -structure on C p L BV r sq b L BV . ω . Specifically, if we consider the shift isomorphism (4.5),then ω induces the (indefinite) inner product x (cid:96) , (cid:96) y : “ p´ q | (cid:96) | L BV ω p σ p (cid:96) q , σ p (cid:96) qq (4.24a)of degree ´ in L BV and of ghost number zero. It is cyclic in the sense that x (cid:96) , µ i p (cid:96) , . . . , (cid:96) i ` qy “ p´ q i ` i p| (cid:96) | L BV `| (cid:96) i ` | L BV q`| (cid:96) i ` | L BV ř ij “ | (cid:96) j | L BV x (cid:96) i ` , µ i p (cid:96) , . . . , (cid:96) i qy , (4.24b)which is a consequence of the vanishing of the Lie derivative of ω along Q BV . This isequivalent to saying that the higher products µ i , with the first i ´ arguments fixed, actas graded derivations on x´ , ´y . Correspondence between actions and L -algebras. Every cyclic L -algebra p L BV , µ i q comes with a homotopy Maurer–Cartan action , cf. Appendix A. In particular,the functional S hMC : “ ÿ i ě p i ` q ! x a, µ i p a, . . . , a qy (4.25)for a P L BV , reproduces the action for the physical fields. Using the superfield a definedin (4.21), we can write down a more general homotopy Maurer–Cartan action S shMC : “ ÿ i ě p i ` q ! x a , µ i p a , . . . , a qy , (4.26a)where we define x f I b e I , f J b e J y : “ p´ q | f I | gh `| f J | gh `| e I | L BV | f J | gh f I f J x e I , e J y (4.26b)for f I , P C p L BV r sq . This superfield version of the homotopy Maurer–Cartan actionis, in fact, the full BV action S BV . Put differently, (4.26a) satisfies the quantum masterequation (4.35) if and only if the µ i in µ i via (4.22b) satisfy the homotopy Jacobi identit-ies (4.23). We shall refer to the action (4.26a) as the superfield homotopy Maurer–Cartanaction of the L -algebra p L BV , µ i q .In summary, the BV formalism provides an equivalence between classical field theoriesand cyclic L -algebras, where the BV operator plays the role of the Chevalley–Eilenbergdifferential of the L -algebra. Clearly, the BV action corresponding to an L -algebra L BV We will, in the bulk of our paper, deviate from this sign convention in order to simplify the signsarising in our double copy formalism.
52s physically only interesting if its degree-one part is non-trivial. To read off the L -algebrafrom a particular action functional, we note that using (4.26b) we have x a , µ i p a , . . . , a qy “ x a I i ` b e I i ` , µ i p a I b e I , . . . , a I i b e I i qy “ ζ p I , . . . , I i q a I i ` a I ¨ ¨ ¨ a I i x e I i ` , µ i p e I , . . . , e I i qy (4.27a)with the sign ζ p I , . . . , I i q given by ζ p I , . . . , I i q : “ p´ q ř ik “ | a I k | gh p i ` k ` ř ij “ k | a I j | gh q . (4.27b)More explicitly, x a , µ p a qy “ p´ q | a I | gh a I a I x e I , µ p e I qy , x a , µ p a , a qy “ p´ q p| a I | gh ` q| a I | gh a I a I a I x e I , µ p e I , e I qy , (4.28)and we shall make use of these formulas later. Remark 4.1.
The exchange of the coordinate functions on field space with the actual fieldscan easily lead to confusion. Let us therefore summarise the situation once more. Actualfields (usually sections of a bundle or connections and their generalisations) are elementsof a graded vector space L BV . The L -algebra structure is defined on the vector space L BV .The symbols appearing in an action S are, technically speaking, not fields but coordinatefunctions on the grade-shifted field space L BV r s , the same way that in differential geometryone writes the metric in terms of the symbols x µ , which are not points in space-time butrather real-valued coordinate functions defined on space-time. Once we evaluate the actionfor particular fields, the coordinate functions are replaced by their values. Similarly, theBV operator, the anti-bracket etc. all act on or take as arguments polynomial functions on L BV r s , which are given by polynomial expressions in the coordinate functions as well astheir derivatives, which are also contained in p L BV r sq ˚ . To simplify notation, the coordin-ate function for a field (e.g. in an action) will be denoted by the same symbol as the field(element of the L -algebra), as commonly done in quantum field theory. Remark 4.2.
The integral defining the action S of a classical field theory is mathematicallyusually not well defined. At a classical level, this does not matter because we are neverinterested in the value of S itself, and we can treat all integrals as formal expressions. Fordefiniteness, mathematicians often drop the action and work with the Lagrangian instead.This can easily be done in the L -algebra picture, working with graded modules over thering of functions instead of graded vector spaces.At quantum level, however, the value of S for particular field configurations does play arole, and one needs to carefully restrict the field space such that all integrals are indeed well-defined, cf. [209]. One suitable restriction offers itself for the perturbative treatment. We plit the field space into interacting fields, F int , which can simply be identified with Schwartzfunctions on Minkowski space S p M d q , and free fields F free , which can be identified withsolutions to the free equations of motion (i.e. fields in the kernel of µ ), which are Schwartztype for any fixed time-slice of Minkowski space, F : “ F int ‘ F free with F int : “ S p M d q and F free : “ ker S p µ q . (4.29) The elements of ker S p µ q are, of course, the states that label the asymptotic on-shell statesin perturbation theory. On the other hand, the fields in S p M d q are the propagating degreesof freedom found on internal lines in Feynman diagrams. The decomposition (4.29) is verymuch in the spirit of the homological perturbation lemma, which can be used to constructthe scattering amplitudes, as we shall discuss below.We note that the wave operator is invertible on S p M d q and the inverse is indeed thepropagator h , as we shall discuss in more detail below. This allows us to define the operators ? l and ? l on S p M d q , which we continue to all of F by mapping elements of ker S p µ q to zero. This fact will play an important role later. Gauge fixing.
The next step in the BV formalism is the implementation of gauge fixing.This is achieved by a canonical transformation S gfBV “ Φ I , ˜Φ ` I ‰ : “ S BV „ Φ I , Φ ` I ` δ Ψ δ Φ I (4.30)which is mediated by a choice of gauge-fixing fermion , the generating functional for thecanonical transformation, which is a function Ψ P C p L BV r sq of ghost number ´ . Theaction (4.30) is then gauge-fixed if its Hessian is invertible. This requires a careful choiceof Ψ : the trivial choice Ψ “ leads back to the original action. When the classical BVaction is only linear in the anti-fields, as is e.g. the case for Yang–Mills theory and all thefield theories we are dealing with, we may set the anti-fields in S gfBV to zero after gauge-fixing,without loss of generality since the BV operator reduces to a BRST operator.Note that to construct the gauge-fixing fermion Ψ of ghost number ´ , we will have tointroduce additional fields of negative ghost number together with their anti-fields, such ase.g. the anti-ghost ¯ c and the Nakanishi–Lautrup field b in the case of Yang–Mills theory. Ifwe do not change the Q BV -cohomology, these new fields do not affect the observables. Thiscan trivially be achieved if Q BV maps one field to another, ¯ c Q BV ÞÝÝÝÑ b , b Q BV ÞÝÝÝÑ , ¯ c ` Q BV ÞÝÝÝÑ , b ` Q BV ÞÝÝÝÑ ´ ¯ c ` , (4.31)cf. (4.10). We shall encounter a number of more involved examples in Section 5.54 uantum master equation and quantum L -algebras. Besides the canonicalsymplectic form (4.15), we also have a canonical second-order differential operator on C p L BV r sq , called the Batalin–Vilkovisky Laplacian , and defined as ∆ F : “ p´ q | Φ I | gh `| F | gh δ Fδ Φ ` I δ Φ I (4.32)for F P C p L BV r sq .The BV Laplacian plays a key role in the path integral quantisation of a theory. Inparticular, the gauge fixing (4.30) is implemented at the path-integral level as Z Ψ : “ ż L BV µ p Φ I , Φ ` I q δ ˆ Φ ` I ´ δ Ψ δ Φ I ˙ e i (cid:126) S (cid:126) qBV r Φ I , Φ ` I s , (4.33)where µ is a measure that is compatible with the symplectic form ω , δ is a functional deltadistribution, (cid:126) is a formal parameter, and S (cid:126) qBV P C p L BV r sq is a functional of ghostnumber zero with S (cid:126) qBV | (cid:126) “ “ S BV . (4.34)For Z Ψ to be independent of the choice of gauge-fixing fermion Ψ , S (cid:126) qBV must satisfy the quantum master equation [166] ∆e i (cid:126) S (cid:126) qBV “ ðñ t S (cid:126) qBV , S (cid:126) qBV u ´ (cid:126) ∆ S (cid:126) qBV “ . (4.35)Consequently, we obtain as generalisation of (4.18) the quantum BRST-BV operator Q qBV : “ t S (cid:126) qBV , ´u ´ (cid:126) ∆ , (4.36)and the quantum master equation (4.35) is equivalent to Q “ . Note that contrary tothe classical version, the quantum version (4.36) is no longer a derivation. Solutions S (cid:126) qBV to (4.35) are called quantum Batalin–Vilkovisky actions . We may now solve (4.35) orderby order in (cid:126) generalising the products µ i in (4.26a) to products µ i,L for L “ , , , . . . toreflect the (cid:126) -dependence with µ i,L “ “ µ i and µ i,L “´ : “ . Consequently, we consider theansatz S qshMC : “ ÿ i ě L ě (cid:126) L p i ` q ! x a , µ i,L p a , . . . , a qy (4.37) Specifically, one requires Z Ψ ` δ Ψ “ Z Ψ for an infinitesimal deformation δ Ψ of Ψ ; the space of gauge-fixing fermions Ψ (whose Hessians may not be invertible) is contractible, so Z Ψ is globally independent of Ψ . µ i,L satisfy the quantum homotopy Jacobi identities [171, 199, 200] ÿ i ` i “ iL ` L “ L ÿ σ P Sh p i ; i q p´ q i χ p σ ; (cid:96) , . . . , (cid:96) i q µ i ` ,L p µ i ,L p (cid:96) σ p q , . . . , (cid:96) σ p i q q , (cid:96) σ p i ` q , . . . , (cid:96) σ p i q q ´´ i µ i ` ,L ´ p e I , e I , (cid:96) , . . . , (cid:96) i q “ (4.38)for (cid:96) , . . . , (cid:96) i P L BV , where the µ i,L are as in (4.22b) via the µ i,L . Here e I : “ e J ω JI , where ω IJ is the inverse of ω IJ of the symplectic form (4.15) when written as ω “ δ a I ^ ω IJ δ a J .Furthermore, (4.22a) generalises to Q qBV a “ ´ ÿ i ě L ě i ! µ i,L p a , . . . , a q . (4.39)The tuple p L BV , µ i,L , ω q with the products µ i,L subject to (4.38) is called a quantum or loop L -algebra . In the classical limit (cid:126) Ñ , the higher products µ i,L for L ą become trivial,and we recover a cyclic L -algebra. Note that for scalar field theory, Yang–Mills theory,and also Chern–Simons theory, the classical BV action also satisfies the quantum masterequation and hence, in those cases, we may set S (cid:126) qBV “ S BV , in which case µ i,L “ for L ą . L -algebras Above, we saw that actions of field theories are encoded in cyclic L -algebras. The sameholds for tree-level scattering amplitudes and loop-level scattering amplitudes are encodedin quantum L -algebras, as we shall explain in this section. Equivalence of field theories.
Classically, two physical theories are equivalent, if theyhave an isomorphic space of observables. Translated to the BV formalism, this implies thatclassically equivalent physical theories have isomorphic Q BV -cohomology. Dually, this im-plies that two physical theories are classically equivalent, if they have quasi-isomorphic L -algebras, which is also mathematically the natural notion of equivalence for L -algebras;see Appendix A for more details.Given two L -algebras p L BV , µ i q and p ˜ L BV , ˜ µ i q constructed from a BV action, a morph-ism φ : L BV Ñ ˜ L BV of L -algebras is a collection of i -linear totally graded anti-symmetric This is weaker than the statement that tree-level scattering amplitudes coincide. To define asymptoticin- and out-states in the same Hilbert space, one needs the additional data of the symplectic form ω . Twoclassical theories have the same tree-level scattering amplitudes if they are related by a quasi-isomorphismscompatible with the cyclic structures. Again, see [177] for some more details. φ i : L BV ˆ ¨ ¨ ¨ ˆ L BV Ñ ˜ L BV of degree ´ i subject to the conditions (A.27).We note that the homotopy Jacobi identities (4.23) imply that µ and ˜ µ are differen-tials. Therefore, we may consider their cohomologies H ‚ µ p L BV q : “ À k P Z H kµ p L BV q and H ‚ ˜ µ p ˜ L BV q : “ À k P Z H k ˜ µ p ˜ L BV q . We also note that the identity (A.27) implies that φ is achain map, that is, ˜ µ i ˝ φ “ φ ˝ µ and thus descends to a map H ‚ µ p L BV q Ñ H ‚ ˜ µ p ˜ L BV q on the cohomologies. Quasi-isomorphisms are those morphisms for which φ induces anisomorphism on cohomology.Under quasi-isomorphisms, the physical theory remains unchanged as is explained inAppendix A, see also [196,195,177,178,210]. In particular, the BV actions S BV and ˜ S BV for L BV and ˜ L BV are related as ˜ S BV “ φ ˚ S BV , where we used the pullback φ ˚ : C p ˜ L BV r sq Ñ C p L BV r sq dual to the morphism φ . Consequently, quasi-isomorphisms constitute thecorrect notion of equivalence .Because the Q BV -cohomologies in ghost numbers different from zero (i.e. dual to L -degree one) are not measurable, one may wonder if the notion of a full quasi-isomorphism isnot too restrictive. For perturbation theory, agreement in H µ p L BV q is certainly sufficient,and this can often be extended to an agreement in further cohomologies, cf. e.g. [264,Appendix C]. Moreover, some fields in L -degree zero, such as e.g. anti-fields of anti-ghosts and Nakanishi–Lautrup fields, are often unphysical, and arise only as internal fieldsin loop diagrams. Therefore their contributions to H µ p L BV q can also be disregarded whenidentifying physical observables. At a technical level, one can restrict these fields suchthat the kernel of the differential operator describing their linearised equations of motionvanishes, cf. Remark 4.2. Tree-level scattering amplitudes.
There is an L -structure µ ˝ i with vanishing differ-ential µ ˝ on the cohomology L ˝ BV : “ H ‚ µ p L BV q of an L -algebra p L BV , µ i q such that L ˝ BV and L BV are quasi-isomorphic. This L -algebra L ˝ BV is called the minimal model of L BV ,cf. Appendix A. The minimal model corresponds to a field theory equivalent to the originalfield theory, but without any propagating degrees of freedom. Its higher products thereforehave to be the tree-level scattering amplitudes [171, 177, 214, 209].The relation between L BV and L ˝ BV can be understood as follows. We start from theunderlying chain complexes and the following diagram: p L BV , µ q p L ˝ BV , q . h pe (4.40a) Here, we are a bit cavalier about the inclusion of the cyclic structure; again, see [177] for some moredetails. p is the obvious projection onto the cohomology, and e is a choice of embedding(involving choices, e.g. a choice of gauge for gauge theories). The quasi-isomorphism alsogives rise to a contracting homotopy h , which is a linear map of degree ´ . The maps e and h can be chosen such that id “ µ ˝ h ` h ˝ µ ` e ˝ p , p ˝ e “ id , p ˝ h “ h ˝ e “ h ˝ h “ , p ˝ µ “ µ ˝ e “ . (4.40b)Mathematically, this is an abstract Hodge–Kodaira decomposition . The map h in L -degree two turns out to be the (Feynman–’t Hooft) propagator of the physical theory inquestion [265–267], see also [268] and references therein.We directly extend the diagram (4.40a) to the Chevalley–Eilenberg picture, where wehave p C p L BV r sq , Q BV , q p C p L ˝ BV r sq , q H E P id “ P ˝ E ` Q BV , ˝ H ` H ˝ Q BV , , E ˝ P “ id , E ˝ H “ H ˝ P “ H ˝ H “ , E ˝ Q BV , “ Q BV , ˝ P “ , (4.41a)where Q BV , is the linear part of Q BV , which encodes the differential µ . The maps E , P ,and H are defined by the ‘tensor trick’ [207] as F “ ÿ i ě i ! p F q i for F P t E , P , H u (4.41b)with p E q i : “ p e ˚ q d i , p P q i : “ p p ˚ q d i , p H q i : “ ÿ k ` l “ i ´ d k d h ˚ d p p ˚ ˝ e ˚ q d l . (4.41c)We can now regard the non-linear part δ : “ Q BV ´ Q BV , (4.42)of Q BV as a perturbation and use the homological perturbation lemma [207, 208], which58sserts that there is a contracting homotopy p C p L BV r sq , Q BV q p C p L ˝ BV r sq , Q ˝ BV q H EP id “ P ˝ E ` Q BV ˝ H ` H ˝ Q BV , E ˝ P “ id , E ˝ H “ H ˝ P “ H ˝ H “ , E ˝ Q BV “ Q ˝ BV ˝ E , Q BV ˝ P “ P ˝ Q ˝ BV (4.43a)in the deformed setting. In particular, E “ E ˝ p id ` δ ˝ H q ´ , H “ H ˝ p id ` δ ˝ H q ´ , P “ P ´ H ˝ δ ˝ P , Q ˝ BV “ E ˝ δ ˝ P , (4.43b)and Q ˝ BV is the Chevalley–Eilenberg differential encoding the higher products of the minimalmodel and thus its tree-level scattering amplitudes. Note that here, the inverse operatorsare to be seen as geometric series. The formula for Q ˝ BV is then recursive, which hasinteresting consequences [209, 201].Translated to the dual picture, the homological perturbation lemma yields the followingformulas for the quasi-isomorphism φ : p L BV , µ i q Ñ p L ˝ BV , µ ˝ i q [195]: φ p (cid:96) ˝ q : “ e p (cid:96) ˝ q ,φ p (cid:96) ˝ , (cid:96) ˝ q : “ ´p h ˝ µ qp φ p (cid:96) ˝ q , φ p (cid:96) ˝ qq , ... φ i p (cid:96) ˝ , . . . , (cid:96) ˝ i q : “ ´ i ÿ j “ j ! ÿ k `¨¨¨` k j “ i ÿ σ P Sh p k ,...,k j ´ ; i q χ p σ ; (cid:96) ˝ , . . . , (cid:96) ˝ i q ζ p σ ; (cid:96) ˝ , . . . , (cid:96) ˝ i q ˆˆ p h ˝ µ j q ` φ k ` (cid:96) ˝ σ p q , . . . , (cid:96) ˝ σ p k q ˘ , . . . , φ k j ` (cid:96) ˝ σ p k `¨¨¨` k j ´ ` q , . . . , (cid:96) ˝ σ p i q ˘˘ , (4.44a) Because we are interested in perturbation theory, we do not have to concern ourselves with convergenceissues. µ ˝ i : L ˝ BV ˆ ¨ ¨ ¨ ˆ L ˝ BV Ñ L ˝ BV are constructed recursively as µ ˝ p (cid:96) ˝ q : “ ,µ ˝ p (cid:96) ˝ , (cid:96) ˝ q : “ p p ˝ µ qp φ p (cid:96) ˝ q , φ p (cid:96) ˝ qq , ... µ ˝ i p (cid:96) ˝ , . . . , (cid:96) ˝ i q : “ i ÿ j “ j ! ÿ k `¨¨¨` k j “ i ÿ σ P Sh p k ,...,k j ´ ; i q χ p σ ; (cid:96) ˝ , . . . , (cid:96) ˝ i q ζ p σ ; (cid:96) ˝ , . . . , (cid:96) ˝ i q ˆˆ p p ˝ µ j q ` φ k ` (cid:96) ˝ σ p q , . . . , (cid:96) ˝ σ p k q ˘ , . . . , φ k j ` (cid:96) ˝ σ p k `¨¨¨` k j ´ ` q , . . . , (cid:96) ˝ σ p i q ˘˘ , (4.44b)where (cid:96) ˝ , . . . , (cid:96) ˝ i P L ˝ BV . Here, χ and ζ are again the Koszul sign (4.23b) and the signfactor (A.27b), respectively.Using the higher products of the minimal model, n -point tree-level scattering amplitudesof the free fields a ˝ , . . . , a ˝ n P H µ p L BV q are then computed using formula [209] (see also [196,195] for the case of string field theory) A n, p a ˝ , . . . , a ˝ n q “ i x a ˝ , µ ˝ n ´ p a ˝ , . . . , a ˝ n qy . (4.45)Furthermore, in [209] it was shown that the recursion relations (4.44a) encode the famousBerends–Giele recursion relations [269] for gluon scattering in Yang–Mills theory. For arelated discussion of the S-matrix in the language of L -algebras, see also [211] as wellas [214, 215] for an interpretation of tree-level on-shell recursion relations in terms of L -algebras. Loop-level scattering amplitudes.
In order to extend the above discussion to recursionrelations for loop-level amplitudes, we follow [200, 201, 210]. Recall that in the transitionfrom the classical to the quantum master equation, the classical BV operator is deformedin powers of (cid:126) according to Q BV : “ t S BV , ´u Ñ Q qBV : “ t S (cid:126) qBV , ´u´ (cid:126) ∆ with S (cid:126) qBV “ S BV ` O p (cid:126) q . (4.46)Consequently, the perturbation δ : “ Q qBV ´ Q qBV , “ Q qBV ´ Q BV , (4.47)between the full and linearised part of Q qBV is now also deformed in powers of (cid:126) . Startingagain from the diagram (4.41a), we use the homological perturbation lemma to obtain a60ontracting homotopy p C p L BV r sq , Q qBV q p C p L ˝ BV r sq , Q ˝ qBV q H EP id “ P ˝ E ` Q qBV ˝ H ` H ˝ Q qBV , E ˝ P “ id , E ˝ H “ H ˝ P “ H ˝ H “ , E ˝ Q qBV “ Q ˝ qBV ˝ E , Q qBV ˝ P “ P ˝ Q ˝ qBV , (4.48a)where E “ E ˝ p id ` δ ˝ H q ´ , H “ H ˝ p id ` δ ˝ H q ´ , P “ P ´ H ˝ δ ˝ P , Q ˝ qBV “ E ˝ δ ˝ P . (4.48b)Note that because δ contains the second order differential operator ∆ , none of the maps willbe algebra morphisms in general; this is just a consequence of the fact that Q ˝ qBV defines aloop homotopy algebra.Importantly, the differential Q ˝ can be written as [270, 200] Q ˝ qBV “ t W (cid:126) qBV , ´u ˝ ´ (cid:126) ∆ ˝ , (4.49)where t´ , ´u ˝ and ∆ ˝ are the anti-bracket and the BV Laplacian on C p L ˝ BV r sq , re-spectively, and W (cid:126) qBV is of the form (4.37) but with µ ˝ ,L “ “ . Altogether, we obtain p L ˝ BV r s , Q ˝ qBV q which corresponds to a quantum L -structure on H ‚ µ ,L “ p L BV q with adifferential that vanishes to zeroth order in (cid:126) .The quantum BV action W (cid:126) qBV is the action that encodes all scattering amplitudesto arbitrary loop order in perturbation theory. In particular, for theories for which theclassical BV action also satisfies the quantum master equation, which includes scalar fieldtheory, Yang–Mills theory, and Chern–Simons theory, L coincides with the loop expansionorder and hence, the products µ ˝ n ´ ,L are the L -loop integrands for the n -point scatteringamplitude. Consequently, (4.45) generalises to A n,L p a ˝ , . . . , a ˝ n q “ i x a ˝ , µ ˝ n ´ ,L p a ˝ , . . . , a ˝ n qy . (4.50)To construct the µ i,L , we note that (4.48) immediately implies E “ E ´ E ˝ δ ˝ H (4.51) One should not confuse the quantum BV action with the one-particle-irreducible effective action orthe Wilsonian effective action, even though it has the form of (cid:126) -corrections to the classical action. E . Hence, we can iterate this equation to obtain E recursivelyand substitute the result into Q ˝ qBV “ E ˝ δ ˝ P from (4.48) with P given in (4.41c). Weconclude, in analogy with (4.39), that Q ˝ qBV a ˝ “ ´ ÿ i ě L ě (cid:126) L i ! µ i,L p a ˝ , . . . , a ˝ q , (4.52)from which the µ i,L and thus the µ ˝ i,L can be read off. We refer to [201, 210] for full details.It is not difficult to see that for (cid:126) Ñ , the recursion relation (4.51) coincides with therecursion relation (4.44a) and (4.52) with that for the products (4.44b) for the minimalmodel at the tree level.
5. Examples of homotopy algebras of field theories
In the following, we review the actions, the BV complexes and the dual L -algebra struc-tures of the field theories relevant to our homotopy algebraic treatment of the double copy.We note that none of the theories we discuss in this section requires the BV formalism forquantisation. As explained before, however, it does make the link to homotopy algebrasevident and clarifies the freedom we have in choosing gauges, an important aspect in ourlater discussion. We start with the simplest relevant field theory, namely that of a biadjoint scalar field theorywith cubic interaction. This theory appeared in the scattering amplitudes and double copyliterature in various incarnations [271, 272, 154, 83, 158, 111, 49, 273, 112, 274, 50, 114, 115, 89,38, 275].In particular, let g and ¯ g be two semi-simple compact matrix Lie algebras. For p g b ¯ g q -valued functions on Minkowski space M d , we define a symmetric bracket and an innerproduct by linearly extending r e b ¯ e , e b ¯ e s g b ¯ g : “ r e , e s g b r ¯ e , ¯ e s ¯ g , x e b ¯ e , e b ¯ e y g b ¯ g : “ tr g p e e q tr ¯ g p ¯ e ¯ e q (5.1)for all e , P g and ¯ e , P ¯ g . BV action and BV operator.
The BV action for biadjoint scalar field theory thenreads as S biadj : “ ż d d x ! x ϕ, l ϕ y g b ¯ g ´ λ x ϕ, r ϕ, ϕ s g b ¯ g y g b ¯ g ) , (5.2)62here λ is a coupling constant, l : “ η µν B µ B ν , and ϕ is a scalar field taking values in g b ¯ g .We write ϕ P p g b ¯ g q b F where F is a suitable function space discussed shortly. Introducingbasis vectors e a and ¯ e ¯ a on g and ¯ g , respectively, we can rewrite this action in componentform S biadj “ ż d d x ! ϕ a ¯ a l ϕ a ¯ a ´ λ f abc f ¯ a ¯ b ¯ c ϕ a ¯ a ϕ b ¯ b ϕ c ¯ c ) , (5.3a)where tr g p e a e b q “ ´ δ ab , tr ¯ g p ¯ e ¯ a ¯ e ¯ b q “ ´ δ ¯ a ¯ b ,f abc : “ ´ tr g p e a r e b , e c s g q , ¯ f ¯ a ¯ b ¯ c : “ ´ tr ¯ g p ¯ e ¯ a r ¯ e ¯ b , ¯ e ¯ c s ¯ g q . (5.3b)Besides the field ϕ , we also have the anti-field ϕ ` and the BV operator (4.18) acts accordingto ϕ a ¯ a Q BV ÞÝÝÝÑ and ϕ ` a ¯ a Q BV ÞÝÝÝÑ l ϕ a ¯ a ´ λ f bca f ¯ b ¯ c ¯ a ϕ b ¯ b ϕ c ¯ c . (5.4) L -algebra. The BV operator (5.4) is the Chevalley–Eilenberg differential of an L -algebra L biadjBV which has the underlying chain complex ˚ ϕ a ¯ a p g b ¯ g q b F looooomooooon L biadjBV , ϕ ` a ¯ a p g b ¯ g q b F looooomooooon L biadjBV , ˚ l (5.5)with cyclic inner product x ϕ, ϕ ` y : “ ż d d x ϕ a ¯ a ϕ ` a ¯ a , (5.6)and the only non-trivial higher product is p ϕ a ¯ a , ϕ b ¯ b q µ ÞÝÝÑ ´ λf bca f ¯ b ¯ c ¯ a ϕ b ¯ b ϕ c ¯ c . (5.7)At this point it is important to recall Remark 4.1 and that we always use the same symbolfor a coordinate function on field space and the corresponding elements of field space.The field space F can roughly be thought of as the smooth functions of Minkowski space C p M d q . More precisely, however, the field space is the direct sum of interacting fields andsolutions to the (colour-stripped) equations of motion, cf. Remark 4.2. A key player in the double copy is Yang–Mills theory on d -dimensional Minkowski space M d with a semi-simple compact matrix Lie algebra g as gauge algebra. The gauge potential A aµ is a one-form on M d taking values in g . Let ∇ be the connection with respect to A .Infinitesimal gauge transformations act according to A aµ ÞÑ ˜ A aµ : “ A aµ ` p ∇ µ c q a for all c P C p M d q b g . (5.8)63 V action and BV operator.
The list of all the fields required in the BV formulationof Yang–Mills theory together with their properties is found in Table 5.1, and the BV actionis [166] S YMBV : “ ż d d x ! ´ F aµν F aµν ` A ` aµ p ∇ µ c q a ` g f bca c ` a c b c c ´ b a ¯ c ` a ) . (5.9)As in Section 4.1, all the fields are rescaled such that the Yang–Mills coupling constant g appears in all interaction vertices. Consequently, the BV operator (4.18) acts as c a Q BV ÞÝÝÝÑ ´ g f bca c b c c , c ` a Q BV ÞÝÝÝÑ ´p ∇ µ A ` µ q a ´ gf bca c b c ` c ,A aµ Q BV ÞÝÝÝÑ p ∇ µ c q a , A ` aµ Q BV ÞÝÝÝÑ p ∇ ν F νµ q a ´ gf bca A ` bµ c c ,b a Q BV ÞÝÝÝÑ , b ` a Q BV ÞÝÝÝÑ ´ ¯ c ` a , ¯ c a Q BV ÞÝÝÝÑ b a , ¯ c ` a Q BV ÞÝÝÝÑ . (5.10) fields anti-fieldsrole | ´ | gh | ´ | L dim | ´ | gh | ´ | L dim c a ghost field 1 0 d ´ c ` a ´ d ` A aµ physical field 0 1 d ´ A ` aµ ´ d ` b a Nakanishi–Lautrup field 0 1 d b ` a ´ d ¯ c a anti-ghost field ´ d ¯ c ` a d Table 5.1: The full set of BV fields for Yang–Mills theory on M d with gauge Lie algebra g , including their ghost numbers, their L -degrees, and their mass dimensions. The massdimension of the coupling constant g is ´ d . L -algebra. The BV operator (5.10) is the Chevalley–Eilenberg differential of an L -algebra which we shall denote by L YMBV . This L -algebra has the underlying complex This complex has been rediscovered several times in the literature. For early references, see [185, 187];more detailed historical references are found in [177]. aµ Ω p M d q b g A ` aµ Ω p M d q b g b a C p M d q b g b ` a C p M d q b g c a C p M d q b g loooooomoooooon “ : L YMBV , ¯ c ` a C p M d q b g loooooomoooooon “ : L YMBV , ¯ c a C p M d q b g loooooomoooooon “ : L YMBV , c ` a C p M d q b g loooooomoooooon “ : L YMBV , ´pB ν B µ ´ δ µν l q ´B µ id ´B µ ´ id (5.11a)We shall label the subspaces L YMBV , i to which the various fields belong by the correspondingsubscripts, that is, L YMBV , “ L YMBV , , c , L YMBV , “ à φ P p
A, b, ¯ c ` q L YMBV , , φ , L YMBV , “ à φ P p A ` , b ` , ¯ c q L YMBV , , φ , L YMBV , “ L YM3 , c ` , (5.11b)and the non-trivial actions of the differential µ in L YMBV , i are c a µ ÞÝÝÑ ´B µ c a P L YMBV , , A , ¨˚˚˝ A aµ b a ¯ c ` a ˛‹‹‚ µ ÞÝÝÑ ¨˚˚˝ ´pB µ B ν ´ δ νµ l q A aν ´ ¯ c ` a b a ˛‹‹‚ P à φ P p A ` , b ` , ¯ c q L YMBV , , φ ,A ` aµ µ ÞÝÝÑ ´B µ A ` aµ P L YMBV , , c ` . (5.11c)The non-vanishing higher products are p c a , c b q µ ÞÝÝÑ gf bca c b c c P L YMBV , , c , p A aµ , c b q µ ÞÝÝÑ ´ gf bca A bµ c c P L YMBV , , A , p A ` aµ , c b q µ ÞÝÝÑ ´ gf bca A ` bµ c c P L YMBV , , A ` , p A aµ , A bν q µ ÞÝÝÑ gf bca ´ B ν p A bν A cµ q ` A bν B r ν A cµ s ¯ P L YMBV , , A ` , p c a , c ` b q µ ÞÝÝÑ gf bca c b c ` c P L YMBV , , c ` , p A aµ , A ` bν q µ ÞÝÝÑ gf bca A bµ A ` cµ P L YMBV , , c ` , p A aµ , A bν , A cκ q µ ÞÝÝÑ g A νc A dν A eµ f edb f bca P L YMBV , , A ` , (5.11d)65nd the general expressions follow from polarisation. One can check that p L YMBV , µ i q formsan L -algebra, and with the inner products x A, A ` y : “ ż d d x A aµ A ` µa , x b, b ` y : “ ż d d x b a b ` a , x c, c ` y : “ ż d d x c a c ` a , x ¯ c, ¯ c ` y : “ ´ ż d d x ¯ c a ¯ c ` a , (5.12)it becomes a cyclic L -algebra. Note that the superfield homotopy Maurer–Cartan ac-tion (4.26a) reduces to the BV action (5.9) when using these higher products and innerproducts together with (4.27). Gauge fixing.
We have discussed the general gauge-fixing procedure in the BV formalismin Section 4.1. Here, to implement R ξ -gauge for some real parameter ξ , we choose thegauge-fixing fermion Ψ : “ ´ ż d d x ¯ c a ` B µ A aµ ` ξ b a ˘ . (5.13)Following (4.30) and (4.33), the Lagrangian of the resulting gauge-fixed BV action is S YM , gfBV “ ż d d x ! ´ F aµν F aµν ´ ¯ c a B µ p ∇ µ c q a ` ξ b a b a ` b a B µ A aµ `` A ` aµ p ∇ µ c q a ` g f bca c ` a c b c c ´ b a ¯ c ` a ) , (5.14)and after putting to zero the anti-fields, we obtain S YMBRST “ ż d d x ! ´ F aµν F aµν ´ ¯ c a B µ p ∇ µ c q a ` ξ b a b a ` b a B µ A aµ ) . (5.15)This is precisely the action appearing in (4.1a). The next theory which we would like to discuss is that of a free two-form gauge potential B P Ω p M d q . It has a three-form curvature given by H µνκ : “ B µ B νκ ` B ν B κµ ` B κ B µν P Ω p M d q (5.16)and transforms under the infinitesimal gauge transformations as B µν ÞÑ ˜ B µν : “ B µν ` B µ Λ ν ´ B ν Λ µ , (5.17)where Λ P Ω p M d q is the one-form gauge parameter. Note that the gauge parametersthemselves transform under a higher gauge symmetry , Λ µ ÞÑ ˜Λ µ : “ Λ µ ` B µ λ , (5.18)where λ P C p M d q is the (scalar) higher gauge parameter.66 elds anti-fieldsrole | ´ | gh | ´ | L dim | ´ | gh | ´ | L dim λ ghost–for–ghost field 2 ´ d ´ λ ` ´ d ` µ ghost field 1 0 d ´ ` µ ´ d ` γ trivial pair partner of ε d ´ γ ` ´ d ` B µν physical field 0 1 d ´ B ` µν ´ d ` α µ Nakanishi–Lautrup field 0 1 d α ` µ ´ d ε trivial pair partner of γ d ´ ε ` ´ d ` µ anti-ghost field ´ d ¯Λ ` µ d ¯ γ trivial pair partner of ¯ λ ´ d ` γ ` d ´ λ trivial pair partner of ¯ γ ´ d ` λ ` d ´ Table 5.2: The full set of BV fields for the free Kalb–Ramond field, including their ghostnumbers, their L -degrees, and their mass dimension. Besides the physical field, the ghostfield, and ghost–for–ghost field, we also introduced trivial pairs p α, ¯Λ q , p γ, ε q , and p ¯ γ, ¯ λ q together with their anti-fields. BV action and BV operator.
The full set of fields required for gauge fixing in the BVformalism is given by what is known as the
Batalin–Vilkovisky triangle [167], see also [177]for a recent review in the notation used here. The complete list of BV fields is given inTable 5.2. Following the discussion of [167], the BV action reads as S KRBV : “ ż d d x ! ´ H µνκ H µνκ ` B ` µν B µ Λ ν ´ Λ ` µ B µ λ ´ ¯Λ ` µ α µ ` ¯ λ ` ¯ γ ` ε ` γ ) , (5.19)where the factor of two has been introduced for later convenience. Consequently, the BVoperator acts (4.18) as λ Q BV ÞÝÝÝÑ , λ ` Q BV ÞÝÝÝÑ B µ Λ ` µ , Λ µ Q BV ÞÝÝÝÑ B µ λ , Λ ` µ Q BV ÞÝÝÝÑ ´ B ν B ` νµ ,γ Q BV ÞÝÝÝÑ , γ ` Q BV ÞÝÝÝÑ ε ` ,B µν Q BV ÞÝÝÝÑ B µ Λ ν ´ B ν Λ µ , B ` µν Q BV ÞÝÝÝÑ B κ H κµν ,α µ Q BV ÞÝÝÝÑ , α ` µ Q BV ÞÝÝÝÑ ¯Λ ` µ ,ε Q BV ÞÝÝÝÑ γ , ε ` Q BV ÞÝÝÝÑ , ¯Λ µ Q BV ÞÝÝÝÑ α µ , ¯Λ ` µ Q BV ÞÝÝÝÑ , ¯ γ Q BV ÞÝÝÝÑ , ¯ γ ` Q BV ÞÝÝÝÑ ¯ λ ` , ¯ λ Q BV ÞÝÝÝÑ ¯ γ , ¯ λ ` Q BV ÞÝÝÝÑ . (5.20)67 -algebra. The BV operator (5.20) is the Chevalley–Eilenberg differential of an L -algebra L KRBV , which has the underlying complex λ C p M d q Λ µ Ω p M d q B µν Ω p M d q B ` µν Ω p M d q Λ ` µ Ω p M d q λ ` C p M d q ¯Λ ` µ Ω p M d q ¯Λ µ Ω p M d q α µ Ω p M d q α ` µ Ω p M d q γ C p M d q ε C p M d q ε ` C p M d q γ ` C p M d q looomooon “ : L KR ´ ¯ λ ` C p M d q looomooon “ : L KRBV , ¯ γ ` C p M d q looomooon “ : L KRBV , ¯ γ C p M d q looomooon “ : L KRBV , ¯ λ C p M d q looomooon “ : L KRBV , looomooon “ : L KRBV , ´B µ B r ν µ B ν ´B µ id ´ idid ´ idid ´ id (5.21a)with L KRBV , ´ “ L KRBV , ´ , λ , L KRBV , “ à φ P p Λ ,γ , ¯ λ ` q L KRBV , , φ , L KRBV , “ à φ P p B, ¯Λ ` , α, ε, ¯ γ ` q L KRBV , , φ , L KRBV , “ à φ P p B ` , ¯Λ , α ` , ε ` , ¯ γ q L KRBV , , φ , L KRBV , “ à φ P p Λ ` , γ ` , ¯ λ q L KRBV , , φ , L KRBV , “ L KRBV , , λ ` , (5.21b)and the non-vanishing action of the differential µ given by λ µ ÞÝÝÑ ´B µ λ P L KRBV , , Λ , ¨˚˚˝ Λ µ γ ¯ λ ` ˛‹‹‚ µ ÞÝÝÑ ¨˚˚˝ ´ B r µ Λ ν s γ ¯ λ ` ˛‹‹‚ P à φ P p
B, ε, ¯ γ ` q L KRBV , , φ , ¨˚˚˝ B µν ¯Λ ` µ α µ ˛‹‹‚ µ ÞÝÝÑ ¨˚˚˝ B κ H κµν α µ ´ ¯Λ ` µ ˛‹‹‚ P à φ P p B ` , ¯Λ , α ` q L KRBV , , φ , ¨˚˚˝ B ` µν ε ` ¯ γ ˛‹‹‚ µ ÞÝÝÑ ¨˚˚˝ B ν B ` µν ´ ε ` ´ ¯ γ ˛‹‹‚ P à φ P p Λ ` , γ ` , ¯ λ q L KRBV , , φ , Λ ` µ µ ÞÝÝÑ ´B µ Λ ` µ P L KRBV , , λ ` , (5.21c)68here are no higher products because the theory is free. The L -algebra L KRBV becomescyclic upon introducing x λ, λ ` y : “ ´ ż d d x λλ ` , x ¯ λ, ¯ λ ` y : “ ´ ż d d x ¯ λ ¯ λ ` , x Λ , Λ ` y : “ ż d d x Λ µ Λ ` µ , x ¯Λ , ¯Λ ` y : “ ´ ż d d x ¯Λ µ ¯Λ ` µ , x B, B ` y : “ ż d d x B µν B ` µν , x α, α ` y : “ ż d d x α µ α ` µ , x ε, ε ` y : “ ż d d x εε ` , x γ, γ ` y : “ ż d d x γγ ` , x ¯ γ, ¯ γ ` y : “ ´ ż d d x ¯ γ ¯ γ ` . (5.22)Again, the superfield homotopy Maurer–Cartan action (4.26a) of L KRBV with higherproducts (4.27) is the BV action (5.19).
Gauge fixing.
Recall the general gauge-fixing procedure in the BV formalism from Sec-tion 4.1. The most general Lorentz covariant linear gauge choices are implemented by thegauge-fixing fermion
Ψ : “ ´ ż d d x ! ¯Λ ν ` B µ B µν ` ζ α ν ˘ ´ ¯ λ ` B µ Λ µ ` ζ γ ˘ ` ε ` B µ ¯Λ µ ` ζ ¯ γ ˘) (5.23)for some real parameters ζ , , . The resulting gauge-fixed action (after putting to zero theanti-fields) is S KRBRST “ ż d d x ! B µν l B µν ` pB µ B µν qpB κ B κν q ´ ¯Λ µ l Λ µ ´´ pB µ ¯Λ µ qpB ν Λ ν q ´ ¯ λ l λ ` ζ α µ α µ ` α ν B µ B µν `` ε B µ α µ ´ p ζ ` ζ q ¯ γγ ` γ B µ ¯Λ µ ´ ¯ γ B µ Λ µ ) . (5.24) The fourth relevant theory is Einstein–Hilbert gravity on a d -dimensional Lorentzian mani-fold M d with metric g P Γ p M d , d T ˚ M d q . Let ∇ be the Levi–Civita connection for g .Recall that infinitesimal gauge transformations of the metric are parametrised by a vectorfield χ and act as g µν ÞÑ ˜ g µν : “ g µν ` p L χ g q µν , (5.25)where L χ denotes the Lie derivative along χ .69 V action and BV operator.
The list of all the fields required in the BV formulationof Einstein–Hilbert gravity together with their properties is found in Table 5.3 and the BVaction (cf. e.g. [276] or [277] for the gauge-fixed version) is S EHBV : “ ż d d x ! ´ κ ?´ g R ` g ` µν p L χ g q µν ` χ ` µ p L χ χ q µ ´ (cid:37) µ ¯ χ ` µ ) , (5.26)where R denotes the Ricci scalar and κ “ πG p d q N Einstein’s gravitational constant.Consequently, the BV operator (4.18) acts as χ µ Q BV ÞÝÝÝÑ ´ p L χ χ q µ , χ ` µ Q BV ÞÝÝÝÑ ´ ∇ ν g ` νµ ` p L χ χ ` q µ ,g µν Q BV ÞÝÝÝÑ p L χ g q µν , g ` µν Q BV ÞÝÝÝÑ ´ κ ?´ g ` R µν ´ g µν R ˘ ` p L χ g ` q µν ,(cid:37) µ Q BV ÞÝÝÝÑ , (cid:37) ` µ Q BV ÞÝÝÝÑ ´ ¯ χ µ , ¯ χ µ Q BV ÞÝÝÝÑ (cid:37) µ , ¯ χ ` µ Q BV ÞÝÝÝÑ , (5.27)where R µν is the Ricci tensor. fields anti-fieldsrole | ´ | gh | ´ | L dim | ´ | gh | ´ | L dim χ µ ghost field 1 0 ´ χ ` µ ´ d ` g µν physical field 0 1 0 g ` µν ´ d(cid:37) µ Nakanishi–Lautrup field 0 1 d (cid:37) ` µ ´ d ¯ χ µ anti-ghost field ´ d ¯ χ ` µ d Table 5.3: The full set of BV fields for Einstein–Hilbert gravity, including their ghostnumbers, their L -degrees, and their mass dimensions. The mass dimension of the couplingconstant κ is ´ d . Note that all fields are tensors and all anti-fields are tensor densities. Perturbation theory.
Let us now restrict to a Lorentzian manifold M d for which themetric can be seen as a fluctuation h µν about the Minkowski metric η µν on M d , that is, g µν “ : η µν ` κh µν . (5.28a)For future reference, we note that g µν “ η µν ´ κh µν ` κ h µρ h ρν ´ κ h µρ h ρσ h σν ` O p κ q , (5.28b)where h µν : “ η νλ h µλ and h µν : “ η µκ η νλ h κλ . Likewise, ?´ g “ ` κ ˚ h ` κ ` ˚ h ´ h µν h ν µ ˘ `` κ ` ˚ h ´ ˚ hh µν h ν µ ` h µν h ν ρ h ρµ ˘ ` O p κ q , (5.28c)70here ˚ h : “ η µν h µν .We also introduce the following rescaled anti-fields and unphysical fields: h ` µν : “ κ ?´ g g ` µν ,X µ : “ κ χ µ , X ` µ : “ κ ?´ g χ ` µ , ¯ X µ : “ ¯ χ µ , ¯ X ` µ : “ ?´ g ¯ χ ` µ ,(cid:36) µ : “ (cid:37) µ , (cid:36) ` µ : “ ?´ g (cid:37) ` µ . (5.29)In addition to these, we introduce two trivial pairs p β, δ q and p π, ¯ β q , together with thecorresponding anti-fields. These do not modify the physical observables; as we shall seelater, however, they do arise rather naturally in the double copy and are crucial once thedilaton enters. We also note that precisely these trivial pairs were also introduced in [278]in order to explain a unimodular gauge fixing in the BV formalism. The full list of fieldsand their properties is given in Table 5.4. fields anti-fieldsrole | ´ | gh | ´ | L dim | ´ | gh | ´ | L dim X µ ghost field 1 0 d ´ X ` µ ´ d ` β trivial pair partner of δ d ´ β ` ´ d ` h µν physical field 0 1 d ´ h ` µν ´ d ` (cid:36) µ Nakanishi–Lautrup field 0 1 d (cid:36) ` µ ´ d π trivial pair partner of ¯ β d ` π ` ´ d ´ δ trivial pair partner of β d ´ δ ` ´ d ` X µ anti-ghost field ´ d ¯ X ` µ d ¯ β trivial pair partner of π ´ d ` β ` d ´ Table 5.4: The full set of BV fields for perturbative Einstein–Hilbert gravity, including theirghost numbers, their L -degrees, and their mass dimension. All the fields are regarded astensors on Minkowski space.The action itself can now be expanded in orders of κ , S eEHBV “ ż d d x ?´ g ! ´ κ R ` ?´ g g ` µν ∇ µ χ ν `` ?´ g χ ` µ p L χ χ q µ ´ ?´ g (cid:36) µ ¯ χ ` µ ` βδ ` ` π ¯ β ` ) “ ż d d x ?´ g ! ´ κ R ` h ` µν ∇ µ X ν ` κ X ` µ p L X X q µ ´ (cid:36) µ ¯ χ ` µ ` βδ ` ` π ¯ β ` ) “ : ż d d x ÿ n “ κ n L eEH n (5.30)71ith indices now raised and lowered with the Minkowski metric. The lowest-order Lag-rangian L is given by the Fierz–Pauli Lagrangian plus the terms containing ghosts andother unphysical fields, L eEH0 “ ´ B µ h νρ B µ h νρ ` B µ h νρ B ν h µρ ´ B µ ˚ h B ν h µν ` B µ ˚ h B µ ˚ h `` h ` µν B µ X ν ´ (cid:36) µ ¯ X ` µ ` βδ ` ` π ¯ β ` , (5.31)cf. e.g. [279]. To first order in κ , we have L eEH1 “ ´ h µν ! B µ h ρσ B ν h ρσ ´ η µν B σ h τρ B σ h τρ ` B ν ˚ h ´ B ρ h µρ ´ B µ ˚ h ¯ `` B ν h µρ B ρ ˚ h ´ B ρ ˚ h B ρ h µν ´ η µν B ρ ˚ h ´ B σ h ρσ ´ B ρ ˚ h ¯ ` B ρ h µν B σ h ρσ ´´ B ν h ρσ B σ h µρ ´ B ρ h νσ B σ h µρ ` B σ h νρ B σ h µρ ` η µν B ρ h τσ B σ h τρ ) `` h ` µν ! h νλ B µ X λ ` pB µ h λν ` B λ h µν ´ B ν h µλ q X λ ) `` X ` µ p L X X q µ ` ˚ h p´ (cid:36) µ ¯ X ` µ ` βδ ` ` π ¯ β ` q . (5.32) L -algebra. The full L -algebra L eEHBV of Einstein–Hilbert gravity has the underlyingcomplex X µ Ω p M d q h µν Ω p M d q h ` µν Ω p M d q X ` µ Ω p M d q ¯ X ` µ Ω p M d q ¯ X µ Ω p M d q (cid:36) µ Ω p M d q (cid:36) ` µ Ω p M d q ¯ β ` C p M d q ¯ β C p M d q π C p M d q π ` C p M d q β C p M d q looomooon “ : L eEHBV , δ C p M d q looomooon “ : L eEHBV , δ ` C p M d q looomooon “ : L eEHBV , β ` C p M d q looomooon “ : L eEHBV , B p ν µ B ν id ´ id ´ ididid ´ id (5.33a)72ith L eEHBV , “ L eEHBV , , X ‘ L eEHBV , , β , L eEHBV , “ à φ P p h, ¯ X ` , (cid:36), ¯ β ` , π, δ q L eEHBV , , φ , L eEHBV , “ à φ P p h ` , ¯ X, (cid:36) ` , ¯ β, π ` , δ ` q L eEHBV , , φ , L eEHBV , “ L eEHBV , , X ` ‘ L eEHBV , , β ` . (5.33b)The L -algebra L eEHBV comes with non-trivial higher products of arbitrarily high order,with µ i encoding the Lagrangian L eEHBV , i ´ . Below, we merely list µ and µ to prepare ourdiscussion of the double copy later on. The differentials are ˜ X µ β ¸ µ ÞÝÝÑ ˜ ´B p µ X ν q β ¸ P à φ P p h, δ q L eEHBV , , φ , ¨˚˚˚˚˚˚˚˝ h µν ¯ X ` µ (cid:36) µ ¯ β ` π ˛‹‹‹‹‹‹‹‚ µ ÞÝÝÑ ¨˚˚˚˚˚˚˚˝“ p δ ρµ δ σν ´ η µν η ρσ q l ´p δ σν η µκ ´ δ σκ η µν qB ρ B κ ‰ h ρσ ´ (cid:36) µ ¯ X ` µ π ´ ¯ β ` ˛‹‹‹‹‹‹‹‚ P à φ P p h ` , ¯ X,(cid:36) ` , ¯ β, π ` q L eEHBV , , φ , ˜ h ` µν δ ` ¸ µ ÞÝÝÑ ˜ ´B ν h νµ ´ δ ` ¸ P à φ P p X ` , β ` q L eEHBV , , φ , (5.33c)and the cubic interactions are encoded in the binary products p X µ , X ν q µ ÞÝÝÑ p L X X q µ P L eEHBV , , X , p X µ , X ` ν q µ ÞÝÝÑ pB µ X ν q X ` ν ` B ν p X ν X ` µ q P L eEHBV , , X ` , p (cid:36), ¯ X ` µ q µ ÞÝÝÑ (cid:36) ρ ¯ X ` ρ η µν P L eEHBV , , h ` , p h µν , (cid:36) q µ ÞÝÝÑ ´ ˚ h(cid:36) µ P L eEHBV , , ¯ X , p h µν , ¯ X ` ρ q µ ÞÝÝÑ ˚ h ¯ X ` µ P L eEHBV , , (cid:36) ` , p β, δ ` q µ ÞÝÝÑ βδ ` η µν P L eEHBV , , h ` , p h µν , β q µ ÞÝÝÑ ˚ hβ P L eEH1 ,δ , p h µν , δ ` q µ ÞÝÝÑ ´ ˚ hδ ` P L eEHBV , , β ` , p π, ¯ β ` q µ ÞÝÝÑ π ¯ β ` η µν P L eEHBV , , h ` , p h µν , π q µ ÞÝÝÑ ´ ˚ hπ P L eEHBV , , ¯ β , p h µν , ¯ β ` q µ ÞÝÝÑ ˚ h ¯ β ` P L eEHBV , , π ` , p X µ , h νρ q µ ÞÝÝÑ ´ h νκ B µ X κ ´ pB µ h κν ` B κ h µν ´ B ν h µκ q X κ P L eEHBV , , h , h ` µν , h ρσ q µ ÞÝÝÑ ´ B κ p h ` κν h νµ q ` h ` κν pB κ h µν ` B µ h κν ´ B ν h κµ q P L eEHBV , , X ` , p h ` µν , X ρ q µ ÞÝÝÑ ´ h ` κµ B κ X ν ` B κ p h ` κν X µ q ` B κ p h ` µν X κ q ´ B κ p h ` µκ X ν q P L eEHBV , , h ` , p h µν , h ρσ q µ ÞÝÝÑ ! B µ h ρσ B ν h ρσ ´ η µν B σ h τρ B σ h τρ ` B ν ˚ h ´ B ρ h µρ ´ B µ ˚ h ¯ `` B ν h µρ B ρ ˚ h ´ B ρ ˚ h B ρ h µν ´ η µν B ρ ˚ h ´ B σ h ρσ ´ B ρ ˚ h ¯ `` B ρ h µν B σ h ρσ ´ B ν h ρσ B σ h µρ ´ B ρ h νσ B σ h µρ `` B σ h νρ B σ h µρ ` η µν B ρ h τσ B σ h τρ ) ``p Ø q P L eEHBV , , h ` , (5.33d)The cyclic structure is given by the following integrals: x X, X ` y : “ ż d d x X µ X ` µ , x ¯ X, ¯ X ` y : “ ´ ż d d x ¯ X µ ¯ X ` µ , x β, β ` y : “ ż d d x ββ ` , x ¯ β, ¯ β ` y : “ ´ ż d d x ¯ β ¯ β ` , x h, h ` y : “ ż d d x h µν h ` µν , x (cid:36), (cid:36) ` y : “ ż d d x (cid:36) µ (cid:36) ` µ , x π, π ` y : “ ż d d x ππ ` , x δ, δ ` y : “ ż d d x δδ ` . (5.34) Gauge fixing.
Gauge fixing proceeds as usual in the BV formalism, but due to our twoadditional trivial pairs, we can now write down a much more general gauge fixing fermion.We restrict ourselves to Ψ : “ ´ ż d d x ! ¯ X ν ` ζ B µ h µν ´ ζ (cid:36) ν ` ζ B ν ˚ h ´ ζ B ν δ ` ζ B ν π l ˘ `` ¯ β ` ζ ˚ h ´ ζ δ ` ζ B µ B ν h µν l ˘) , (5.35)and this is the freedom required for the discussion of the double copy. The resultingLagrangian, to lowest order in κ , reads as L eEH , gf0 “ h µν l h µν ` pB µ h µν q ` ˚ h B µ B ν h µν ´ ˚ h l ˚ h `` ζ (cid:36) ν B µ h µν ´ ζ (cid:36) µ (cid:36) µ ` ζ (cid:36) µ B µ ˚ h ´ ζ (cid:36) µ B µ δ ` ζ (cid:36) µ B µ π l ´´ πζ ˚ h ` ζ πδ ´ ζ π B µ B ν h µν l `` ζ pB µ ¯ X ν ` B ν ¯ X µ qB µ X ν ` ζ pB κ ¯ X κ q ´ ζ ¯ β B µ X µ ´ ζ B µ B ν ¯ β l B µ X ν `´ ζ β B ν ¯ X ν ´ ζ ¯ ββ , (5.36)after putting to zero the anti-fields. 74 .5. N “ supergravity The actions for the free Kalb–Ramond field and Einstein–Hilbert gravity are combined andcoupled to an additional scalar field ϕ , the dilaton, in N “ supergravity. This is thecommon, or universal, Neveu–Schwarz-Neveu–Schwarz sector of the α Ñ limit of closedstring theories, and the action reads as S N “ : “ ż d d x ?´ g ! ´ κ R ´ d ´ B µ ϕ B µ ϕ ´ e ´ κd ´ ϕ H µνκ H µνκ ) . (5.37)The solutions of the associated equations of motions give backgrounds (with vanishingcosmological constant) around which strings can be quantised to lowest order in α andstring coupling. They also ensure conformal invariance of the string is non-anomalous incritical dimensions.We note that the free part of N “ supergravity is a sum of the free Kalb–Ramondtwo form, Einstein–Hilbert gravity and a free scalar field. Therefore, the free parts of theBV formalism as well as the L -algebra description just add in a straightforward manner.The interaction terms then consist of the interaction terms of Einstein–Hilbert gravity aspresented in the previous section, as well as additional terms of arbitrary order involvingthe dilaton and the Kalb–Ramond two-form. These are readily read off (5.37), but theirexplicit form will not be of relevance to us.
6. Factorisation of homotopy algebras and colour ordering
A key point of our discussion of the double copy is the factorisation of the L -algebras ofYang–Mills theory and N “ supergravity into common factors. In this section, we discussthe basics of tensor products of homotopy algebras and introduce a twisted generalisationthat will prove to be the key to understanding the double copy from a homotopy algebraicperspective. Let
Ass , Com , and
Lie denote (thecategories of) associative, commutative, and Lie algebras, respectively. Schematically, wehave tensor products of the form b : Ass ˆ Ass Ñ Ass , b : Com ˆ Ass Ñ Ass , b : Ass ˆ Com Ñ Ass , b : Com ˆ Com Ñ Com , b : Com ˆ Lie Ñ Lie , b : Lie ˆ Com Ñ Lie . (6.1)In particular, let A and B be two algebras from this list for which there is a tensor product.The vector space underlying the tensor product algebra A b B is simply the ordinary tensor75roduct of vector spaces and the product m A b B is given by m A b B p a b b , a b b q : “ m A p a , a q b m B p b , b q (6.2)for a , a P A and b , b P B .On the other hand, the tensor product of two chain complexes p A , m A q and p B , m B q isdefined as the tensor product of the underlying (graded) vector spaces A and B , A b B “ à k P Z p A b B q k with p A b B q k : “ à i ` j “ k A i b B j , (6.3a)cf. (4.2). The differential m A b B is defined as m A b B p a b b q : “ m A p a q b b ` p´ q | a | A a b m B p b q (6.3b)for a P A and b P B .Strict A -, C -, and L -algebras are nothing but differential graded associative, com-mutative, and Lie algebras, respectively. For such algebras A and B , the above formulascombine to m A b B p a b b q : “ m A p a q b b ` p´ q | a | A a b m B p b q , m A b B p a b b , a b b q : “ p´ q | b | B | a | A m A p a , a q b m B p b , b q (6.4)for a , a P A and b , b P B . If, in addition, the two differential graded algebras carrycyclic inner products x´ , ´y A and x´ , ´y B , then the tensor product carries the cyclic innerproduct x a b b , a b b y A b B : “ p´ q | b | B | a | A ` s p| a | A `| a | A q x a , a y A x b , b y B (6.5)for a , a P A and b , b P B . Here, s : “ |x´ , ´y B | B is the degree of the inner product on B . Tensor products of general homotopy algebras.
There is a simple argument thatextends the above tensor product of strict homotopy algebras to general homotopy algebras,using not much more than the homological perturbation lemma. Let us therefore also brieflyconsider this case, even though we will only make use of it in passing when discussing colour-stripping of Yang–Mills amplitudes.An extension from the strict case to the general case can be performed as follows. Recallthat the strictification theorem asserts that every homotopy algebra is quasi-isomorphic to astrict homotopy algebra, see Appendix A.4 for details. Using this theorem, we first strictifyeach of the factors A and B in the tensor product A b B we wish to define. We thencompute the tensor product A st b B st of the strictified factors. This is a homotopy algebra76hose underlying chain complex Ch p A st b B st q is quasi-isomorphic to the tensor product Ch p A q b Ch p B q of the two differential complexes underlying the factors A and B . We canthen use the homological perturbation lemma, most readily in the form used e.g. in [201]for the coalgebra formulation of homotopy algebras, to transfer the full homotopy structurefrom Ch p A st b B st q to Ch p A q b Ch p B q along the quasi-isomorphism between the chaincomplexes. This yields a homotopy algebra structure on Ch p A q b Ch p B q together witha quasi-isomorphism to the tensor product of the strictified factors. We stress that thistransfer is not unique and depends on the choice of contracting homotopy (essentially, achoice of gauge).We stress that the fact that the tensor products (6.1) lift to corresponding tensorproducts of homotopy algebras is found in the literature for special cases, see e.g. [280, 281]for the case of A -algebras, as well as [282, Appendix B] for the case of tensor products of C -algebras with Lie algebras. Tensor products of matrix and Lie algebras with homotopy algebras.
To capturethe colour decomposition of amplitudes in Yang–Mills theory, it suffices to consider thetensor product between homotopy algebras and matrix (Lie) algebras. In particular, givena matrix algebra (or, more generally, an associative algebra) a and an A -algebra p A , m i q ,then the tensor product a b A is equipped with the higher products m a b A i p e b a , . . . , e i b a i q : “ e ¨ ¨ ¨ e i b m i p a , . . . , a i q (6.6)for all e , . . . , e i P a and a , . . . , a i P A and i P N ` . Evidently, these formulas can also beapplied to the tensor product between a matrix algebra and a C -algebra, however, theresult will, in general, be an A -algebra rather than a C -algebra as, for instance, thebinary product on the tensor product will not necessarily be graded commutative.Next, we may consider the tensor product g b C between a Lie algebra p g , r´ , ´sq and a C -algebra p C , m i q . We obtain an L -algebra p L , µ i q with L : “ g b C , however, the higherproducts µ i are less straightforward than the ones in (6.6) for A -algebras. Nevertheless,they can be computed iteratively, and we obtain for the lowest products µ p e b c q : “ e b m p c q ,µ p e b c , e b c q : “ r e , e s b m p c , c q , (6.7a) As detailed in (A.15), the graded anti-symmetrisation of any A -algebra yields an L -algebra, and sothe form of the higher products can be gleaned from the graded anti-symmetrisation of (6.6). µ p e b c , e b c , e b c q : “ r e , r e , e ss b m p c , c , c q ´´ p´ q | c | C | c | C r e , r e , e ss b m p c , c , c q `` p´ q | c | C | c | C rr e , e s , e s b m p c , c , c q ,µ p e b c , e b c , e b c , e b c q : “ r e , r e , r e , e sss b m p c , c , c , c q ´´ p´ q | c | C | c | C r e , r e , r e , e sss b m p c , c , c , c q ´´ p´ q | c | C | c | C r e , r e , r e , e sss b m p c , c , c , c q `` p´ q | c | C p| c | C `| c | C q rrr e , e s , e s , e s b m p c , c , c , c q ´´ p´ q p| c | C `| c | C q| c | C rr e , r e , e ss , e s b m p c , c , c , c q ´´ p´ q | c | C p| c | C `| c | C q`| c | C | c | C rrr e , e s , e s , e s b m p c , c , c , c q ... (6.7b)for all e , . . . , e P g and c , . . . , c P C . We list these formulas here as they are usefulin colour-stripping in Yang–Mills theory and we have not been able to find them in theliterature. As an example of the above factorisations, let us discuss colour-stripping in Yang–Millstheory and show that this is nothing but a factorisation of homotopy algebras. For con-creteness, let us consider the gauge-fixed action (5.14) and the corresponding L -algebra L YM , gfBV .If the gauge Lie algebra g is a matrix Lie algebra, then the L -algebra L YM , gfBV is thetotal anti-symmetrisation via (A.15) of a family of A -algebras. One of these is special inthat it is totally graded anti-symmetric [201] and thus is also a C -algebra.More generally, we can factorise L YM , gfBV into a gauge Lie algebra g and a colour C -algebras C YM , gfBV using formula (6.6), L YM , gfBV “ g b C YM , gfBV (6.8)78xplicitly, the C -algebra C YM , gfBV has the underlying chain complex A µ Ω p M d q A ` µ Ω p M d q b C p M d q b ` C p M d q c C p M d q looomooon “ : C YM , gfBV , ¯ c ` C p M d q looomooon “ : C YM , gfBV , ¯ c C p M d q looomooon “ : C YM , gfBV , c ` C p M d q looomooon “ : C YM , gfBV , ´pB ν B µ ´ δ µν l q´B µ ´B µ ξ B µ ´B µ ´ l ´ l (6.9a)where we label subspaces again by the fields parametrising them C YM , gfBV , “ C YM , gfBV , , c , C YM , gfBV , “ à φ P p
A, b, ¯ c ` q C YM , gfBV , , φ , C YM , gfBV , “ à φ P p A ` , b ` , ¯ c q C YM , gfBV , , φ , C YM , gfBV , “ C YM , gfBV , , c ` . (6.9b)The non-trivial actions of the differential m are c m ÞÝÝÑ ¨˚˚˝ ´B µ c ´ l c ˛‹‹‚ P à φ P p
A, b, ¯ c ` q C YM , gfBV , , φ , ¨˚˚˝ A µ b ¯ c ` ˛‹‹‚ m ÞÝÝÑ ¨˚˚˝ ´pB µ B ν ´ δ νµ l q A ν ´ B µ b B µ A µ ` ξb ˛‹‹‚ P à φ P p A ` , b ` , ¯ c q C YM , gfBV , , φ , ¨˚˚˝ A ` µ b ` ¯ c ˛‹‹‚ m ÞÝÝÑ ´B µ p A ` µ ` B µ ¯ c q P C YM , gfBV , , c ` , (6.9c)79he binary product m acts as p c , c q m ÞÝÝÑ gc c P C YM , gfBV , , c , ¨˚˚˝ c, ¨˚˚˝ A µ b ¯ c ` ˛‹‹‚˛‹‹‚ m ÞÝÝÑ g ¨˚˚˝ ´ cA µ ´B µ p cA µ q ˛‹‹‚ P à φ P p
A, b, ¯ c ` q C YM , gfBV , , φ , ¨˚˚˝ c, ¨˚˚˝ A ` µ ¯ cb ` ˛‹‹‚˛‹‹‚ m ÞÝÝÑ g ¨˚˚˝ c p A ` µ ` B µ ¯ c q ˛‹‹‚ P à φ P p A ` , b ` , ¯ c q C YM , gfBV , , φ , p c, c ` q m ÞÝÝÑ gcc ` P C YM , gfBV , , c ` , ¨˚˚˝¨˚˚˝ A µ b ¯ c ` ˛‹‹‚ , ¨˚˚˝ A ` ν ¯ cb ` ˛‹‹‚˛‹‹‚ m ÞÝÝÑ gA µ p A ` µ ` B µ ¯ c q P C YM , gfBV , , c ` , ¨˚˚˝¨˚˚˝ A µ b ¯ c ` ˛‹‹‚ , ¨˚˚˝ A ν b ¯ c ` ˛‹‹‚˛‹‹‚ m ÞÝÝÑ g ¨˚˚˝ B ν p A r ν A µ s q ` A ν B r ν A µ s ´ B r ν A µ s A ν ˛‹‹‚ P à φ P p A ` , b ` , ¯ c q C YM , gfBV , , φ , (6.9d)and the ternary product m acts as ¨˚˚˝¨˚˚˝ A µ b ¯ c ` ˛‹‹‚ , ¨˚˚˝ A ν b ¯ c ` ˛‹‹‚ , ¨˚˚˝ A κ b ¯ c ` ˛‹‹‚˛‹‹‚ m ÞÝÝÑ g ¨˚˚˝ A ν A r µ A ν s ´ A r µ A ν s A ν ˛‹‹‚ P à φ P p A ` ,b ` , ¯ c q C YM , gf2 , φ . (6.9e)It is a straightforward exercise to check that these higher products do indeed satisfy the C -algebra relations (A.1) and (A.10).The factorisation (6.8) descends to the minimal model L YM , gf ˝ BV , L YM , gf ˝ BV “ g b C YM , gf ˝ BV , (6.10)and the higher products in the C -algebra C YM , gf ˝ describes the colour-ordered tree-levelscattering amplitudes. We set A n, p , . . . , n q “ : i ÿ σ P S n { Z n tr p e a σ p q ¨ ¨ ¨ e a σ p n q q A n, p σ p q , . . . , σ p n qq , (6.11)80nd we have the formula A n, p , . . . , n q “ x n, m ˝ n ´ p , . . . , n ´ qy , (6.12)where the numbers , . . . , n represent the external free fields. The symmetry of the colour-stripped amplitude is reflected in the graded anti-symmetry of the higher products m ˝ i inthe C -algebra C YM , gf ˝ , because all fields are of degree one. The factorisation of the L -algebras corresponding to the field theories involved in thedouble copy is a twisted factorisation and we define our notion of twisted tensor productsin the following. Chain complexes.
A graded vector space is a particular example of a chain complex withtrivial differential. In our situation, we would like the vector space to act as an Abelian Liealgebra on the chain complex. We therefore generalise the usual tensor product as follows.Given a graded vector space V together with a chain complex p A , m q , we define a twistdatum τ to be a linear map τ : V Ñ V b End p A q ,v ÞÑ τ p v q : “ ÿ π τ π, p v q b τ π, p v q , (6.13)where the index π labels the summands in the twist element τ p v q . Given a twist datum τ , the twisted differential is defined by m τ p v b a q : “ ÿ π p´ q | τ π, p v q| V τ π, p v q b m p τ π, p v qp a qq (6.15)for v b a P V b A . This rather cumbersome formula describes a rather simple procedureand it will become fully transparent in concrete examples. Evidently, there are constraintson admissible twist data. Firstly, m τ has to be differential and satisfy m τ ˝ m τ “ , (6.16)and secondly, m τ has to be cyclic with respect to the inner product (6.5) on the tensorproduct V b A . We note that as it stands, the twisted tensor product is not necessarily In Sweedler notation, popular e.g. in the context of Hopf algebras, we would simply write τ p v q : “ τ p q p v q b τ p q p v q . (6.14) Example 6.1.
Consider the graded vector space V and the chain complex p A , m q definedby V : “ M d ‘ R looomooon “ : V and A : “ ` C p M d q looomooon “ : A id ÝÝÑ C p M d q looomooon “ : A ˘ . (6.17) For a basis p v µ , n q of M d ‘ R , a choice of twist datum is given by τ p v µ q : “ b and τ p n q : “ v µ b BB x µ . (6.18) The complex V b τ A with the twisted differential m τ is then V b τ A “ ¨˚˚˚˝ Ω p M d q – M d b C p M d q Ω p M d q “ M d b C p M d q‘ ‘ C p M d q – R b C p M d q C p M d q “ R b C p M d q d ˛‹‹‹‚ (6.19) Hence, we obtain a description of the chain complex p C p M d q ‘ Ω p M d q , d q , albeit withsome amount of redundancy. Differential graded algebras.
Twisted tensor products for unital algebras were dis-cussed in various places in the literature, e.g., in [283]. We would like to twist the ordinarytensor product of differential graded algebras introduced in Section 6.1, by extending thenotion of twist datum from chain complexes as follows. Given a graded vector space V anda differential graded algebra p A , m , m q , a twist datum is a pair of maps, one linear andthe other one bilinear, τ : V Ñ V b End p A q ,v ÞÑ τ p v q : “ ÿ π τ π, p v q b τ π, p v q , (6.20a)and τ : V b V Ñ V b End p A q b End p A q ,v b v ÞÑ τ p v , v q : “ ÿ π τ π, p v , v q b τ π, p v , v q b τ π, p v , v q , (6.20b)82here we again label summands in the tensor product by π . The twisted tensor producthas then higher maps m τ p v b a q : “ ÿ π p´ q | τ π, p v q| V τ π, p v q b m p τ π, p v qp a qq , m τ p v b a , v b a q : “ : “ p´ q | v | V | a | A ÿ π τ π, p v , v q b m p τ π, p v , v qp a q , τ π, p v , v qp a qq . (6.21)Note that in general, one may want to insert an additional sign p´ q | τ π, p v ,v q| V | a | A intothis equation; all our twist, however, satisfy | τ π, p v , v q| V “ .Clearly, not every twist datum leads to a valid homotopy algebra, and just as in the caseof chain complexes, one has to check that this works for a given twist by hand. We also notethat the twist datum relevant for the double copy will be able to mix types of homotopyalgebras, that is, for A an L -algebra, we obtain a C -algebra and for A a C -algebra, weobtain again an L -algebra.Altogether, our twisted tensor products are a way of factorising strict homotopy algebrasin a unique fashion as necessary for the double copy. However, it remains to be seen if ourconstruction in its present form is mathematically interesting in a wider context.
7. Factorisation of free field theories and free double copy
In this section, we factorise the chain complexes of the L -algebras of biadjoint scalar fieldtheory, Yang–Mills theory, and N “ supergravity into common factors. This exposes thefactorisation of field theories underlying the double copy at the linearised level. Summary.
Recall that the unary product µ in any L -algebra is a differential. Con-sequently, any L -algebra p L , µ i q naturally comes with an underlying chain complex Ch p L q : “ ` ¨ ¨ ¨ L L L L ¨ ¨ ¨ ˘ . µ µ µ µ µ (7.1)In an L -algebra corresponding to a field theory, the chain complex Ch p L q is the L -algebraof the free theory with all coupling constants put to zero. In each factorisation, we thusexpose the field content as well as the free fields that parametrise the theory’s scatteringamplitudes.We will obtain the following factorisations of chain complexes isomorphic to the chaincomplexes underlying the L -algebras of biadjoint scalar field theory, Yang–Mills theory in83 ξ -gauge, and gauge-fixed N “ supergravity: Ch p L biadjBRST q “ Ch p ˜ L biadjBRST q “ g b p ¯ g b Ch p Scal qq , Ch p L YMBRST q – Ch p ˜ L YMBRST q “ g b p Kin b τ Ch p Scal qq , Ch p L N “ q – Ch p ˜ L N “ q “ Kin b τ p Kin b τ Ch p Scal qq , (7.2)where g and ¯ g are semi-simple compact matrix Lie algebras corresponding to the colourfactors, Kin is a graded vector space and
Scal is the L -algebra of a scalar field theory. Wesee that the chain complex Ch p ˜ L N “ q is fully determined by the factorisation of Ch p ˜ L YMBRST q ,which is nothing but the double copy at the linearised level.There are two points to note concerning the factorisations of all those field theoriesbut that of biadjoint scalar field theory. Firstly, these factorisations are most convenientlyperformed in particular field bases. We explain the required changes of basis, which arecanonical transformations on the relevant BV field spaces. Secondly, these factorisationsare twisted factorisation of chain complexes of the type introduced in Section 6.3, withcommon twist datum τ , as indicated in (7.2). Let us start with the case of biadjoint scalar field theory as introduced in Section 5.1. Thiscase is particularly simple as its chain complex Ch p L biadjBRST q factorises as an ordinary tensorproduct. Factorisation of the chain complex.
We can factor out the colour Lie algebras g and ¯ g leaving us with the L -algebra Scal of a plain scalar theory, Ch p L biadjBRST q “ g b p ¯ g b Ch p Scal qq , (7.3)where Scal is a homotopy algebra of cubic scalar field theory which we will fully identifylater in (9.3). The natural chain complex is Ch p Scal q : “ ¨˚˝ s x F r´ s loomoon Scal l ÝÝÑ s ` x F r´ s loomoon Scal ˛‹‚ , (7.4)concentrated in degrees one and two, cf. [177, 209]. Here, s x and s ` x are basis vectors forthe function spaces F r´ s and F r´ s with F given in (4.29). Their inner product is givenby x s x , s ` x y : “ δ p d q p x ´ x q . (7.5)84elds anti-fields | ´ | gh | ´ | L dim | ´ | gh | ´ | L dim s x d ´ s ` x ´ d ` Table 7.1: The basis vectors of
Scal with their L -degrees, their ghost numbers, and theirmass dimensions. fields anti-fieldsfactorisation | ´ | gh | ´ | L dim factorisation | ´ | gh | ´ | L dim ϕ “ e a ¯ e ¯ a s x ϕ a ¯ a p x q d ´ ϕ ` “ e a ¯ e ¯ a s ` x ϕ ` a ¯ a p x q ´ d ` Table 7.2: Factorisation of the BV fields in the theory of biadjoint scalars. Note that wesuppressed the integrals over x and the tensor products for simplicity.The L -degrees correspond to the evident ghost numbers and the differential inducesmass dimensions, and both are summarised in Table 7.1. The factorisation of the BV fieldsis listed in Table 7.2. The differential µ : L biadjBRST , Ñ L biadjBRST , is given by (6.3b) for theuntwisted tensor product, µ p ϕ q “ µ ˆ e a b ¯ e ¯ a b ż d d x s x ϕ a ¯ a p x q ˙ “ e a b ¯ e ¯ a b µ Scal ˆż d d x s x ϕ a ¯ a p x q ˙ “ l ϕ , (7.6)where µ Scal is the product appearing in (7.4). Furthermore, the inner product is x ϕ, ϕ ` y “ tr g p e a e b q tr ¯ g p ¯ e ¯ a ¯ e ¯ b q ż d d x ż d d x x s x , s ` x y ϕ a ¯ a p x q ϕ ` b ¯ b p x q“ ż d d x ϕ a ¯ a p x q ϕ ` a ¯ a p x q . (7.7)In conclusion, we have thus verified the factorisation of the chain complex (7.3). The case of Yang–Mills theory is more involved than the previous one. We start with thegauge fixed BV action (5.15) and perform a canonical transformation on BV field space,which then allows for a convenient factorisation of the resulting chain complex Ch p ˜ L YMBRST q .For the following discussion, recall the gauge-fixing procedure and the gauge-fixed actionfrom Section 5.2. See (4.4) for the notation F r k s . anonical transformation. We note that the term B µ A aµ will vanish for physical statesdue to the polarisation condition p ¨ ε “ where p µ is the momentum and ε µ is thepolarisation vector for A aµ . Off-shell, and at the level of the action, our gauge fixing termsallow us to absorb quadratic terms in B µ A aµ in a field redefinition of the Nakanishi–Lautrupfield b a . We further rescale the field b a in order to homogenise its mass dimension with thatof A aµ , which will prove useful in our later discussion. Explicitly, we perform the fieldredefinitions ˜ c a : “ c a , ˜ c ` a : “ c ` a , ˜ A aµ : “ A aµ , ˜ A ` aµ : “ A ` aµ ` ´ ? ´ ξξ B µ b ` a , ˜ b a : “ c ξ l ˆ b a ` ´ ? ´ ξξ B µ A aµ ˙ , ˜ b ` a : “ c l ξ b ` a , ˜¯ c a : “ ¯ c a , ˜¯ c ` a : “ ¯ c ` a . (7.8)Under these field redefinitions, the action (5.15) S YMBRST “ ż d d x ! A aµ l A aµ ` pB µ A aµ q ´ ¯ c a l c a ` ξ b a b a ` b a B µ A aµ ) ` S YM , intBRST , (7.9)where S YM , intBRST represents the interaction terms, turns into ˜ S YMBRST : “ ż d d x ! ˜ A aµ l ˜ A aµ ´ ˜¯ c a l ˜ c a ` ˜ b a l ˜ b a ` ˜ ξ ˜ b a ? l B µ ˜ A aµ ) ` ˜ S YM , intBRST , (7.10)where we rewrote the gauge-fixing parameter as ˜ ξ : “ d ´ ξξ . (7.11)Note that at the level of the BV field space, the redefinitions (7.8) constitute a canonicaltransformation. For a more detailed analytical discussion, including the precise meaning ofthe inverses of the l operator, see Appendix B. The redefinition of the anti-fields preserves the cyclic structure of the L -algebra; it is mostly irrelevantfor our discussion. -algebra. The action (7.10) is now the superfield homotopy Maurer–Cartan ac-tion (4.26b) for an L -algebra ˜ L YMBRST . The complex underlying ˜ L YMBRST is given as ˜ A aµ Ω p M d q b g ˜ A ` aµ Ω p M d q b g ˜ b a C p M d q b g ˜ b ` a C p M d q b g ˜ c a C p M d q b g loooooomoooooon “ : ˜ L YMBRST , ˜¯ c ` a C p M d q b g loooooomoooooon “ : ˜ L YMBRST , ˜¯ c a C p M d q b g loooooomoooooon “ : ˜ L YMBRST , ˜ c ` a C p M d q b g loooooomoooooon “ : ˜ L YMBRST , l ´ ˜ ξ ? l B µ l ˜ ξ ? l B µ ´ l ´ l (7.12a)with ˜ L YMBRST , “ ˜ L YMBRST , , ˜ c , ˜ L YMBRST , “ à φ P p ˜ A, ˜ b, ˜¯ c ` q ˜ L YMBRST , , φ , ˜ L YMBRST , “ à φ P p ˜ A ` , ˜ b ` , ˜¯ c q ˜ L YMBRST , , φ , ˜ L YMBRST , “ ˜ L YMBRST , , ˜ c ` . (7.12b)The differential µ acts on the various fields as follows p ˜ c a q µ ÞÝÝÑ ´ l ˜ c a P ˜ L YMBRST , , ˜¯ c ` , ˜ ˜ A aµ ˜ b a ¸ µ ÞÝÝÑ ˜ l ˜ A aµ ´ ˜ ξ ? l B µ ˜ b a l ˜ b a ` ˜ ξ ? l B µ ˜ A aµ ¸ P à φ P p ˜ A ` , ˜ b ` q ˜ L YMBRST , , φ , p ˜¯ c a q µ ÞÝÝÑ ´ l ˜¯ c a P ˜ L YMBRST , , ˜ c ` (7.12c)with all other actions trivial. The non-vanishing images of the higher products µ and µ are p ˜ A aµ , ˜ c b q µ ÞÝÝÑ ´ gf bca B µ p ˜ A bµ ˜ c c q P ˜ L YMBRST , , ˜¯ c ` , p ˜ c a , ˜¯ c b q µ ÞÝÝÑ ´ gf bca ˜ c b B µ ˜¯ c c P ˜ L YMBRST , , ˜ A ` , p ˜ A aµ , ˜ A bν q µ ÞÝÝÑ gf bca B ν p ˜ A bν ˜ A cµ q P ˜ L YMBRST , , ˜ A ` , p ˜ A aµ , ˜¯ c b q µ ÞÝÝÑ ´ gf bca ˜ A bµ B µ ˜¯ c c P ˜ L YMBRST , , ˜ c ` , p ˜ A aµ , ˜ A bν , ˜ A cκ q µ ÞÝÝÑ ´ g f bca f deb ˜ A νc ˜ A dν ˜ A eµ P ˜ L YMBRST , , ˜ A ` , (7.12d)and the general expressions follow from anti-symmetrisation and polarisation. We note thatthe formulas (4.27) are useful in the derivation of the explicit form of these higher products.87y construction, p ˜ L YMBRST , µ i q forms an L -algebra, and with the inner products x ˜ A, ˜ A ` y : “ ż d d x ˜ A aµ ˜ A ` µa , x ˜ b, ˜ b ` y : “ ż d d x ˜ b a ˜ b ` a , x ˜ c, ˜ c ` y : “ ż d d x ˜ c a ˜ c ` a , x ˜¯ c, ˜¯ c ` y : “ ´ ż d d x ˜¯ c a ˜¯ c ` a , (7.13)it is cyclic.We stress that the Chevalley–Eilenberg differential of the L -algebra ˜ L YMBRST is not theusual gauge-fixed BV operator ˜ Q YM , gfBV : “ (cid:32) ˜ S YM , gfBV , ´ (ˇˇ ˜Φ ` I “ , (7.14)where ˜ S YM , gfBV is the gauge-fixed BV action that is obtained from (4.30) by the canonicaltransformation determined by the gauge fixing fermion (5.13). Instead, we are merelyusing the general correspondence between Lagrangians and L -algebras as pointed out inSection 4.2. This is reflected in the images of all higher products of (7.12a) lying in spacesparametrised by anti-fields. Factorisation of the chain complex.
As explained in Section 6.2, we may factor outthe gauge Lie algebra g , and we are left with a C -algebra. This C -algebra can be furtherfactorised into a twisted tensor product, extending Example 6.1, and we obtain Ch p ˜ L YMBRST q “ g b p Kin b τ Ch p Scal qq . (7.15)Here, g is the colour Lie algebra, Ch p Scal q is the chain complex (7.4), and Kin is the gradedvector space Kin : “ g R r s loomoon “ : Kin ´ ‘ ` v µ M d ‘ n R ˘loooomoooon “ : Kin ‘ a R r´ s loomoon “ : Kin , (7.16)where the typewriter letters label basis elements of the corresponding vector spaces. Thenatural degree-zero inner product on Kin is given by x g , a y : “ ´ , x v µ , v ν y : “ η µν , x n , n y : “ . (7.17)The element of Kin also carry mass dimensions, which are listed in Table 7.3.We summarise the factorisation of individual Yang–Mills fields in Table 7.4. A fewremarks about the structure of the factorisation are in order. Whilst fields always have afactor of s x , anti-fields always have a factor of s ` x . This guarantees that the inner product is Here, | ˜Φ ` I “ is again the restriction to the subspace of the BV field space where all anti-fields are zero. See (4.4) for the notation R r k s , etc. ´ | gh | ´ | L dim g ´ ´ v µ n a ´ Table 7.3: The elements of
Kin with their L -degrees, their ghost numbers, and their massdimensions.indeed that of the factorisation: (7.13) is reproduced correctly using the factorisations givenin Table 7.4 and (7.17) complemented by the inner product x e a , e b y “ ´ tr p e a e b q “ δ ab on g : x ˜ c, ˜ c ` y “ B e a b g b ż d d x s x ˜ c a p x q , e b b a b ż d d x s ` x ˜ c ` b p x q F “ ´x e a , e b y x g , a y ż d d x ż d d x δ p d q p x ´ x q ˜ c a p x q ˜ c ` b p x q“ ż d d x ˜ c a p x q ˜ c ` a p x q , x ˜ A, ˜ A ` y “ B e a b v µ b ż d d x s x ˜ A aµ p x q , e b b v ν b ż d d x s ` x ˜ A ` bν p x q F “ x e a , e b y x v µ , v ν y ż d d x ż d d x δ p d q p x ´ x q ˜ A aµ p x q ˜ A ` bν p x q“ ż d d x ˜ A aµ p x q ˜ A ` µa p x q , x ˜ b, ˜ b ` y “ B e a b n b ż d d x s x ˜ b a p x q , e b b n b ż d d x s ` x ˜ b ` b p x q F “ x e a , e b y x n , n y ż d d x ż d d x δ p d q p x ´ x q ˜ c a p x q ˜ c ` b p x q“ ż d d x ˜ b a p x q ˜ b ` a p x q , x ˜¯ c, ˜¯ c ` y “ B e a b a b ż d d x s x ˜¯ c a p x q , e b b g b ż d d x s ` x ˜¯ c ` b p x q F “ ´x e a , e b y x a , g y ż d d x ż d d x δ p d q p x ´ x q ˜¯ c a p x q ˜¯ c ` b p x q“ ´ ż d d x ˜¯ c a p x q ˜¯ c ` a p x q . (7.18)Note that the kinematic factor Kin essentially arranges the fields in a quartet: the physicalfield has a ghost, a Nakanishi–Lautrup field, and an anti-ghost. These patterns reoccur inthe double copy.To extend this factorisation of graded vector spaces to a factorisation of chain complexes,89 elds anti-fieldsfactorisation | ´ | gh | ´ | L dim factorisation | ´ | gh | ´ | L dim ˜ c “ e a gs x ˜ c a p x q d ´ c ` “ e a as ` x ˜ c ` a p x q ´ d ` A “ e a v µ s x ˜ A aµ p x q d ´ A ` “ e a v µ s ` x ˜ A ` aµ p x q ´ d ` b “ e a ns x ˜ b a p x q d ´ b ` “ e a ns ` x ˜ b ` a p x q ´ d ` c “ e a as x ˜¯ c a p x q ´ d ˜¯ c ` “ e a gs ` x ˜¯ c ` a p x q d Table 7.4: Factorisation of the redefined BV fields for Yang–Mills theory from Table 5.1after the field redefinitions (7.8). Here, e a denote the basis vectors of g . Likewise, g , n , v µ ,and a denote the basis vectors of Kin defined in (7.16). Furthermore, s x and s ` x are thebasis vectors of Scal from Table 7.1. Note that we suppressed the integrals over x and thetensor products for simplicity.we introduce the twist datum τ given by τ p g q : “ g b id , τ p v µ q : “ v µ b id ` ˜ ξ n b ? l B µ ,τ p n q : “ n b id ´ ˜ ξ v µ b ? l B µ , τ p a q : “ a b id , (7.19)and we shall use the convenient shorthand notation τ p v µ , n q ˜ş d d x s x ˜ A aµ p x q ş d d x s x ˜ b a p x q ¸ “ p v µ , n q b ¨˝ id ´ ˜ ξ ? l B µ ˜ ξ ? l B µ id ˛‚˜ş d d x s x ˜ A aµ p x q ş d d x s x ˜ b a p x q ¸ . (7.20)The twisted differentials on g b p Kin b τ Scal q are now indeed those of (7.12c): µ p ˜ c q “ µ ˆ e a b g b ż d d x s x ˜ c a p x q ˙ “ ´ e a b g b µ Scal ˆż d d x s x ˜ c a p x q ˙ “ e a b g b ż d d x s ` x (cid:32) ´ l ˜ c a p x q ( , (7.21a) µ ˜ ˜ A ˜ b ¸ “ µ ˜ e a b p v µ , n q b ˜ş d d x s x ˜ A aµ p x q ş d d x s x ˜ b a p x q ¸¸ “ e a b p v µ , n q b µ Scal ¨˝¨˝ id ´ ˜ ξ ? l B µ ˜ ξ ? l B µ id ˛‚˜ş d d x s x ˜ A aµ p x q ş d d x s x ˜ b a p x q ¸˛‚ “ e a b p v µ , n q b ˜ş d d x s ` x (cid:32) l ˜ A aµ p x q ´ ˜ ξ ? l B µ ˜ b a p x q (ş d d x s ` x (cid:32) l ˜ b a p x q ` ˜ ξ ? l B µ ˜ A aµ p x q (¸ “ e a b ˜ v µ b ş d d x s ` x (cid:32) l ˜ A aµ p x q ´ ˜ ξ ? l B µ ˜ b a p x q ( n b ş d d x s ` x (cid:32) l ˜ b a p x q ` ˜ ξ ? l B µ ˜ A aµ p x q ( ¸ , (7.21b)90 p ˜¯ c q “ µ ˆ e a b a b ż d d x s x ˜¯ c a p x q ˙ “ ´ e a b a b µ Scal ˆż d d x s x ˜¯ c a p x q ˙ “ e a b a b ż d d x s ` x (cid:32) ´ l ˜¯ c a p x q ( . (7.21c)Altogether, we saw that the factorisation (7.15) is valid for twist datum τ . To keep our discussion manageable, we shall discuss the canonical transformations for thefree Kalb–Ramond two-form and Einstein–Hilbert gravity separately. For the followingdiscussion, recall the gauge-fixing procedure and the gauge-fixed action from Section 5.3.
Canonical transformation.
Analogously to the case of Yang–Mills theory, we can nowperform a field redefinition in order to eliminate the quadratic terms that would vanishon-shell in Lorenz gauge due to contractions between momenta and polarisation tensors.We also insert inverses of the Laplace operator to match the mass dimensions of fields of L -degree one. The field redefinitions are ˜ λ : “ λ , ˜ λ ` : “ λ ` , ˜Λ µ : “ Λ µ , ˜Λ ` µ : “ Λ ` µ ` ´ ? ´ ξξ B µ γ ` , ˜ γ : “ c ξ l ˆ γ ` ´ ? ´ ξξ B µ Λ µ ˙ , ˜ γ ` : “ c l ξ γ ` , ˜ B µν : “ ˜ B µν , ˜ B ` µν : “ B ` µν ` ´ ? ´ ξξ B r µ α ` ν s , ˜ α µ : “ c ξ l ˆ α µ ´ B µ ε ´ ˜ α ` µ : “ c l ξ ˆ α ` µ ` ´ ξ l B µ ε ` ˙ , ´ ´ ξ l B µ B ν α ν `` ´ ? ´ ξξ B ν B νµ ˙ , ˜ ε : “ ε ` ´ ξ l B µ α µ , ˜ ε ` : “ ` ξ ε ` ´ B µ α ` µ , ˜¯Λ µ : “ ¯Λ µ , ˜¯Λ ` µ : “ ¯Λ ` µ ` ´ ? ´ ξξ B µ ¯ γ ` , ˜¯ γ : “ c ξ l ˆ ¯ γ ` ´ ? ´ ξξ B µ ¯Λ µ ˙ , ˜¯ γ ` : “ c l ξ ¯ γ ` , ˜¯ λ : “ ¯ λ , ˜¯ λ ` : “ ¯ λ ` , (7.22a)91ith ξ : “ ξ “ ξ ´ ξ . (7.22b)These redefinitions constitute canonical transformations on the BV field space. Upon ap-plying these transformations to the action (5.24), we obtain ˜ S KRBRST : “ ż d d x ! ˜ B µν l ˜ B µν ´ ˜¯Λ µ l ˜Λ µ ` ˜ α µ l ˜ α µ ´ ˜ ξ pB µ ˜ α µ q ` ˜ ε l ˜ ε ´ ˜¯ λ l ˜ λ ´´ ˜¯ γ l ˜ γ ` ˜ ξ ˜ α ν ? l B µ ˜ B µν ` ˜ ξ ˜ γ ? l B µ ˜¯Λ µ ´ ˜ ξ ˜¯ γ ? l B µ ˜Λ µ ) , (7.23)where we have again used the shorthand ˜ ξ : “ b ´ ξξ , cf. (7.11). L -algebra. The action (7.23) is the superfield homotopy Maurer–Cartan action (4.26b)of an L -algebra, denoted by ˜ L KRBRST , that is given by ˜ ε C p M d q ˜ ε ` C p M d q ˜Λ µ Ω p M d q ˜¯Λ ` µ Ω p M d q ˜¯Λ µ Ω p M d q ˜Λ ` µ Ω p M d q ˜ γ C p M d q ˜¯ γ ` C p M d q ˜¯ γ C p M d q ˜ γ ` C p M d q ˜ λ C p M d q ˜¯ λ ` C p M d q ˜ B µν Ω p M d q ˜ B ` µν Ω p M d q ˜¯ λ C p M d q ˜ λ ` C p M d q looomooon “ : ˜ L KRBRST , ´ looomooon “ : ˜ L KRBRST , ˜ α µ Ω p M d q looomooon “ : ˜ L KRBRST , ˜ α ` µ Ω p M d q looomooon “ : ˜ L KRBRST , looomooon “ : ˜ L KRBRST , looomooon “ : ˜ L KRBRST , l ´ l ´ ˜ ξ ? l B µ ´ l ´ ˜ ξ ? l B µ ´ l ˜ ξ ? l B µ ´ l ˜ ξ ? l B µ l l ´ ˜ ξ ? l B ν ll ˜ ξ B ν B µ ˜ ξ ? l B r ν (7.24a)with ˜ L KRBRST , ´ “ ˜ L KRBRST , ´ , ˜ λ , ˜ L KRBRST , “ à φ P p ˜Λ , ˜ γ, ˜¯ λ ` q ˜ L KRBRST , , φ , ˜ L KRBRST , “ à φ P p ˜ ε, ˜¯Λ ` , ˜¯ γ ` , ˜ B, ˜ α q ˜ L KRBRST , , φ , ˜ L KRBRST , “ à φ P p ˜ ε ` , ˜¯Λ , ˜¯ γ, ˜ B ` , ˜ α ` q ˜ L KRBRST , , φ , ˜ L KRBRST , “ à φ P p ˜Λ ` , ˜ γ ` , ˜¯ λ q ˜ L KRBRST , , φ , ˜ L KRBRST , “ ˜ L YMBRST , , ˜ λ ` , (7.24b)92nd the non-vanishing differential p ˜ λ q µ ÞÝÝÑ l ˜ λ P ˜ L KRBRST , , ˜¯ λ ` , ˜ ˜Λ µ ˜ γ ¸ µ ÞÝÝÑ ´ ˜ l ˜Λ µ ´ ˜ ξ ? l B µ ˜ γ l ˜ γ ` ˜ ξ ? l B µ ˜Λ µ ¸ P à φ P p ˜¯Λ ` , ˜¯ γ ` q ˜ L KRBRST , , φ , ˜ ˜ B µν ˜ α µ ¸ µ ÞÝÝÑ ˜ l ˜ B µν ´ ξ ? l B r µ ˜ α ν s l ˜ α µ ` ˜ ξ ? l B ν ˜ B νµ ` ˜ ξ B µ B ν ˜ α ν ¸ P à φ P p ˜ B ` , ˜ α ` q ˜ L KRBRST , , φ , ˜ ˜¯Λ µ ˜¯ γ ¸ µ ÞÝÝÑ ´ ˜ l ˜¯Λ µ ´ ˜ ξ ? l B µ ˜¯ γ l ˜¯ γ ` ˜ ξ ? l B µ ˜¯Λ µ ¸ P à φ P p ˜Λ ` , ˜ γ ` q ˜ L KRBRST , , φ , p ˜¯ λ q µ ÞÝÝÑ l ˜¯ λ P ˜ L KRBRST , , ˜ λ ` . (7.24c)There are no additional higher products because the theory is free. The expressions x ˜ λ, ˜ λ ` y : “ ´ ż d d x ˜ λ ˜ λ ` , x ˜¯ λ, ˜¯ λ ` y : “ ´ ż d d x ˜¯ λ ˜¯ λ ` , x ˜Λ , ˜Λ ` y : “ ż d d x ˜Λ µ ˜Λ ` µ , x ˜¯Λ , ˜¯Λ ` y : “ ´ ż d d x ˜¯Λ µ ˜¯Λ ` µ , x ˜ B, ˜ B ` y : “ ż d d x ˜ B µν ˜ B ` µν , x ˜ α, ˜ α ` y : “ ż d d x ˜ α µ ˜ α ` µ , x ˜ ε, ˜ ε ` y : “ ż d d x ˜ ε ˜ ε ` , x ˜ γ, ˜ γ ` y : “ ż d d x ˜ γ ˜ γ ` , x ˜¯ γ, ˜¯ γ ` y : “ ´ ż d d x ˜¯ γ ˜¯ γ ` (7.25)define a cyclic inner product on p ˜ L YMBRST , µ q . The case of Einstein–Hilbert gravity with dilaton is now more involved that of the freeKalb–Ramond field. For the following discussion, recall the gauge-fixing procedure and thegauge-fixed action from Section 5.4.
Canonical transformations.
We start from the Lagrangian (5.36) but add a scalarkinetic term for the dilaton ϕ , L eEHD , gf0 : “ L eEH , gf0 ` ϕ l ϕ . (7.26)We perform a field redefinition analogous to the case of Yang–Mills theory and the Kalb–Ramond field, absorbing various terms that vanish on-shell, as well as the trace of h µν in δ and ensuring that all fields come with the right propagators. For the fields of non-vanishing93host number, the transformation read as ˜ X µ : “ X µ , ˜ X ` µ : “ X ` µ , ˜ β : “ ? l β , ˜ β ` : “ ? l β ` ˜¯ X µ : “ ¯ X µ , ˜¯ X ` µ : “ ¯ X ` µ ´ ´ ? ´ ξ ? ξ B µ ¯ β ` , ˜¯ β : “ ? l ˆ ¯ β ´ ´ ? ´ ξ ? ξ B µ ¯ X µ ˙ , ˜¯ β ` : “ ? l ¯ β ` , (7.27a)where we worked in the special gauge ζ “ , ζ “ ´ ? ´ ξ ? ξ , ζ “ ´ , ζ “ ´ p ξ ` ? ´ ξξ ´ ? ´ ξ ´ q? ξ p ξ ´ q ,ζ “ p ´ ξ q ? ξ ´ p ` a ´ ξ q ´ ξ ` p ` a ´ ξ q`` ξ p´ ´ a ´ ξ ` p ` a ´ ξ q ξ q ˘¯ ,ζ “ , ζ “ ξ ` ? ´ ξ ´ , ζ “ . (7.27b)From the expressions for ζ and ζ , it is already apparent that the field redefinitions wewould like to perform here are much more involved than in the case of the Kalb–Ramondfield. Because the resulting expressions for the fields of ghost number zero are too involvedand not very illuminating, we restrict ourselves to the case ξ “ corresponding to Feynmangauge in Yang–Mills theory. Here, we have the inverse field transformations h µν “ ˜ h µν ` B µ B ν ˚˜ h l ´ B µ B κ ˜ h κν l ´ B µ ˜ (cid:36) ν ` B ν ˜ (cid:36) µ ? l ,(cid:36) µ “ ´B µ ˜ δ ´ B κ ˜ h µκ ´ ? l ˜ (cid:36) µ ,π “ ´ l ˜ δ ` l ˜ π ´ B µ B ν ˜ h µν ,δ “ ˜ δ ` ˜ π ` B µ B ν ˜ h µν l ,ϕ “ ? h ´ ? l B µ B ν ˜ h µν (7.27c)with readily computed antifield transformations. Jumping a bit ahead of our story, we notethat the field redefinition for ϕ agrees precisely with the expectation of how the dilatonshould be extracted from the double copied metric perturbation ˜ h . We suspect that there is a simpler field redefinition in a simpler gauge which we have not been ableto identify yet. ξ , the total Lagrangian to lowest order in κ , reads as ˜ L eEHDBRST , “ ˜ h µν l ˜ h µν ` ˜ (cid:36) µ l ˜ (cid:36) µ ` ˜ ξ pB µ ˜ (cid:36) µ q ` ˜ ξ ˜ (cid:36) ν ? l B µ ˜ h µν ´´ ˜ δ l ˜ δ ` ˜ π l ˜ π ` ˜ ξ ˜ π ? l B µ ˜ (cid:36) µ ` ˜ ξ ˜ π B µ B ν ˜ h µν ´´ ˜¯ X µ l ˜ X µ ´ ˜¯ β l ˜ β ` ˜ ξ ˜ β ? l B µ ˜¯ X µ ´ ˜ ξ ˜¯ β ? l B µ ˜ X µ . (7.28)This is the quadratic part of the Lagrangian of the superfield homotopy Maurer–Cartanaction (4.26b) for an L -algebra ˜ L eEHDBRST . The latter has underlying complex ˜ ϕ C p M d q ˜ ϕ ` C p M d q ˜ δ C p M d q ˜ δ ` C p M d q ˜ X µ Ω p M d q ˜¯ X ` µ Ω p M d q ˜¯ X µ Ω p M d q ˜ X ` µ Ω p M d q ˜ β C p M d q ˜¯ β ` C p M d q ˜¯ β C p M d q ˜ β ` C p M d q ˜ h µν Ω p M d q ˜ h ` µν Ω p M d q ˜ (cid:36) µ Ω p M d q ˜ (cid:36) ` µ Ω p M d q looomooon “ : ˜ L eEHDBRST , ˜ π C p M d q looomooon “ : ˜ L eEHDBRST , ˜ π ` C p M d q looomooon “ : ˜ L eEHDBRST , looomooon “ : ˜ L eEHDBRST , l ´ l ´ l ´ l ´ lll ´ ˜ ξ B µ B ν l (7.29a)with ˜ L eEHDBRST , “ à φ P p ˜ β, ˜ X q ˜ L eEHDBRST , , φ , ˜ L eEHDBRST , “ à φ P p ˜ δ, ˜¯ X ` , ˜¯ β ` , ˜ h, ˜ (cid:36), ˜ π q ˜ L eEHDBRST , , φ , ˜ L eEHDBRST , “ à φ P p ˜ β ` , ˜ X ` q ˜ L eEHDBRST , , φ , ˜ L eEHDBRST , “ à φ P p ˜ δ ` , ˜¯ X, ˜¯ β, ˜ h ` , ˜ (cid:36) ` , ˜ π ` q ˜ L eEHDBRST , , φ , (7.29b)95nd the lowest non-vanishing products ˜ ˜ X µ ˜ β ¸ µ ÞÝÝÑ ´ ˜ l ˜ X µ ´ ˜ ξ ? l B µ ˜ β l ˜ β ` ˜ ξ ? l B µ ˜ X µ ¸ P à φ P p ˜¯ X ` , ˜¯ β ` q ˜ L eEHDBRST , , φ , ¨˚˚˝ ˜ h µν ˜ (cid:36) µ ˜ π ˛‹‹‚ µ ÞÝÝÑ ¨˚˚˝ l ˜ h µν ´ ξ ? l B µ ˜ (cid:36) ν ` ˜ ξ B µ B ν ˜ π l ˜ (cid:36) µ ` ˜ ξ ? l B µ ˜ h µν ´ ˜ ξ ? l B µ ˜ π ´ ˜ ξ B µ B ν ˜ (cid:36) ν l ˜ π µ p x q ` ξ ? l B µ ˜ (cid:36) µ p x q ` ˜ ξ B µ B ν ˜ h µν ˛‹‹‚ P à φ P p ˜ h ` , ˜ (cid:36) ` , ˜ π ` q ˜ L eEHDBRST , , φ , ˜ ˜¯ X µ ˜¯ β ¸ µ ÞÝÝÑ ´ ˜ l ˜¯ X µ ´ ˜ ξ ? l B µ ˜¯ β l ˜¯ β ` ˜ ξ ? l B µ ˜¯ X µ ¸ P à φ P p ˜ X ` , ˜ β ` q ˜ L eEHDBRST , , φ . (7.29c)The ˜ L eEHDBRST algebra is endowed with the following cyclic structure: x ˜ X, ˜ X ` y : “ ż d d x ˜ X µ ˜ X ` µ , x ˜¯ X, ˜¯ X ` y : “ ´ ż d d x ˜¯ X µ ˜¯ X ` µ , x ˜ β, ˜ β ` y : “ ż d d x ˜ β ˜ β ` , x ˜¯ β, ˜¯ β ` y : “ ´ ż d d x ˜¯ β ˜¯ β ` , x ˜ h, ˜ h ` y : “ ż d d x ˜ h µν ˜ h ` µν , x ˜ (cid:36), ˜ (cid:36) ` y : “ ż d d x ˜ (cid:36) µ ˜ (cid:36) ` µ , x ˜ π, ˜ π ` y : “ ż d d x ˜ π ˜ π ` , x ˜ δ, ˜ δ ` y : “ ´ ż d d x ˜ δ ˜ δ ` . (7.30) N “ supergravity The factorisation of the chain complex of the L -algebra for Yang–Mills theory now fixescompletely the factorisation of the chain complex of the L -algebra of N “ supergravity.In view of (7.15), it thus merely remains to verify that Ch p ˜ L N “ q “ Kin b τ p Kin b τ Ch p Scal qq (7.31)at the level of chain complexes, where Kin is given in (7.16) and Ch p Scal q in (7.4). Fur-thermore, the twist in the outer tensor product of (7.31) will only affect Ch p Scal q andcommute with the other factor of Kin . Let us stress that we could have allowed for twodifferent twist parameters for each of the tensor products. This, however, would make ourdiscussion unnecessarily involved.
Factorisation of fields.
It is not surprising that the identification works at the level ofgraded vector spaces for the physical fields. This is merely the statement that a rank-two(covariant) tensor decomposes into its symmetric part and its anti-symmetric part. The96 elds anti-fieldsfactorisation | ´ | gh | ´ | L dim factorisation | ´ | L dim ˜ λ “ ´r g , g s s x ˜ λ p x q ´ d ´ λ ` “ ´r a , a s s ` x ˜ λ ` p x q d ` “ r g , v µ s s x ? ˜Λ µ p x q d ´ ` “ r a , v µ s s ` x ? ˜Λ ` µ d ` γ “ r g , n s s x ? ˜ γ p x q d ´ γ ` “ r a , n s s ` x ? ˜ γ ` p x q d ` B “ r v µ , v ν s s x ? ˜ B µν p x q d ´ B ` “ r v µ , v ν s s ` x ? ˜ B ` µν p x q d ` α “ r n , v µ s s x ? ˜ α µ p x q d ´ α ` “ r n , v µ s s ` x ? ˜ α ` µ p x q d ` ε “ ´r g , a s s x ? ˜ ε p x q d ´ ε ` “ ´r g , a s s ` x ? ˜ ε ` p x q d ` “ r a , v µ s s x ? ˜¯Λ µ p x q ´ d ˜¯Λ ` “ r g , v µ s s ` x ? ˜¯Λ ` µ p x q d ˜¯ γ “ r a , n s s x ? ˜¯ γ p x q ´ d ˜¯ γ ` “ r g , n s s ` x ? ˜¯ γ ` p x q d ˜¯ λ “ ´r a , a s s x ˜¯ λ p x q ´ d ` λ ` “ ´r g , g s s ` x ˜¯ λ ` p x q d ´ X “ p g , v µ q s x ? ˜ X µ p x q d ´ X ` “ p a , v µ q s ` x ? ˜ X ` µ p x q d ` β “ p g , n q s x ? ˜ β p x q d ´ β ` “ p a , n q s ` x ? ˜ β ` p x q d ` h “ p v µ , v ν q s x ? ˜ h µν p x q d ´ h ` “ p v µ , v ν q s ` x ? ˜ h ` µν p x q d ` (cid:36) “ ´p n , v µ q s x ? ˜ (cid:36) µ p x q d ´ (cid:36) ` “ ´p n , v µ q s ` x ? ˜ (cid:36) ` µ p x q d ` π “ p n , n q s x ? ˜ π p x q d ´ π ` “ p n , n q s ` x ? ˜ π ` p x q d ` δ “ ´p g , a q s x ? ˜ δ p x q d ´ δ ` “ ´p g , a q s ` x ? ˜ δ ` p x q d ` X “ p a , v µ q s x ? ˜¯ X µ p x q ´ d ˜¯ X ` “ p g , v µ q s ` x ? ˜¯ X µ p x q d ˜¯ β “ p a , n q s x ? ˜¯ β p x q ´ d ˜¯ β ` “ p g , n q s ` x ? ˜¯ β ` p x q d Table 7.5: Factorisation of the redefined BV fields for N “ supergravity. Just as inthe case of Yang–Mills theory, all fields have a factor of s x , while all anti-fields have afactor of s ` x . Here, we again suppressed the integrals over x and we used the notation r x , y s : “ x b y ´ p´ q | x | | y | y b x and p x , y q : “ x b y ` p´ q | x | | y | y b x for x , y P Kin .symmetric part splits further into the trace, which can be identified with the dilaton, andthe remaining components, which describe gravitational modes. More interesting is thesector of unphysical fields, and the complete factorisation of all fields is given in Table 7.5.The elements of
Kin form a quartet, which is reflected in the well-known quartet of fieldsin the gauge-fixed Yang–Mills action: nv µ g a ÝÑ b a A aµ c a ¯ c a (7.32)Each field in Ch p ˜ L N “ q thus lives in the tensor product of two such quartets. This tensor97roduct further splits into (graded) symmetric, anti-symmetric, and trace parts, whichbelong to the two-form B µν , the graviton modes h µν , and the dilaton ϕ . Because theproduct of two ghosts g ˜ g is automatically anti-symmetric, only the B -field has a ghost forghost λ . On the graviton/dilaton side, we do not have the higher gauge transformations,but contrary to Yang–Mills theory, the ghost is a vector. We can summarise the relationsbetween the fields in the following two diagrams: α µ γ B µν ¯ γ Λ µ ¯Λ µ λ ε ¯ λ π(cid:36) µ β h µν ¯ βX µ ¯ X µ δ (7.33)where upper, lower left, and lower right arrows point to fields where a vector factor v µ hasbeen replaced by a factor n , g , and a , respectively. The L -degrees of the fields are thesame in each column, increasing from left to right by one. Factorisation as cyclic complex.
From Table 7.5, it is clear that the tensorproduct (7.31) is indeed correct at the level of graded vector spaces. The inner productstructure on the anti-symmetric part is given by x ˜ λ, ˜ λ ` y “ B ´ g b g b ż d d x s x ˜ λ p x q , ´ a b a b ż d d x s ` x ˜ λ ` p x q F “ ´x g , a yx g , a y ż d d x ż d d x δ p d q p x ´ x q ˜ λ p x q ˜ λ ` p x q“ ´ ż d d x ˜ λ p x q ˜ λ ` p x q , (7.34a)Similarly, x ˜Λ , ˜Λ ` y “ ż d d x ˜Λ µ p x q ˜Λ ` µ p x q , x ˜¯Λ , ˜¯Λ ` y “ ´ ż d d x ˜¯Λ µ p x q ˜¯Λ ` µ p x q , x ˜ γ, ˜ γ ` y “ ż d d x ˜ γ p x q ˜ γ ` p x q , x ˜¯ γ, ˜¯ γ ` y “ ´ ż d d x ˜¯ γ p x q ˜¯ γ ` p x q , x ˜ B, ˜ B ` y “ ż d d x ˜ B µν p x q ˜ B ` µν p x q , x ˜ ε, ˜ ε ` y “ ż d d x ˜ ε p x q ˜ ε ` p x q , x ˜ α, ˜ α ` y “ ż d d x ˜ α µ p x q ˜ α ` µ p x q , x ˜¯ λ, ˜¯ λ ` y “ ´ ż d d x ˜¯ λ p x q ˜¯ λ ` p x q . (7.34b)98n the symmetric part, we have analogously x ˜ X, ˜ X ` y “ ż d d x ˜ X µ p x q ˜ X ` µ p x q , x ˜ π, ˜ π ` y “ ż d d x ˜ π p x q ˜ π ` p x q , x ˜ β, ˜ β ` y “ ż d d x ˜ β p x q ˜ β ` p x q , x ˜ δ, ˜ δ ` y “ ´ ż d d x ˜ δ p x q ˜ δ ` p x q , x ˜ h, ˜ h ` y “ ż d d x ˜ h µν p x q ˜ h ` µν p x q , x ˜¯ X, ˜¯ X ` y “ ´ ż d d x ˜¯ X µ p x q ˜¯ X ` µ p x q , x ˜ (cid:36), ˜ (cid:36) ` y “ ż d d x ˜ (cid:36) µ p x q ˜ (cid:36) ` µ p x q , x ˜¯ β, ˜¯ β ` y “ ´ ż d d x ˜¯ β p x q ˜¯ β ` p x q . (7.34c)Next, we compute the action of the differential µ , which is completely fixed by thetensor product Kin b τ p Kin b τ Scal q , cf. definition (6.15). We have, for example, µ p ˜ λ q “ µ ˆ ´r g , g s b ż d d x s x ˜ λ p x q ˙ “ ´r g , g s b µ ˆż d d x s x ˜ λ p x q ˙ “ l ˜ λ ,µ ˜ ˜Λ˜ γ ¸ “ µ ¨˝ pr g , v µ s , r g , n sq b ¨˝ş d d x s x ? ˜Λ µ p x q ş d d x s x ? ˜ γ p x q ˛‚˛‚ “ ´pr g , v µ s , r g , n sq b µ ¨˝˜ id ´ ˜ ξ l ´ B µ ˜ ξ l ´ B µ id ¸ ¨˝ş d d x s x ? ˜Λ µ p x q ş d d x s x ? ˜ γ p x q ˛‚˛‚ “ ´pr g , v µ s , r g , n sq b ¨˝ş d d x s ` x ? t l ˜Λ µ p x q ´ ˜ ξ ? l B µ ˜ γ p x qu ş d d x s ` x ? t l ˜ γ p x q ` ˜ ξ ? l B µ ˜Λ µ p x qu ˛‚ ,µ ˜ ˜ B ˜ α ¸ “ µ ¨˝ pr v µ , v ν s , r n , v µ sq b ¨˝ş d d x s x ? ˜ B µν p x q ş d d x s x ? ˜ α µ p x q ˛‚˛‚ “ pr v µ , v ν s , r n , v µ sq b ¨˝ ş d d x s ` x ? t l ˜ B µν p x q ´ ˜ ξ ? l B µ ˜ α ν p x qu ş d d x s ` x ? t l ˜ α µ p x q ` ˜ ξ ? l B ν ˜ B νµ p x q ` ˜ ξ B µ B ν ˜ α ν p x qu ˛‚ ,µ ¨˚˚˝ ˜ h ˜ (cid:36) ˜ π ˛‹‹‚ “ µ ¨˚˚˝ pp v µ , v ν q , p n , v µ q , p n , n qq b ¨˚˚˝ ş d d x s x ? ˜ h µν p x q ş d d x s x ´ ´ ? ˜ (cid:36) µ p x q ¯ş d d x s x ? ˜ π µ p x q ˛‹‹‚˛‹‹‚ “ pp v µ , v ν q , p n , v µ q , p n , n qq b M (7.35a)with M : “ ¨˚˚˝ ş d d x s ` x t ? l ˜ h µν p x q ´ ? ˜ ξ ? l B µ ˜ (cid:36) ν p x q ` ? ˜ ξ B µ B ν ˜ π p x qu ş d d x s ` x t´ ? l ˜ (cid:36) µ p x q ´ ? ˜ ξ ? l B µ ˜ h µν p x q ` ? ˜ ξ ? l B µ ˜ π p x q ` ? ˜ ξ B µ B ν ˜ (cid:36) ν p x qu ş d d x s ` x t ? l ˜ π µ p x q ` ? ˜ ξ ? l B µ ˜ (cid:36) µ p x q ` ? ˜ ξ B µ B ν ˜ h µν u ˛‹‹‚ . (7.35b)99urthermore, we have µ ˜ ˜¯Λ˜¯ γ ¸ “ ´pr a , v µ s , r a , n sq b ¨˝ş d d x s ` x ? t l ˜¯Λ µ p x q ´ ˜ ξ ? l B µ ˜¯ γ p x qu ş d d x s ` x ? t l ˜¯ γ p x q ` ˜ ξ ? l B µ ˜¯Λ µ p x qu ˛‚ ,µ p ˜ ε q “ l ˜ ε ,µ p ˜¯ λ q “ l ˜¯ λ ,µ ˜ ˜ X ˜ β ¸ “ ´pp g , v µ q , p g , n qq b ¨˝ş d d x s ` x ? t l ˜ X µ p x q ´ ˜ ξ ? l B µ ˜ β p x qu ş d d x s ` x ? t l ˜ β p x q ` ˜ ξ ? l B µ ˜ X µ p x qu ˛‚ ,µ ˜ ˜¯ X ˜¯ β ¸ “ ´pr a , v µ s , r a , n sq b ¨˝ş d d x s ` x ? t l ˜¯ X µ p x q ´ ˜ ξ ? l B µ ˜¯ β p x qu ş d d x s ` x ? t l ˜¯ β p x q ` ˜ ξ ? l B µ ˜¯ X µ p x qu ˛‚ ,µ p ˜ δ q “ l ˜ δ . (7.35c)The resulting superfield homotopy Maurer–Cartan action (4.26a) for the superfield a “ ˜ λ ` ˜Λ ` ¨ ¨ ¨ ` ˜ B ` ˜ h is ˜ S DC0 : “ ż d d x ! ˜ B µν l ˜ B µν ´ ˜¯Λ µ l ˜Λ µ ` ˜ α µ l ˜ α µ ´ ˜ ξ pB µ ˜ α µ q ` ˜ ε l ˜ ε ´ ˜¯ λ l ˜ λ ´´ ˜¯ γ l ˜ γ ` ˜ ξ ˜ α ν ? l B µ ˜ B µν ` ˜ ξ ˜ γ ? l B µ ˜¯Λ µ ´ ˜ ξ ˜¯ γ ? l B µ ˜Λ µ `` ˜ h µν l ˜ h µν ´ ˜¯ X µ l ˜ X µ ` ˜ (cid:36) µ l ˜ (cid:36) µ ` ˜ ξ pB µ ˜ (cid:36) µ q ´´ ˜ δ l ˜ δ ` ˜ π l ˜ π ´ ˜¯ β l ˜ β ` ˜ ξ ˜ (cid:36) ν ? l B µ ˜ h µν ` ˜ ξ ˜ π ? l B µ ˜ (cid:36) µ `` ˜ ξ ˜ π B µ B ν ˜ h µν ` ˜ ξ ˜ β ? l B µ ˜¯ X µ ´ ˜ ξ ˜¯ β ? l B µ ˜ X µ ) . (7.36)This is action is precisely the sum of the transformed Kalb–Ramond action (7.23) and ofthe transformed zeroth-order gravity action augmented by a dilaton kinetic term (7.28).Consequently, we see that our double copy prescription, arising from the factorisation ofthe L -algebras of Yang–Mills theory and N “ supergravity into three factors, works atthe level of chain complexes.
8. Quantum field theoretic preliminaries
Having completed the discussion at the free, linear level, we are almost ready to turn tothe factorisation in the full, interacting picture, which is, perhaps not surprisingly, veryinvolved, cf. [9].Firstly, as explained in Section 2, the double copy of amplitudes is based on a reformu-lation of the underlying Feynman diagrams in terms of diagrams with exclusively trivalentvertices. At the level of the action, this means that we need to strictify the field theory,100.e. to replace it by a physically equivalent one with exclusively cubic interaction terms. Inthis section, we will be relatively explicit, at least to lowest orders in the amplitude legsand coupling constants.Secondly, it is clear that the double copy of the factorisation of interacting Yang–Millstheory will be some form of strictified N “ supergravity. We will not work out detailedexpressions for this action but merely show that the produced action is quantum equivalentto N “ supergravity.To this end, we shall need a number of quantum field theoretic observations alreadymade in [9]. This section contains both a review and a much more detailed explanation ofthese observations than [9].In the following, we shall always clearly distinguish between scattering amplitudes A p¨ ¨ ¨ q and correlation functions x¨ ¨ ¨ y . Correlation functions, contain operators that cre-ate and annihilate arbitrary fields without any constraints. Scattering amplitudes, on theother hand, are labelled by external fields , which usually are physical fields with on-shellmomenta and physical polarisations. For our arguments, it is convenient to lift the restric-tion to physical polarisations and work with the BRST-extended Hilbert space of externalfields which, in the case of Yang–Mills theory, includes gluons of arbitrary polarisations aswell as the ghosts and anti-ghosts as explained next. The tree-level scattering amplitudes of Yang–Mills theory are parametrised by degree oneelements of the minimal model of the L -algebra (7.12). These are the physical, on-shellstates. A convenient set of coordinates for these are the gluon’s momentum p µ as well as adiscrete label indicating the gluon’s helicity. More conveniently, we can replace the discretelabels by a linearly independent set of polarisation vectors ε µ that satisfy p ε µ q “ ˜ (cid:126)ε ¸ , (cid:126)p ¨ (cid:126)ε “ , and | (cid:126)ε | “ . (8.1) BRST-extended Hilbert space.
We can extend this conventional Hilbert space of ex-ternal fields to the full BRST field space H YMBRST as done, e.g., in [284]. We thus have twoadditional, unphysical polarisations of the gluon, called forward and backward and denotedby A Ò aµ and A Ó aµ , respectively. We can be a bit more explicit for general gluons with light-like momenta. Here, the polarisation vector ε Ò µ is proportional to the momentum p µ and101he backwards polarisation vector ε Ò µ is obtained by reversing the spatial part, p ε Ò µ q “ ? | (cid:126)p | ˜ p (cid:126)p ¸ and p ε Ó µ q “ ? | (cid:126)p | ˜ p ´ (cid:126)p ¸ , (8.2a)so that ε Ò ¨ ε Ò “ , ε Ó ¨ ε Ó “ , and ε Ò ¨ ε Ó “ ´ . (8.2b)We also have ghost and anti-ghost states. All scattering amplitudes we shall consider willbe built from the Hilbert space H YMBRST . We note that the S-matrix of the physical Hilbertspace H YMphys is then the restriction of the S-matrix for the BRST extended Hilbert space H YMBRST . Although there are scattering amplitudes producing unphysical particles in H YMBRST from physical gluons in H YMphys , this is consistent, because the restricted S-matrix is unitary.This is a consequence of the full S-matrix on H YMBRST being unitary and BRST symmetry,cf. [285, Section 16.4].Evidently, H YMBRST carries an action of the linearisation of the BRST operator, denoted by Q linBRST , cf. again [284] or the discussion in [285, Section 16.4]. Note that after gauge-fixing,the full BRST transformations are given by the restriction of the BV transformations (5.10)since the gauge-fixing fermion is assumed to be independent of the anti-fields. We have c a Q YMBRST
ÞÝÝÝÝÝÑ ´ gf bca c b c c , ¯ c a Q YMBRST
ÞÝÝÝÝÝÑ b a ,A aµ Q YMBRST
ÞÝÝÝÝÝÑ p ∇ µ c q a , b a Q YMBRST
ÞÝÝÝÝÝÑ , (8.3)and p Q YMBRST q “ off-shell. In momentum space, it is then easy to see that the trans-versely-polarised or physical gluon states A K aµ are singlets under the action of the linearisedBRST operator, Q YM , linBRST A K aµ “ . The remaining four states arrange into two doublets, A Ò aµ Q YM , linBRST ÞÝÝÝÝÝÑ B µ c a and ¯ c a Q YM , linBRST ÞÝÝÝÝÝÑ b a “ ξ B µ A Ó aµ ` ¨ ¨ ¨ , (8.4)where the ellipsis indicates terms that would arise from the shift of the gauge-fixing fermionin (8.20). Connected correlation functions.
In our later analysis of the double copy, we shallcompare correlation functions at the tree level. Recall that the partition function Z andthe free energy W : “ log p Z q are the generating functionals for the correlation functions andthe connected correlation functions, respectively. Evidently, this implies that the connec-ted correlation functions can be written as linear combinations of products of correlationfunctions. This simplifies our analysis as we can restrict ourselves to the contributions ofconnected Feynman diagrams to correlation functions.102 bservation 8.1. The set of connected correlation functions is BRST-invariant becausethe connected correlation functions can be written as linear combinations of products ofcorrelation functions.
Ward identities for scattering amplitudes.
In order to translate colour–kinematicsduality for scattering amplitudes from gluons to ghosts, we shall use supersymmetric on-shell Ward identities, cf. [20, 21], and we focus on the supersymmetry generated by thelinearised BRST operator Q YM , linBRST acting on the BRST-extended Hilbert space H YMBRST ,whose elements label our scattering amplitudes.The free vacuum is certainly invariant under the action of Q YM , linBRST , cf. again [284]or [285, Section 16.4]. We therefore have the on-shell Ward identity “ x |r Q YM , linBRST , O ¨ ¨ ¨ O n s| y . (8.5)In order to use this Ward identity to link scattering amplitudes with k ghost–anti-ghostpairs to amplitudes with k ` such pairs, we consider the special case O ¨ ¨ ¨ O n “ A Ò ¯ c p c ¯ c q k A K ¨ ¨ ¨ A K n ´ k ´ , (8.6)where the gluon A Ò aµ is forward polarised while all other gluons have physical polarisation.In this special case, the on-shell Ward identity (8.5) becomes x |p c ¯ c q k ` A K ¨ ¨ ¨ A K n ´ k ´ | y ` x | A Ò p c ¯ c q k bA K ¨ ¨ ¨ A K n ´ k ´ | y “ . (8.7) Observation 8.2.
Any amplitude with k ` ghost–anti-ghost pairs and all gluons trans-versely polarised is given by a sum of amplitudes with k ghost pairs. The simplest non-trivial concrete example to illustrate Observation 8.2 is the case n “ , k “ in Yang–Mills theory (the three-point scattering amplitudes vanish). We may thenidentify x | ˆ A Ò a p p q ˆ b b p p q ˆ A K c p p q ˆ A K d p p q| y ““ p A AAAA ` ε Ò p p q , p , a ; ε Ó p p q , p , b ; ε K p p q , p , c ; ε K p p q , p , d ˘ (8.8a)and x | ˆ c a p p q ˆ¯ c b p p q ˆ A K c p p q ˆ A K d p p q| y ““ p A c ¯ cAA ` p , a ; p , b ; ε K p p q , p , c ; ε K p p q , p , d ˘ , (8.8b)where A AAAA and A c ¯ cAA denote the four-gluon and two-ghost–two-gluon scattering amp-litudes, respectively, with external particles labelled by polarisation vectors, momenta, and103olour indices. The hat indicates the Fourier transform. A standard Feynman diagramcomputation then shows that p A AAAA “ f ade f ebc ? ! p ε ¨ ε q “ p p ¨ ε q ` p p ¨ ε q ‰ ´ p ε ¨ ε q “ p p ¨ ε q ` p p ¨ ε q ‰ ´´ p p p ¨ ε qp p ¨ ε q? ` p p ¨ p q ` p p ¨ p q ˘ ´ p ε ¨ ε qp p ¨ ε q´ p ε ¨ ε qp p ¨ ε q ´ ? p p ε ¨ ε q ) `` f abe f ecd ? ! ´ p ? p p ¨ p q “ p p ¨ ε qp p ¨ ε q ´ p p ¨ ε qp p ¨ ε q ‰ ´´ p ? p p ¨ p q “ p p ¨ p q ´ p p ¨ p q ‰ p ε ¨ ε q ´´ p ε ¨ ε q “ p p ¨ ε q ` p p ¨ ε q ‰ `` p ε ¨ ε q “ p p ¨ ε q ` p p ¨ ε q ‰ ´´ p ε ¨ ε q “ p p ¨ ε q ` p p ¨ ε q ‰) `` f ace f ebd ? ! p p p ¨ ε qp p ¨ ε q? p p ¨ p q ` p ε ¨ ε q “ p p ¨ ε q ` p p ¨ ε q ‰ ´´ p ε ¨ ε q “ p p ¨ ε q ` p p ¨ ε q ‰ `` p ε ¨ ε q ` p p ¨ ε q ` p p ¨ ε q ‰ ` ? p p ε ¨ ε q ) (8.9a)and p A c ¯ cAA “ f ace f ebd p p p ¨ ε qp p ¨ ε q p p ¨ p q` f abe f ecd p p p ¨ p q ! p p ¨ ε qp p ¨ ε q ´ p p ¨ ε qp p ¨ ε q `` “ p p ¨ p q ` p p ¨ p q ‰ p ε ¨ ε q ) ´´ f ade f ebc p p p ¨ ε qp p ¨ ε q “ p p ¨ p q ` p p ¨ p q ‰ . (8.9b)The sum of both terms vanishes, p A AAAA ` p A c ¯ cAA “ , (8.10)upon using momentum conservation, transversality of the physically polarised gluons, theexplicit form of the on-shell polarisation vectors (8.2), and the Jacobi identity. That is, the s -, t -, and u -channels are not related separately. This is not very surprising: as indicated inSection 2.1, the four-point gluon vertex can be distributed in different ways to the various104hannels and each distribution would imply a different relation between the channels of thetwo amplitudes. If we ensured colour–kinematics duality for the four-point vertex, however,then the relation between the two amplitudes would hold for each individual channel.When we come to discussing the double copy theory, we will be able to ensure BRSTinvariance of the action only on-shell. However, from the construction of correlators fromFeynman diagrams it is clear that the action of Q YM , linBRST on the on-shell BRST-extendedHilbert space will still be preserved, and we again have (8.5) with the corresponding linkbetween scattering amplitudes with different number of ghost–anti-ghost pairs: Observation 8.3.
Suppose that Q YMBRST S YMBRST “ and p Q YMBRST q “ only on-shell. Then,we still have an identification of scattering amplitudes with k ` ghost–anti-ghost pairs andall gluons transversely polarised and a sum of amplitudes with k ghost–anti-ghost pairs. Off-shell Ward identities.
BRST invariance of the action, being a global symmetry,induces an off-shell Ward identity for correlation functions, xpB µ j µ p x qq O p x q ¨ ¨ ¨ O n p x n qy “ n ÿ i “ ¯ δ p d q p x ´ x i q C p Q BRST O i p x i qq ź j ‰ i O j p x j q G , (8.11)where j µ is the BRST current and the sign is the Koszul sign arising from permutingoperators of non-vanishing ghost number. Note that in general, Q YMBRST is the renormalisedBRST operator of the full quantum theory, cf. [286, Chapter 17.2]. As we will always discusstree-level correlators, however, we can restrict ourselves to the classical BRST operator withaction (8.3). We note that the left-hand side of (8.11) vanishes after integration over x andthe Ward identity simplifies to “ n ÿ i “ ˘ C p Q YMBRST O i p x i qq ź j ‰ i O j p x j q G . (8.12)When applying Ward identities to correlation functions, we can use Observation 8.1 torestrict the correlation functions to purely connected correlators, i.e. the contribution arisingfrom connected Feynman diagrams. Moreover, we can restrict the correlation functions toa particular order in the coupling constant g . This implies that for operators linear in thefields we can truncate the action of the BRST operator Q YMBRST to the Abelian action.As a short explicit example, let us consider (8.12) for the special case n “ with ˆ O “ ˆ A a Ò µ p p q , ˆ O “ ˆ¯ c b p p q , ˆ O “ ˆ A c Ò µ p p q , (8.13)and we switched to momentum space for simplicity. We obtain the identity P Ò µµ p p q P Ò ν ν p p q ´ x ˆ A a Ò µ p p q ˆ b b p p q ˆ A c Ò ν p p qy ` x p µ ˆ c a p p q ˆ¯ c b p p q ˆ A c Ò ν p p qy ´´ x ˆ A a Ò µ p p q ˆ¯ c b p p q p ν ˆ c c p p qy ¯ “ , (8.14)105here P Ò µµ p p q is the projector onto (off-shell) forward polarised gluons. Explicitly, P Ò µν p p q : “ p µ p p ¨ ˜ p qp p ¨ ˜ p q ´ p p ¨ p q „ ˜ p ν ´ p p ¨ p qp p ¨ ˜ p q p ν , P Ó µν p p q : “ ˜ p µ p p ¨ ˜ p qp p ¨ ˜ p q ´ p p ¨ p q „ p ν ´ p p ¨ p qp p ¨ ˜ p q ˜ p ν , (8.15)where ˜ p µ is p µ with spatial components reverted.The relevant vertices are clearly the cubic gluon vertex to which ˆ b b p p q is linked by apropagator, as well as the ghost–anti-ghost–gluon vertex. At tree-level, we thus obtain P Ò µµ p p q P Ò ν ν p p q x ˆ A a Ò µ p p q ˆ b b p p q ˆ A c Ò ν p p qy ““ f abc P Ò µµ p p q P Ò ν ν p p q “ p µ p ν ´ p µ p ν ` η µ ν p p ´ p q ¨ p P Ó p p q ¨ p q ‰ , (8.16a) P Ò µµ p p q P Ò ν ν p p q x p µ ˆ c a p p q ˆ¯ c b p p q ˆ A c Ò ν p p qy “ f abc P Ò µµ p p q P Ò ν ν p p q p µ p ν , (8.16b)and P Ò µµ p p q P Ò ν ν p p q x ˆ A a Ò µ p p q ˆ¯ c b p p q p ν ˆ c c p p qy “ f cba P Ò µµ p p q P Ò ν ν p p q p ν p µ . (8.16c)The sum of these three terms is f abc P Ò µµ p p q P Ò ν ν p p q η µ ν “ p p ´ p q ¨ p P Ó p p q ¨ p q ‰ , (8.17)which vanishes after inserting the explicit expressions (8.15).We conclude with the following observation. Observation 8.4.
We have Ward identities between tree-level correlation functions for thelinearised BRST operator.
Let us now leave the special case of Yang–Mills theory for a moment and reconsider notionsof equivalence between field theories in general. As discussed in Section 4.3, two fieldtheories are classically equivalent if they are quasi-isomorphic and thus have a commonminimal model. In the same section, it was explained how the minimal model of a fieldtheory is constructed using the homological perturbation lemma.
Perturbative quantum equivalence.
For the full quantum equivalence at the perturb-ative level, we have the following evident statement.106 bservation 8.5.
Two field theories are perturbatively quantum equivalent if all correlatorsbuilt from polynomials of fields and their derivatives agree to any finite order in couplingconstant and loop level. Since correlators can be glued together from tree-level correlators(up to regularisation issues), it suffices if the tree level correlators agree.
We stress that we are only interested in the integrands of scattering amplitudes, whichallows us to ignore all issues related to regularisation.To provide a link between the double-copied action and the action of N “ supergravity,we will need to perform a sequence of field redefinitions. The field content of the theorieswill be the same from the outset, and we choose to work with the same path integralmeasure in both cases. We are therefore interested in field redefinitions that leave the pathintegral measure invariant.There are large classes of such field redefinitions. The most evident such class of fieldredefinitions is φ ÞÑ ˜ φ : “ φ ` f p φ , . . . , φ n q , (8.18)where f is a polynomial function of a set of fields t φ , . . . , φ n u and their derivatives with φ Rt φ , . . . , φ n u . Under such a field redefinition, the path integral measure remains unchanged;this becomes evident when imagining the finite-dimensional analogue of volume forms anda coordinate shifted by a function of different coordinates.More subtle is the fact that field redefinitions of the form φ ÞÑ ˜ φ : “ φ ` O p φ q , (8.19)where O p φ q denotes local polynomial functions in arbitrary fields and their derivativeswhich are at least of quadratic order in φ can also be considered as leaving the path integralmeasure invariant.Invariance of the S-matrix under (8.19) without derivatives is captured by the Chisholm–Kamefuchi–O’Raifeartaigh–Salam equivalence theorem [287, 288]. A proof using the BVformalism of perturbative quantum equivalence for local field redefinitions of the form (8.19)allowing for derivatives was given in [289]. This is sufficient for our purposes as we are onlyconcerned with the integrands of scattering amplitudes. Note, however, the well-knownneed to choose the counter-terms consistently, as emphasised in [289]. With this in mind,the simplest approach is to use dimensional regularisation, since (8.19) produces a Jacobianwhich is then regulated to unity, see [290, 291] as well as [292, Sections 18.2.3–4].We sum up the above discussion as follows. Observation 8.6.
A shift of a field by products of fields and their derivatives which do notinvolve the field itself does not change the path integral measure. Local field redefinitions hat are trivial at linear order are quantum mechanically safe as they produce a Jacobianthat can be regulated to unity in dimensional regularisation.
Nakanishi–Lautrup field shifts and changes of gauge.
Besides field redefinitions,we also adjust our choice of gauge to link equivalent field theories. In particular, we canshift the usual choice (5.13) for R ξ -gauge to Ψ ÞÑ Ψ ` Ξ with Ξ : “ ż d d x ¯ c a Y a . (8.20)Here, Y a is of ghost number zero, and we limit ourselves to terms Y a that are independ-ent of the Nakanishi–Lautrup field. The shift (8.20) leads to a shift of the gauge-fixedLagrangian (5.15) given by L YMBRST ÞÑ L YMBRST ` δ Ξ δA aµ p ∇ µ c q a ` g f bca δ Ξ δc a c b c c ´ b a δ Ξ δ ¯ c a . (8.21)Evidently, this new Lagrangian is quantum-equivalent to the one with Y a “ , as we merelychose to work in a different gauge.Subsequently, we may perform the shift b a ÞÑ b a ` Z a (8.22)in the Nakanishi–Lautrup field with Z a polynomials in the fields and their derivatives. Thecombination of this shift and (8.20) results in L YMBRST ÞÑ L YMBRST ` δ Ξ δA aµ p ∇ µ c q a ` g f bca δ Ξ δc a c b c c `` ξ Z a Z a ` Z a p ξb a ` B µ A aµ q ´ p b a ` Z a q δ Ξ δ ¯ c a . (8.23)We shall assume that Z a is independent of the Nakanishi–Lautrup field as this will yield aquantum-equivalent Lagrangian by Observation 8.6. We shall also assume that Z a dependsat least quadratically on the other fields and their derivatives to preserve the linearisedBRST action on the BRST-extended Hilbert space introduced in Section 8.1. Interaction terms linear in the Nakanishi–Lautrup fields.
Let us now consider thefollowing special case: suppose that we are in R ξ -gauge and that our Lagrangian contains aterm Z a B µ A aµ with Z a independent of the Nakanishi–Lautrup field and at least quadratic inthe fields and their derivatives. On the physical Hilbert space with transversely polarisedgluons, such expressions vanish. Off-shell, we can still remove such terms by the shifts (8.22).108iven the need to shift by Z a , we can then iteratively construct a Y a which cancels anynew terms linear in b a , as is clear from (8.23). Explicitly, we solve the equation “ ξZ a ´ δ Ξ δ ¯ c a “ ξZ a ´ Y a ` ¯ c b B Y b B ¯ c a ` ¨ ¨ ¨ , (8.24)where the ellipsis denotes terms containing partial derivatives with respect to derivativesof the anti-ghost field ¯ c b . Clearly, for consistency, Y a needs to be at least quadratic in thefields and their derivatives because Z a is. We are left with the terms ´ ξ Z a Z a ` δ Ξ δA aµ p ∇ µ c q a ` g f bca δ Ξ δc a c b c c , (8.25)which are either at least quartic in the fields or at least cubic in the fields, containing ghostfields. The ability to remove any terms of the form Z a pB µ A µ q a through local shifts of theNakanishi–Lautrup field, absorbing them into b a , and a compensating gauge choice is the‘off-shell’ Lagrangian analogue of being able to impose that the on-shell external gluons inan amplitude are transverse. We summarise as follows. Observation 8.7.
Interaction terms in the Lagrangian of degree n ě of the form Z a pB µ A µ q a with Z a independent of the Nakanishi–Lautrup field can be removed in R ξ -gaugeby shifting the Nakanishi–Lautrup field according to (8.22) . This creates the additionalterms (8.25) which do no modify the scattering amplitudes by Observation 8.6 and, in ad-dition, contribute only to interaction vertices of degree n with more ghost–anti-ghost pairsor to interaction vertices of degree greater than n . We also note that a shift of the gauge-fixing fermion by itself (8.20) allows us to absorbphysical terms proportional to the Nakanishi–Lautrup field without further affecting thephysical sector.
Observation 8.8.
Terms in the action that are proportional to the Nakanishi–Lautrup fieldcan be absorbed by choosing a suitable term Y a . This leaves the physical sector invariant butit may modify the ghost sector. Because Nakanishi–Lautrup fields appear via trivial pairs inthe BV action, this extends to general gauge theories, e.g. with several Nakanishi–Lautrupfields and ghosts–for–ghosts. Actions related by field redefinitions.
Let us return to a general setting. Supposethat we are given two classical field theories which are specified by local actions S and ˜ S ,as power series in the fields and their derivatives, whose corresponding L -algebras havethe same minimal model, the same field content and the same kinetic parts.Consider the cubic interaction terms L and ˜ L in S and ˜ S . Since the three-point amp-litudes agree, these interaction terms can differ at most in terms that vanish on external109elds. Therefore, these terms have to be proportional to either the on-shell equation foran external field or to a field with unphysical polarisation which is not contained in theexternal fields. Both types of terms can be cancelled by a local field redefinition which shiftsthe discrepancy into the quartic and higher interaction terms. Such field redefinitions con-stitute a quasi-isomorphism of L -algebras and leaves the tree-level scattering amplitudesunmodified. We are left with two theories with the same tree-level scattering amplitudesand which agree to cubic order in the interaction terms.The discrepancy between the total quartic terms of both field theories after the abovefield redefinition is again invisible at the level of external fields, because the tree-levelscattering amplitudes still agree. We then compensate again by further field redefinitions,shifting the discrepancy into quintic and higher interaction terms. In this way, we canremove the differences between the Lagrangians order by order in the interaction vertices,field-redefining the difference away to higher order interaction vertices. Since we are merelyinterested in perturbation theory, agreements to arbitrary finite orders are completely suf-ficient.Altogether, we can conclude that for the purpose of perturbative quantum field theory,we can regard the actions S and ˜ S to be related by local field redefinitions. In certain casesit is even possible to give closed all order expression for (part of) the field redefinitions,providing a formal non-perturbative equivalence. Observation 8.9.
If two field theories have the same tree-level scattering amplitudes, thenthe minimal models of the corresponding L -algebras coincide, cf. [177, 209]. If also theassociated actions are local and given by power series of the fields and their derivatives, andhave the same field content and kinetic parts, then they are related by local (invertible) fieldredefinitions. The explicit example of Yang–Mills theory may be instructive. Consider the ac-tion (7.10) of Yang–Mills theory in R ξ -gauge with the field redefinitions (7.8) implementedas in Section 7.2 and consider an action ˜ S with the same fields, the same kinematic partsand identical tree-level scattering amplitudes. The discrepancies in the interaction verticesat each order are proportional to (at least) one of the terms ˜ A Ò aµ , ? l ˜ b a ` ˜ ξ B µ ˜ A aµ , l ˜ A aµ , l ˜ c a , l ˜¯ c a , and l ˜ b a . (8.26)Given the BRST invariance, we can always exclude terms proportional to ˜ A Ò aµ , as these canbe absorbed by residual gauge transformations. Terms proportional to ? l ˜ b a ` ˜ ξ B µ ˜ A aµ canbe absorbed by a field redefinition of the Nakanishi–Lautrup field due to Observation 8.7.All remaining differences are sums of terms proportional to l ˜ A aµ , l ˜ c a , l ˜¯ c a , or l ˜ b a ,110nd they can be absorbed by iterative field redefinitions, starting with the three-pointamplitudes. There is an evident field redefinition of the relevant field, quadratic in the fieldsand their derivatives, such that the kinetic term of redefined Yang–Mills theory producesthe difference in kinetic terms. Since such a field redefinition is a quasi-isomorphism ofthe corresponding L -algebras, it preserves the minimal model and thus the tree-levelamplitudes. Moreover, such a field redefinition is clearly local. An important structure theorem for homotopy algebras is the strictificationtheorem, cf. Appendix A.4. In particular, it implies that any L -algebra is quasi-isomorphicto a strict L -algebra, i.e. an L -algebra with µ i “ for i ě , better known as a differentialgraded Lie algebra.From a field theory perspective, this implies that any classical field theory is equivalentto a classical field theory with interaction terms which are all cubic in the fields. Gener-ically, a strictifying quasi-isomorphism may produce non-local terms, but there is alwaysa systematic choice of strictification that is entirely local. This is quite evident for the in-teractions of scalar fields, since we can ‘blow up’ n -ary vertices to cubic graphs with edgescorresponding to propagating auxiliary fields, cf. e.g. the discussions in [177, 209].As a simple example of a strictification, consider the first-order formalism of Yang–Millstheory on four-dimensional Euclidean space R [249], in which an additional self-dual twoform B ` P Ω ` p R q b g in the adjoint representation of the gauge Lie algebra is added tothe field content, S YM : “ ż d x ! ε µνκλ F aµν B a ` κλ ` ε µνκλ B ` aµν B a ` κλ ) . (8.27)The L -algebra corresponding to the full BV complex of this theory is indeed strict; see [192,177] for a quasi-isomorphism between this L -algebra and that of the ordinary, second-orderformulation of Yang–Mills theory.Note, however, that the full strictification of gauge theories including ghosts is a bitmore involved: the equations of motion of the introduced auxiliary fields would be atleast quadratic in other fields, and if these transform in the adjoint representation or asconnections, the gauge transformations of auxiliary fields are at least cubic in fields andghosts, leading to quartic or higher terms in the BV action. The strictification theorem stillguarantees the existence of an equivalent formulation as a field theory with cubic interactionvertices, but we may have to extend our field space not merely by adding fields, but byswitching e.g. to its loop space. This is due to the fact that cubic gauge transformations in111n L -algebra are encoded in a µ , which in turn corresponds to a particular three-cocycle.The latter can be transgressed to a two-cocycle over loop space, which merely correspondsto a Lie algebra extension and thus, is turned into a higher product µ . For fully gauge-fixedactions, however, this problem never arises.We also note that the factorisation in the double copy is most easily performed in aspecific strictification , which is not the first order formulation (8.27). Its precise form isdiscussed in the following. Colour–kinematics-dual form and cubic diagrams.
Recall from Section 2.2 that thetree-level Yang–Mills amplitudes can be rearranged in colour–kinematics-dual form, whichis by now a well-established fact [8, 7, 293, 25, 234, 294, 26, 28, 29].
Observation 8.10.
The tree amplitudes of Yang–Mills theory can be written in colour–kinematics-dual form.
Explicitly, one can construct a Lagrangian whose Feynman diagrams generate colour–kinematics-dual tree-level amplitudes of physical (transverse) gluons in Yang–Mills theory,making colour–kinematics duality manifest at the Lagrangian level. This is achieved byadding non-local interaction terms O p A n q , for all n ą , to the action that vanish identicallydue to the colour Jacobi identity. The necessary terms were first constructed in [6] up tosix points. The algorithm of Tolotti–Weinzierl [80] is a prescription of how to find thenecessary terms to arbitrary order.Since the new terms are identically zero they obviously leave the theory and amplitudesinvariant, but nonetheless change the individual kinematic numerators to realise colour–kinematics duality. Moreover, the new terms can be rendered cubic and local through theintroduction of auxiliary fields [9], as demonstrated explicitly at five points in [6]. Roughlyspeaking, one starts from Yang–Mills theory and strictifies the already present quarticinteraction vertex by inserting an auxiliary field, redistributing the contributions to ensurecolour–kinematics duality for four-point amplitudes. The colour–kinematics duality of thefive-point amplitudes then requires a new interaction term O p A q which vanishes due tothe Jacobi identity. This vertex is then strictified by inserting further auxiliary fields, etc.The overall action is thus trivially equivalent to Yang–Mills theory. We note that the formof the strictification is encoded in the action produced by the Tolotti–Weinzierl algorithm.We shall be completely explicit below, but let us first summarise the situation. Observation 8.11.
Given tree-level physical gluon amplitudes in colour–kinematics-dualform, there is a corresponding purely cubic Lagrangian whose Feynman diagrams (summed It is actually a family of strictifications. ver identical topologies) produce kinematic numerators satisfying the kinematic Jacobi iden-tities.
To illustrate the strictification, let us consider the four- and five-point contributions,which were already computed in [6]: L p q „ tr (cid:32) r A µ , A ν sr A µ , A ν s ( “ ´ η µν η κρ η λσ B µ B ν tr (cid:32) r A κ , A λ sr A ρ , A σ s ( l , L p q „ tr " r A ν , A ρ s l ˆ„ rB µ A ν , A ρ s , ll A µ `` „ r A ρ , A µ s , ll B µ A ν ` „ r A µ , B µ A ν s , ll A ρ ˙* . (8.28)We immediately note that L p q vanishes by the colour Jacobi identity. Its presence, how-ever, is required for the kinematic Jacobi identity to hold after factorisation.As explained in Section 2.4, these terms reflect a ‘blow up’ of n -point interaction verticesinto trees with trivalent vertices and all symmetries taken into account: n “ , , ,n “ , , . . . (8.29)Here, an internal wavy line comes with a propagator in Feynman gauge l , while a dashedline corresponds to the identity operator ll .The general Lagrangian at n -th order is then of the form L p n q “ f M ¨¨¨ M k E M D p E M D p E M D ¨ ¨ ¨ qq , (8.30)where D i stands for either l or ll and the M i s are Lorentz multi-indices. Note that allthe E i s are polynomials of degree one or two in the fields. In the tree picture, the waveoperators in the denominator correspond precisely to the edges in the trees. Strictification.
To strictify the non-local action, we now iteratively insert auxiliary fields G Mn, Γ ,i and ¯ G n, Γ ,iM for each operator D i . If we are dealing with an operator of the form ll ,113e first use partial integration E M l E M l “ ´ pB µ E M qpB µ E M q l , (8.31)where E Mi is an arbitrary expression in the fields, derivatives, and auxiliary fields. We thenuse the fact that the Lagrangians E M l E M (8.32a)and ´ G Mn, Γ ,i l ¯ G n, Γ ,iM ` G Mn, Γ ,i E M ` E M ¯ G n, Γ ,iM (8.32b)are physically equivalent after integrating out the auxiliary fields G Mn, Γ ,i and ¯ G n, Γ ,iM . Weiterate this process until all the inverse wave operators have been replaced in this manner.We note that in each iteration, E M and E M are both polynomials of degree at leasttwo in the fields. Introducing the auxiliary fields reduces the polynomial degree at leastby one, and in the end, the action has indeed only cubic interaction terms and thus is astrictification of the original action. We also note that two auxiliary fields can be combinedinto one if they have identical equations of motion. Homotopy algebraic perspective.
The strictification L YM , stBRST of the L -algebra L YMBRST or, equivalently, of the colour–kinematics-dual action is nothing but a quasi-isomorphism(see Appendix A.3) φ : L YMBRST Ñ L YM , stBRST , (8.33)and the map φ is given by A st ` ÿ n, Γ ,i G n, Γ ,i “ φ p A q ` φ p A, A q ` ¨ ¨ ¨ “ ÿ k ě k ! φ k p A, . . . , A q , (8.34)where A st is the gauge potential in L YM , st , A st “ φ p A q , (8.35)and the higher maps are such that G n, Γ ,i are given by their equations of motion, fullyreduced to expressions in the original gauge potential A .Let us work out the details for the example of the fourth- and fifth-order terms (8.28).The explicit form of the corresponding strictified Lagrangian is already found in [6], L YM , st : “ tr (cid:32) A µ l A µ ( ` L YM , st4 ` L YM , st5 , (8.36a)114ith L YM , st4 : “ tr ! ´ G µνκ , Γ , l G , Γ , µνκ ´ g pB µ A ν ` ? B κ G , Γ , κµν qr A µ , A ν s ) , L YM , st5 : “ tr ! G µν , Γ , l ¯ G , Γ , µν ` G µνκ , Γ , l ¯ G , Γ , µνκ ` G µνκλ , Γ , l ¯ G , Γ , µνκλ `` gG µν , Γ , r A µ , A ν s ` g B µ G µνκ , Γ , r A ν , A κ s ´ g B µ G µνκλ , Γ , rB r ν A κ s , A λ s `` g ¯ G µν , Γ , ` rB κ ¯ G , Γ , κλµ , B λ A ν s ` rB κ ¯ G , Γ , κλν r µ , A λ s ˘) . (8.36b)Consequently, the resulting quasi-isomorphism reads as φ p A q ` φ p A, A q ` φ p A, A, A q ““ ¨˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˝ A sµ G , Γ , µνκ G , Γ , µν ¯ G , Γ , µν G µνκ , Γ , ¯ G , Γ , µνκ G µνκλ , Γ , ¯ G , Γ , µνκλ ˛‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‚ “ ¨˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˝ A µg l B µ r A ν , A κ s´ g l ` rr A λ , A µ s , B λ A ν s ´ rrB r λ A ν s , A µ s , A λ s ˘ ´ g l r A µ , A ν s´ g l B µ “ B ν A λ , l r A κ , A λ s ‰ g l B µ r A ν , A κ s´ g l B µ “ A ν , l r A λ , A κ s ‰ ´ g l B µ rB r ν A κ s , A λ s ˛‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‚ . (8.37)Note that the decomposition into the images of the maps φ i corresponds to the decompos-ition of the image into monomials of power i in the fields. Tree-level double copy.
As reviewed in Section 2.3, the double copy of the kinematicnumerators in the scattering amplitudes of the strictified Yang–Mills theory produces thetree-level scattering amplitudes of N “ supergravity [4–6]. Observation 8.12.
Double copying the Yang–Mills tree-level scattering amplitudes of phys-ical gluons in colour–kinematics-dual form yields the physical tree-level scattering amplitudesof N “ supergravity. Compatibility with quantisation.
It is clear that quantisation does not commute withquasi-isomorphisms: classically equivalent field theories can have very different quantumfield theories. A simple example making this evident is the L -algebra of Yang–Mills theory L YMBRST and one of its quasi-isomorphic minimal models L YM ˝ BRST . The vector space of L YM ˝ BRST issimply the free fields labelling external states in Yang–Mills scattering amplitudes, togetherwith some irrelevant cohomological remnants in the ghosts, Nakanishi–Lautrup fields, andanti-ghosts. The tree-level scattering amplitudes of L YMBRST are given by the higher products115f L YM ˝ BRST . They are also the tree-level scattering amplitudes of L YM ˝ BRST since there areno propagating degrees of freedom left. Clearly, however, there are loop-level scatteringamplitudes in Yang–Mills theory which L YMBRST can describe but which are absent in L YM ˝ BRST .Thus, the quantum theories described by the quasi-isomorphic L -algebras L YMBRST and L YM ˝ BRST differ.Certainly, there are quasi-isomorphisms which are compatible with quantisation. Inparticular, any quasi-isomorphism that corresponds to integrating out fields which appearsat most quadratically in the action are of this type: we can simply complete the squarein the path integral and perform the Gaußian integral. This amounts to replacing eachauxiliary field by the equation of motion.This is precisely the case in the above strictification of Yang–Mills theory, and theoriginal formulation is quantum equivalent to its strictification. This is also clear at thelevel of Feynman diagrams: as the kinematic terms are all of the form ´ G Mn, Γ ,i l ¯ G n, Γ ,iM , eachauxiliary field propagates into precisely one other auxiliary field. Moreover, each auxiliaryfield G appears in precisely one type of vertex and then only as one leg. That is, once apropagator ends in one of the auxiliary fields, the continuation of the diagram at this end inunique until all the remaining open legs are non-auxiliaries. There are no loops consistingof purely auxiliary fields. All loops containing at least one gluon propagator are simplycontracted to gluon loops. It is thus clear that the degrees of freedom running aroundloops in the strictified theory are the same as those running around in ordinary Yang–Millstheory. The action and factorisation we have presented so far are the complete data to double copytree-level gauge theory amplitudes to gravity amplitudes. For the full double copy at theloop level, however, we need to work a bit harder, as explained in our previous paper [9].So far, colour–kinematics duality is only ensured for all on-shell gluon states with phys-ical polarisation. Our goal will be to double copy arbitrary tree-level correlators, whichcan have unphysical polarisations of gluons as well as ghost states on external legs. Wetherefore need to ensure that colour–kinematics duality holds more generally. In order toestablish the off-shell double copy it is sufficient to guarantee colour–kinematics duality foron-shell states in the BRST-extended Hilbert space from Section 8.1.
Unphysical states.
Colour–kinematics duality is not affected by forward-polarisedgluons, as these can be absorbed by residual gauge transformations. Furthermore, colour–116inematics duality for backward-polarised gluons can be achieved by adding new termsto the action, which are physically irrelevant since they are introduced only through thegauge-fixing fermion. Colour–kinematics duality for ghosts is then achieved by transferringcolour–kinematics duality for longitudinal gluons to the ghost sector by Observation 8.2 viathe BRST Ward identities. We now explain the procedure in detail.We perform the corrections order by order in the degree n of the vertices and for eachdegree order by order in the number k of ghost–anti-ghost pairs. The first vertex to consideris n “ , and we start at k “ . Colour–kinematics duality for four on-shell gluons in theBRST-extended Hilbert space can only be violated by terms proportional to ξb a ` B µ A aµ and we can introduce a vertex compensating these violations in the Lagrangian. We do thisdirectly in a BRST-invariant fashion, and a short calculation shows that the appropriateaddition to the Lagrangian is L YM , comp n “ , k “ “ ´ ξ " b b A cµ l “ pB ν A dµ q A eν ‰ ´ ¯ c b Q BRST ˆ A cµ l “ pB ν A dµ q A eν ‰˙* f eda f acb . (8.38)Here, the first term compensates the colour–kinematics duality violating term for fourgluons and the second term renders the compensation BRST-invariant, thus ensuring Q BRST L YM , corr n “ , k “ “ . (8.39)To show that these terms are indeed unphysical and that they do not modify the tree-levelcorrelation functions, we use Observations 8.7 and 8.8: these terms are produced by ashift (8.22) of the form Z a : “ ´ A cµ l “ pB ν A dµ q A eν q ‰ f edb f bca and Y a : “ ξ Z a . (8.40)We note that the terms in L YM , comp n “ , k “ come with a canonical strictification given by thecolour structure. This strictification then yields colour–kinematics-dual four-point gluonamplitudes.The next case to consider is n “ , k “ . We now use Observation 8.4 to relatethe four-gluon correlation function to this correlation function, and, correspondingly, thefour-gluon tree-level correlator to the two gluon, one ghost-anti-ghost pair correlator. Weobtain colour–kinematics duality for amplitudes consisting of a ghost–anti-ghost pair aswell as two physically polarised gluons. Generalising the latter to two arbitrary gluonsin the BRST-extended Hilbert space, we expect colour–kinematics duality violating termsproportional to ξb a ` B µ A aµ . It turns out that these terms happen to vanish and there isnothing left to do. Note that if these terms had not vanished, we would have compensated117or them again by inserting physically irrelevant terms to the action in a BRST-invariantfashion.Observation 8.4 now immediately implies that the amplitudes for n “ , k “ arecolour–kinematics-dual, because those for n “ , k “ are.So far, we constructed a strict Lagrangian for Yang–Mills theory with the same tree-levelscattering amplitudes for the BRST-extended Hilbert space as ordinary Yang–Mills theory,but with a manifestly colour–kinematics-dual factorisation of the four-point amplitudes.We now turn to n “ , k “ and iterate our procedure in the evident fashion:Step 1) Identify the colour–kinematics duality violating terms. They are necessarily pro-portional to ξb a ` B µ A aµ .Step 2) Compensate by inserting a corresponding non-local vertex. Complete the com-pensating term to a BRST-invariant one, which may be deduced directly via thegauge-fixing fermion.Step 3) The colour structure of the vertices induces a canonical strictification, implementthis strictification.Step 4) Use Observation 8.4 to transfer colour–kinematics duality to tree level correlatorswith one more ghost–anti-ghost pair, but all other gluons physically polarised.Step 5) Continue with Step 1), if there is room for backward-polarised gluons. Otherwiseturn to the next higher n -point scattering amplitudes.The outcome of this construction is a strictified BRST action for Yang–Mills theorywhich is perturbatively quantum equivalent to ordinary Yang–Mills theory and whose scat-tering amplitudes come canonically factorised in colour–kinematics-dual form.We note that this action comes with a BRST operator which is cubic in the fields ofthe BRST-extended Hilbert space, but of higher order in its action on the auxiliary fieldsintroduced in strictification.
9. Double copy from factorisation of homotopy algebras
We now turn to the factorisation of the full, interacting theories. In this case, the doublecopy procedure is implied by the factorisations ˜ L YM , stBRST “ g b p Kin st b τ Scal q and ˜ L N “ , stBRST “ Kin st b τ p Kin st b τ Scal q , (9.1)which now hold at the level of strict homotopy algebras. In order to establish these factor-isations, we use the definition of twisted tensor products of differential graded algebras wepresented in (6.21). 118 .1. Biadjoint scalar field theory Let us start with a brief consideration of the factorisation of biadjoint scalar field theory,cf. Section 5.1. This theory does not require any twists, and we lift the factorisation ofchain complexes (7.3) to the factorisation L biadjBRST “ g b p ¯ g b Scal q (9.2)into (strict) L -algebras. In general, a tensor product between a Lie algebra and an L -algebra is not well-defined; in particular, it is not a homotopy version of any of the productsin the list (6.1). However, for nilpotent L -algebras, i.e. L -algebras with µ i ˝ µ j “ , theproduct exists and yields a C -algebra. The latter is then further tensored by a Lie algebrain the canonical way as explained in Section 6.1, leading to an L -algebra. L -algebra Scal . Explicitly, the L -algebra Scal is built from the chain complex (7.4),
Scal : “ ˜ s x F r´ s l ÝÝÑ s ` x F r´ s ¸ , (9.3a)and the only non-vanishing higher product beyond the differential µ Scal is µ Scal ˆż d d x s x ϕ p x q , ż d d x s x ϕ p x q ˙ : “ λ ż d d x s ` x ϕ p x q ϕ p x q . (9.3b)Evidently, Scal is nilpotent.
Factorisation.
Following the prescription for the untwisted tensor product of strict ho-motopy algebras from Section 6.1, we obtain the binary product µ p e a b ¯ e ¯ a b s x , e b b ¯ e ¯ b b s x q “ r e a , e b s b r ¯ e ¯ a , ¯ e ¯ b s b λδ p d q p x ´ x q s ` x , (9.4)which, together with the differential µ p e a b ¯ e ¯ a b s x q “ e a b ¯ e ¯ a b l s ` x , (9.5)and the cyclic structure x ϕ, ϕ ` y “ ż d d x ϕ a ¯ a p x q ϕ ` a ¯ a p x q , (9.6)forms the cyclic L -algebra L biadjBRST . The homotopy Maurer–Cartan action of this L -algebra is then the action (5.2) of biadjoint scalar field theory, S biadj “ x ϕ, µ p ϕ qy ` x ϕ, µ p ϕ, ϕ qy“ ż d d x ! ϕ a ¯ a l ϕ a ¯ a ´ λ f abc f ¯ a ¯ b ¯ c ϕ a ¯ a ϕ b ¯ b ϕ c ¯ c ) , (9.7)which verifies (9.2). 119 .2. Strictified Yang–Mills theoryGeneral considerations. The strictification of Yang–Mills theory formulated in Sec-tion 8.3 is now readily extended to a BV action, which can then be gauge fixed and convertedinto a strict L -algebra ˜ L YM , stBRST .The full strictification of Yang–Mills theory involves an infinite number of additionalauxiliary fields and corresponding interaction terms in the Lagrangian. Thus, our discussioncannot be fully explicit and has to remain somewhat conceptual, but as before, we shall giveexplicit lowest order terms to exemplify our discussion. Recall, however, that for computing n -point correlation function at the tree-level, only a finite number of auxiliary fields andinteraction terms are necessary. Moreover, for computing n -point scattering amplitudes upto (cid:96) loops, only a finite number of correlators is necessary. Therefore, we can always truncatethe Yang–Mills action to finitely many auxiliary fields to perform our computations.We note that gauge fixing of Yang–Mills theory is fully equivalent to gauge fixing ofthe strictified theory. Moreover, the additional interaction vertices that arise from the BVformalism are all cubic, except for the terms involving anti-fields of the auxiliary fields; thelatter, however, will not contribute.The last point implies that the L -algebra ˜ L YM , stBRST for the strictified and gauge-fixedform of Yang–Mills theory contains the chain complex of the L -algebra ˜ L YMBRST which wehave computed in Section 5.2. This chain complex is enlarged by the kinematic termsfor all the auxiliary fields. We then have additional binary products encoding the cubicinteractions. L -algebra of Yang–Mills theory. For concreteness, we consider the strictification upto quartic terms, as explained in Section 8.3. By the arguments given there, however, itis clear that our discussion trivially generalises to strictifications up to an arbitrary order.The Lagrangian, including the strictification of the colour–kinematics duality producingterms (8.38), reads as L YM , stBRST , “ ˜ A aµ l ˜ A µa ´ ˜¯ c a l ˜ c a ` ˜ b a l ˜ b a ` ˜ ξ ˜ b a ? l B µ ˜ A µa ´ gf abc ¯ c a B µ p A bµ c c q ´´ ˜ G µνκa l ˜ G aµνκ ` gf abc ´ B µ A aν ` ? B κ ˜ G aκµν ¯ A µb A νc ´´ ˜ K µ a l ˜¯ K aµ ´ ˜ K µ a l ˜¯ K aµ ´´ gf abc ! ˜ K aµ pB ν ˜ A bµ q ˜ A cν ` ”´ ´ b l ξ ˜ b a ` ´? ´ ξ ? ξ B κ ˜ A aκ ¯ ˜ A bµ ´ ˜¯ c a B µ ˜ c b ı ˜¯ K cµ ) `` gf abc ! ˜ K aµ ” pB ν B µ ˜ c b q ˜ A cν ` pB ν ˜ A bµ qB ν ˜ c c ı ` ˜¯ c a ˜ A bµ ˜¯ K cµ ) , (9.8)120here K aµi and ¯ K aiµ are auxiliary g -valued one-forms, strictifying L YM , compBRST , n “ , k “ , and weused the shorthand ˜ G aµνκ : “ ˜ G ,γ, ,aµνκ . Note that K aµ and ¯ K a µ are of ghost number zero,while K aµ and ¯ K a µ carry ghost numbers ´ and ` , respectively. The L -algebra ˜ L YM , stBRST to quartic order has underlying chain complex p ˜ K aµ , ˜¯ K aµ q R b Ω p M d q b g p ˜ K ` aµ , ˜¯ K ` aµ q R b Ω p M d q b g ˜ G aµνκ b Ω p M d q b g ˜ G ` aµνκ b Ω p M d q b g ˜ A aµ Ω p M d q b g ˜ A ` aµ Ω p M d q b g ˜ b a C p M d q b g ˜ b ` a C p M d q b g ˜¯ K aµ Ω p M d q b g ˜ K ` aµ Ω p M d q b g ˜ K aµ Ω p M d q b g ˜¯ K ` aµ Ω p M d q b g ˜ c a C p M d q b g loooooomoooooon “ : ˜ L YM , stBRST , ˜¯ c ` a C p M d q b g loooooomoooooon “ : ˜ L YM , stBRST , ˜¯ c a C p M d q b g loooooomoooooon “ : ˜ L YM , stBRST , ˜ c ` a C p M d q b g loooooomoooooon “ : ˜ L YM , stBRST , lll ˜ ξ ? l B µ l ´ ˜ ξ ? l B µ ´ l ´ l ´ l ´ l (9.9a)Besides the differentials in (9.9a), we also have the following higher products ¨˚˚˚˚˚˚˚˝¨˚˚˚˚˚˚˚˝ ˜ K aµ ˜¯ K aµ ˜ G aµνκ ˜ A aµ ˜ b ˛‹‹‹‹‹‹‹‚ , ˜ ˜¯ K aµ ˜ c a ¸˛‹‹‹‹‹‹‹‚ µ ÞÝÝÑ gf bca ˜ pB ν ˜ A bµ qB ν ˜ c c ´ ˜ A bν B ν B µ ˜ c c ´B µ p ˜ A bµ ˜ c c q ´ ˜¯ K b µ pB µ ˜ c c q ` ˜ A bµ ˜¯ K cµ ¸ P à φ P p ˜ K ` , ˜¯ c ` q ˜ L YM , stBRST , , φ , (9.9b)121 ˜ ˜¯ K aµ ˜ c a ¸ , ˜ ˜ K aµ ˜¯ c a ¸¸ µ ÞÝÝÑ gf bca ˜ ´pB µ ˜ c b q ˜¯ c c ´ ˜¯ K bµ ˜¯ c c ` pB µ B ν ˜ c b q ˜ K cν ` B ν pB ν ˜ c b ˜ K c µ q ´ ˜ c b B µ ˜¯ c c ¸ P à φ P p ˜¯ K ` , ˜ A ` q ˜ L YM , stBRST , , φ , (9.9c) ¨˚˚˚˚˚˚˚˝¨˚˚˚˚˚˚˚˝ ˜ K aµ ˜¯ K aµ ˜ G aµνκ ˜ A aµ ˜ b a ˛‹‹‹‹‹‹‹‚ , ¨˚˚˚˚˚˚˚˝ ˜ K aµ ˜¯ K aµ ˜ G aµνκ ˜ A aµ ˜ b a ˛‹‹‹‹‹‹‹‚˛‹‹‹‹‹‹‹‚ µ ÞÝÝÑ gf bca ¨˚˚˚˚˚˚˚˚˝ pB ν ˜ A bµ q ˜ A cν ´? ´ ξ ? ξ pB κ ˜ A bκ q ˜ A cµ ` b l ξ p ˜ A bµ ˜ b c q? B µ p ˜ A bν ˜ A cκ q R ˜ A ` bcµ ´ b l ξ ` ˜¯ K bµ ˜ A cµ ˘ ˛‹‹‹‹‹‹‹‹‚ P à φ P p ˜ K ` , ˜¯ K ` , ˜ G ` , ˜ A ` , ˜ b ` q ˜ L YM , stBRST , , φ ,R ˜ A ` bcµ : “ ´ B ν p ˜ A bν ˜ A cµ q ´ ? A νb B κ ˜ G cκνµ ´ K bν B µ ˜ A cν ´´ ´ ? ´ ξ ? ξ pB κ ˜ A bκ q ˜¯ K cµ ` K bµ c l ξ ˜ b c , (9.9d)and ¨˚˚˚˚˚˚˚˝¨˚˚˚˚˚˚˚˝ ˜ K aµ ˜¯ K aµ ˜ G aµνκ ˜ A aµ ˜ b a ˛‹‹‹‹‹‹‹‚ , ˜ ˜ K aµ ˜¯ c a ¸˛‹‹‹‹‹‹‹‚ µ ÞÝÝÑ gf bca ˜ ˜ A bµ ˜¯ c c ´ ˜ A bµ B µ ˜¯ c c ´ B µ p ˜¯ K bµ ˜¯ c c q ` B ν B µ p ˜ A bν ˜ K cµ q ¸ P à φ P p ˜¯ K ` , ˜ c ` q ˜ L YM , stBRST , , φ , (9.9e)122nd the cyclic structure is given by x ˜ A, ˜ A ` y : “ ż d d x ˜ A aµ ˜ A ` µa , x ˜ b, ˜ b ` y : “ ż d d x ˜ b a ˜ b ` a , x ˜ c, ˜ c ` y : “ ż d d x ˜ c a ˜ c ` a , x ˜¯ c, ˜¯ c ` y : “ ´ ż d d x ˜¯ c a ˜¯ c ` a , x ˜ K , ˜ K ` y : “ ´ ż d d x ˜ K aµ ˜ K ` aµ , x ˜¯ K , ˜¯ K ` y : “ ´ ż d d x ˜¯ K aµ ˜¯ K ` µa , x ˜ K , ˜ K ` y : “ ´ ż d d x ˜ K aµ ˜ K ` aµ , x ˜¯ K , ˜¯ K ` y : “ ż d d x ˜¯ K aµ ˜¯ K ` µa , x ˜ G, ˜ G ` y : “ ´ ż d d x ˜ G aµνκ ˜ G ` µνκa . (9.9f) Factorisation and twist datum.
We factorise this L -algebra as ˜ L YM , stBRST “ g b p Kin st b τ Scal q , (9.10)where g is the usual colour Lie algebra, Kin st the graded vector space Kin st : “ ¨˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˝ ¯ t µ M d ‘ g R r s loomoon “ : Kin st ´ ‘ t µ , ¯ t µ M d ‘ M d ‘ t µνκ M d b p M d ^ M d q‘ v µ M d ‘ R n looooooooooomooooooooooon “ : Kin st0 ‘ t µ M d ‘ a R r´ s looomooon “ : Kin st1 ˛‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‚ , (9.11)and Scal the L -algebra defined in (9.3). This L -algebra is cyclic with the inner productsgiven by (7.17) together with x t µ , ¯ t ν y : “ ´ δ νµ , x ¯ t ν , t µ y : “ ´ δ νµ , x t µ , ¯ t ν y : “ δ νµ , x ¯ t ν , t µ y : “ δ νµ , x t µνκ , t λρσ y : “ ´ η µλ p η νρ η κσ ´ η νσ η κρ q . (9.12)The twist datum τ , see (6.20) for the general definition, in the factorisation (9.10) isthen given by the maps τ p g q : “ g b id , τ p t iµ q : “ t iµ b id , τ p ¯ t µi q : “ ¯ t µi b id ,τ p t µνκ q : “ t µνκ b id ,τ p v µ q : “ v µ b id ` ˜ ξ n b l ´ B µ ,τ p n q : “ n b id ´ ˜ ξ v µ b l ´ B µ , τ p a q : “ a b id (9.13a)123 elds anti-fieldsfactorisation | ´ | gh | ´ | L dim factorisation | ´ | gh | ´ | L dim ˜ c “ e a gs x ˜ c a p x q d ´ c ` “ e a as ` x ˜ c ` a p x q ´ d ` A “ e a v µ s x ˜ A aµ p x q d ´ A ` “ e a v µ s ` x ˜ A ` aµ p x q ´ d ` b “ e a ns x ˜ b a p x q d ´ b ` “ e a ns ` x ˜ b ` a p x q ´ d ` c “ e a as x ˜¯ c a p x q ´ d ˜¯ c ` “ e a gs ` x ˜¯ c ` a p x q d ˜ K “ e a t µ s x ˜ K µ p x q d ´ K ` “ e a t µ s ` x ˜ K ` aµ p x q ´ d ´ K “ e a ¯ t µ s x ˜¯ K aµ p x q d ´ K ` “ e a ¯ t µ s ` x ˜¯ K ` aµ p x q ´ d ´ K “ e a t µ s x ˜ K µ p x q ´ d ´ K ` “ e a t µ s ` x ˜ K ` aµ p x q d ´ K “ e a ¯ t µ s x ˜¯ K aµ p x q d ´ K ` “ e a ¯ t µ s ` x ˜¯ K ` aµ p x q ´ d ´ G “ e a t µνκ s x ˜ G aµνκ p x q d ´ G ` “ e a t µνκ s ` x ˜ G ` aµνκ p x q ´ d ´ Table 9.1: Factorisation of the fields in the L -algebra corresponding to the Lagrangian L YM , stBRST , . Note that we suppressed the integrals over x and the tensor products for simpli-city.and τ p g , v µ q : “ ´ g b p id b B µ ` B µ b id q ` t µ b pB ν b B ν ´ B µ B ν b id q ,τ p v µ , g q : “ g b p id b B µ ` B µ b id q ´ t µ b pB ν b B ν ´ id b B µ B ν q ,τ p g , ¯ t µ q : “ ´ g b B µ b id ,τ p ¯ t µ , g q : “ g b id b B µ ,τ p ¯ t µ , v ν q : “ η µν g b id b id ,τ p v µ , ¯ t ν q : “ ´ η µν g b id b id ,τ p g , a q : “ v µ b id b B µ ` ¯ t µ b B µ b id ,τ p a , g q : “ ´ v µ b B µ b id ´ ¯ t µ b id b B µ ,τ p ¯ t µ , a q : “ v µ b id b id ,τ p a , ¯ t µ q : “ ´ v µ b id b id ,τ p g , t µ q : “ ´ v ν b B ν B µ b id ´ v µ b l b id ´ v µ b B ν b B ν ,τ p t µ , g q : “ v ν b id b B ν B µ ` v µ b id b l ` v µ b B ν b B ν ,τ p v µ , a q : “ ´ ¯ t µ b id b id ` a b id b B µ ,τ p a , v µ q : “ ¯ t µ b id b id ´ a b B µ b id ,τ p ¯ t µ , a q : “ a b pB µ b id ` id b B µ q , (9.13b) τ p a , ¯ t µ q : “ ´ a b pB µ b id ` id b B µ q ,τ p v µ , t ν q : “ ´ a b pB µ B ν b id ` B µ b B ν ` B ν b B µ ` id b B µ B ν q , p t µ , v ν q : “ a b pB µ B ν b id ` B µ b B ν ` B ν b B µ ` id b B µ B ν q ,τ p v µ , v ν q : “ t µ b B ν b id ´ t ν b id b B µ `` ´ ? ´ ξ ? ξ ` ¯ t ν b B µ b id ´ ¯ t µ b id b B ν ˘ ´´ ” v ν b pB µ b id ` id b B µ q ´ v µ b pB ν b id ` id b B ν q ı `` ? ` t κµν b B κ b id ` t κµν b id b B κ ˘ ,τ p v µ , n q : “ ¯ t µ b c l ξ p q b c l ξ p q ,τ p n , v µ q : “ ´ ¯ t µ b c l ξ p q b c l ξ p q ,τ p v µ , t νκλ q : “ ´ ? ` η µκ v λ b id b B ν ´ η µλ v κ b id b B ν ˘ ,τ p t νκλ , v µ q : “ ? ` η µκ v λ b B ν b id ´ η µλ v κ b B ν b id ˘ ,τ p t µ , v ν q : “ ´ η µν v κ b id b B κ ,τ p v ν , t µ q : “ η µν v κ b B κ b id ,τ p v ν , ¯ t µ q : “ ´ ´ ? ´ ξ ? ξ v µ b B ν b id ` η µν n b c l ξ p q b c l ξ p q ,τ p ¯ t µ , v ν q : “ ´ ? ´ ξ ? ξ v µ b id b B ν ´ η µν n b c l ξ p q b c l ξ p q ,τ p ¯ t µ , n q : “ v µ b id b c l ξ ,τ p n , ¯ t µ q : “ ´ v µ b c l ξ b id , where we defined ˜c l ξ p q b c l ξ p q ¸ p f b g q : “ c l ξ p f g q . (9.13c)We note that the twisted tensor product Kin st b τ Scal is a (strict) C -algebra, which be-comes an L -algebra after the tensor product with the colour Lie algebra g ; see Section 6.1for details. A key to showing that our double copy prescription based on factorisations of the L -algebras of gauge-fixed BRST Lagrangians is that not only the action but also the BRSToperator double copies. This fact guarantees that the double copy creates the appropriategauge-fixing sectors which is crucial in considering the double copy at the loop level. In the125ollowing, we give a general discussion of what we called the BRST Lagrangian double copy in [9].
Strictification of BRST-invariant actions.
As discussed in Section 8.3, any field the-ory can be strictified to a classically equivalent field theory with purely cubic interactionterms, and this equivalence extends to the quantum level. Consider a general strictifiedfield theory S “
12 Φ I g IJ Φ J `
13! Φ I f IJK Φ J Φ K , (9.14)where g IJ and f IJK are some structure constants. As in Section 4.1,
I, J, . . . are DeWittindices that include labels for the field species, the gauge and Lorentz representations, aswell as the space-time position.Let us now consider a theory which is invariant under a gauge symmetry. We extend theaction of this theory to its BV form by including ghosts, anti-ghosts, and the Nakanishi–Lautrup field, as done in Section 5. We then strictify the full BV action to an action withcubic interaction vertices. Restricting to gauge-fixing fermions which are quadratic in thefields guarantees that the action remains cubic after gauge fixing. The resulting BRSToperator Q BRST , given by (4.17c), is then automatically at most quadratic in the fields,and we can write Φ I Q
BRST
ÞÝÝÝÝÝÑ Q IJ Φ J ` Q IJK Φ J Φ K (9.15)for some structure constants Q IJ and Q IJK . Factorisation of structure constants.
As indicated previously, the key to the doublecopy is the factorisation of the field space L into L : “ V b ¯ V b C p M d q , (9.16)where V and ¯ V are two (graded) vector spaces. In our preceeding discussion, we haveencountered the three examples in Table 9.2. Consequently, in our formulas, we shall splitthe multi indices into triples, that is, I “ p α, ¯ α, x q , and write (see e.g. (4.12b)) p L r sq ˚ b L Q a “ Φ I b e I “ ż d d x Φ α ¯ α p x q b p e α b ¯ e ¯ α b s x q . (9.17)We also demand that the structure constants g IJ and f IJK that appear in the action (9.14)as well as the structure constants q IJ and q IJK that appear in the BRST operator (9.15)are local in the sense that they vanish unless all the space-time points in the multi-indicesagree. This is the case for all explicit gauge-fixing fermions used in this paper. g IJ “ : g αβ ¯ g ¯ α ¯ β l , (9.18)where g αβ and ¯ g ¯ α ¯ β are differential operators, mapping C p M d q to itself. In more detail,we have g IJ Φ J ” ż d d y g p α, ¯ α,x q ; p β, ¯ β,y q Φ β ¯ β,y “ ż d d y ż d d z g αβ p x, y q ¯ g ¯ α ¯ β p y, z q l Φ β ¯ β,z , (9.19a)where the integral kernels are of the fom g αβ p x, y q “ δ p d q p x ´ y q g αβ p x q and ¯ g ¯ α ¯ β p y, z q “ δ p d q p y ´ z q ¯ g ¯ α ¯ β p y q (9.19b)due to our assumption about locality, and we assume that g αβ p x q is invertible.Analogously, we write f IJK “ f p α, ¯ α,x q ; p β, ¯ β,y q ; p γ, ¯ γ,z q “ : p f αβγ ¯ f ¯ α ¯ β ¯ γ , (9.19c)where f αβγ and ¯ f ¯ α ¯ β ¯ γ are bi-differential operators C p M d qb C p M d q Ñ C p M d qb C p M d q and p : C p M d q b C p M d q Ñ C p M d q (9.19d)is the natural diagonal product of functions. For the integral kernels of f αβγ and ¯ f ¯ α ¯ β ¯ γ wehave again the locality condition f αβγ p x , x ; y , y q “ δ p d q p x ´ y q δ p d q p x ´ y q δ p d q p x ´ y q f αβγ p x q , ¯ f ¯ α ¯ β ¯ γ p x , x ; y , y q “ δ p d q p x ´ y q δ p d q p x ´ y q δ p d q p x ´ y q ¯ f ¯ α ¯ β ¯ γ p x q . (9.19e)We note that there is some ambiguity in the definition (9.19c) due to the projection ontothe diagonal involved in p , but this redundancy never arises in any formula. To give aclearer picture of what the above construction is doing, we can expand the f αβγ and the ¯ f ¯ α ¯ β ¯ γ further in a basis of differential operators B M for M a Lorentz multiindex, and we have p p f αβγ ¯ f ¯ α ¯ β ¯ γ qp Φ b Φ q “ f αβM γM ¯ f ¯ α ¯ βN ¯ γN pB M B N Φ β ¯ β qpB M B N Φ γ ¯ γ q . (9.20)For convenience, we also introduce the operators f αβγ and ¯ f ¯ α ¯ β ¯ γ by p f αβγ “ : g αδ p f δβγ and p ¯ f ¯ α ¯ β ¯ γ “ : ¯ g ¯ α ¯ δ p ¯ f ¯ δ ¯ β ¯ γ , (9.21)which is possible due to the invertibility of g αβ and ¯ g ¯ α ¯ β as well as the form of the integralkernels (9.19e) . Evidently, f αβγ and ¯ f ¯ α ¯ β ¯ γ are again bi-differential operators, just as f αβγ and ¯ f ¯ α ¯ β ¯ γ . 127 ¯ V Biadjoint scalar field theory g ¯ g Yang–Mills theory g Kin N “ supergravity Kin Kin
Table 9.2: Factors appearing in the field space factorisation (9.16) with
Kin given in (7.16)and g and ¯ g the colour Lie algebras.With the factorisation restriction, the action (9.14) becomes S “ ż d d x "
12 Φ α ¯ α g αβ ¯ g ¯ α ¯ β l Φ β ¯ β `
13! Φ α ¯ α p p f αβγ ¯ f ¯ α ¯ β ¯ γ qp Φ β ¯ β b Φ γ ¯ γ q * . (9.22)For the BRST operator Q BRST , the factorisation of indices and the linearity of Q BRST implythe decomposition Q BRST “ : q BRST ` ¯ q BRST , (9.23)where q BRST and ¯ q BRST are BRST operators acting in a non-trivial way on the factors V b C p M d q and ¯ V b C p M d q in the factorisation (9.17), respectively. By this, we mean thatthe structure constants Q IJ and Q IJK decompose as Q IJ Ñ p q IJ , ¯ q IJ q and Q IJK
Ñ p q IJK , ¯ q IJK q .More explicitly, q p α, ¯ α,x qp β, ¯ β,y q “ δ p d q p x ´ y q q αβ p x q δ ¯ α ¯ β , q p α, ¯ α,x qp β, ¯ β,y q ; p γ, ¯ γ,z q “ δ p d q p x ´ y q δ p d q p x ´ z q q αβγ p x q ¯ f ¯ α ¯ β ¯ γ p x q , ¯ q p α, ¯ α,x qp β, ¯ β,y q “ δ p d q p x ´ y q δ αβ ¯ q ¯ α ¯ β p x q , ¯ q p α, ¯ α,x qp β, ¯ β,y q ; p γ, ¯ γ,z q “ δ p d q p x ´ y q δ p d q p x ´ z q f αβγ p x q ¯ q ¯ α ¯ β ¯ γ p x q , (9.24)where q αβ and ¯ q ¯ α ¯ β are differential operators and q αβγ and ¯ q ¯ α ¯ β ¯ γ are again bi-differentialoperators, just as f αβγ and ¯ f ¯ α ¯ β ¯ γ , with locality again restricting their integral kernels.Note that in this splitting, the association of terms of the form δ p d q p x ´ y q δ αβ δ ¯ α ¯ β and δ p d q p x ´ y q δ p d q p x ´ z q f αβγ p y, z q ¯ f ¯ α ¯ β ¯ γ p y, z q is not unique; we assign half of each of these termsto p q IJ , q IJK q and half to p ¯ q IJ , ¯ q IJK q . Example.
To make our rather abstract discussion more concrete, let us briefly considerthe case of Yang–Mills theory (5.9). We refrain from discussing the details of the strictific-ation of the BV action, but it is clear that V “ g and V “ Kin with Kin some extension of Kin allowing for auxiliary fields, similar to
Kin st defined in (9.12). It is then also clear that g αβ and f αβγ are the Killing form and the structure constants of the gauge Lie algebra g .On Kin , the integral kernel for the differential operator ¯ g µν is given by ¯ g µν “ η µν ´ l B µ B ν . (9.25)128e note that q αβ “ and ¯ q ¯ α ¯ β is only non-trivial for ¯ α and ¯ β labelling a factor of the gaugepotential and the factor of the ghost, respectively. Working out all other structure constantsis a straightforward but tedious process; since no more insights would be obtained from it,we refrain from listing them here. We only note that for Yang–Mills theory, the ambiguityin assigning terms to q and ¯ q is absent. Double copy.
We now note that the decomposition of the Lagrangian matches preciselythe decomposition of scattering amplitudes in the discussion of colour–kinematics duality,cf. Section 2.1, which is the starting point for the double copy. We merely extended thefactorisation of the interaction vertices to a factorisation of the whole BRST structure.In the usual double copy, we start from the factorisation for Yang–Mills theory andreplace the colour factor by a kinematic factor. More generally, however, we can certainlyreplace any one of the (graded) vector spaces V and ¯ V and the corresponding structureconstants with (graded) vector spaces and structure constants from other theories. Thisgives us a new action, which we shall denote by ˜ S DCBRST . The corresponding BRST operator ˜ Q DCBRST is obtained by replacing one set of kinematic structure constants in the decomposi-tion of the BRST operator (9.22) with those from the new factor.
BRST Lagrangian double copy.
In order to obtain a consistent and quantisable theory,we demand the new BRST structure to be consistent. Specifically, ˜ Q DCBRST ˜ S DCBRST “ and p ˜ Q DCBRST q “ . (9.26)By construction, we have again a decomposition ˜ Q DCBRST “ : ˜ q DCBRST ` ˜¯ q DCBRST . The condition Q “ implies q “ , and we decompose the latter into linear, quadratic, andcubic terms in the fields, q Φ ¨¨¨ “ : q p , q ` q p , q ` q p , q , (9.27)and analogously for ¯ q , p ˜ q DCBRST q , and p ˜¯ q DCBRST q , respectively. Schematically, the sum-mands read as q p , q “ ¨ ¨ ¨ q αβ q βγ ¨ ¨ ¨ ,q p , q “ ¨ ¨ ¨ p q αδ q δβγ ` q δβ q αδγ ˘ q δγ q αβδ q ¯ f ¯ α ¯ β ¯ γ ¨ ¨ ¨ ,q p , q “ ¨ ¨ ¨ p q εβγ q αεδ ¯ f ¯ ε ¯ β ¯ γ ¯ f ¯ α ¯ ε ¯ δ ˘ q εβγ q αδε ¯ f ¯ ε ¯ β ¯ γ ¯ f ¯ α ¯ δ ¯ ε q ¨ ¨ ¨ , (9.28a)and ˜ q p , q “ ¨ ¨ ¨ q αβ q βγ ¨ ¨ ¨ , ˜ q p , q “ ¨ ¨ ¨ p q αδ q δβγ ` q δβ q αδγ ˘ q δγ q αβδ q ˜¯ f ¯ α ¯ β ¯ γ ¨ ¨ ¨ , ˜ q p , q “ ¨ ¨ ¨ p q εβγ q αεδ ˜¯ f ¯ ε ¯ β ¯ γ ˜¯ f ¯ α ¯ ε ¯ δ ˘ q εβγ q αδε ˜¯ f ¯ ε ¯ β ¯ γ ˜¯ f ¯ α ¯ δ ¯ ε q ¨ ¨ ¨ , (9.28b)129here ˜ f αβγ and ˜¯ f ¯ α ¯ β ¯ γ denote the kinematic constants in ˜ S DCBRST . It is now clear that ˜ q p , q and ˜ q p , q vanish if q “ and thus, q p , q and q p , q vanish on arbitrary fields.So far, our discussion was fairly general and nothing singled out colour–kinematics-dualtheories from other theories. This changes with the condition that q p , q “ must imply ˜ q p , q “ . Vanishing of q p , q relies on a transfer of the symmetry properties of the openindices of ¯ f ¯ ε ¯ β ¯ γ ¯ f ¯ α ¯ ε ¯ δ and ¯ f ¯ ε ¯ β ¯ γ ¯ f ¯ α ¯ δ ¯ ε via the contracting fields (in which the expression is totallysymmetric) to q εβγ q αεδ and q εβγ q αδε . It follows that if the symmetry properties of the openindices in the terms quadratic in ¯ f ¯ α ¯ β ¯ γ are the same as for the terms quadratic in ˜¯ f ¯ α ¯ β ¯ γ that ˜ q p , q “ . The colour–kinematics duality provides such a condition.The same argument shows that p ˜¯ q DCBRST q “ , and we can directly turn to the crossterms and split them again into linear, quadratic, and cubic pieces, p q BRST ¯ q BRST ` ¯ q BRST q BRST q Φ ¨¨¨ “ : q p , q ` q p , q ` q p , q , (9.29a)and p ˜ q DCBRST ˜¯ q DCBRST ` ˜¯ q DCBRST ˜ q DCBRST q Φ ¨¨¨ “ : ˜ q p , q ` ˜ q p , q ` ˜ q p , q . (9.29b)We note that the conditions q p , q “ and ˜ q p , q “ are implied directly when q and ¯ q and ˜ q and ˜¯ q anti-commute, respectively, which is always the case in the theories we study.Moreover, we have, again schematically, the conditions q p , q “ ¨ ¨ ¨ q αβγ p ¯ q ¯ α ¯ δ ¯ f ¯ δ ¯ β ¯ γ ˘ ¯ q ¯ δ ¯ β ¯ f ¯ α ¯ δ ¯ γ ˘ ¯ q ¯ δ ¯ γ ¯ f ¯ α ¯ β ¯ δ q ¨ ¨ ¨ ` ¨ ¨ ¨ ¯ q ¯ α ¯ β ¯ γ p q αδ f δβγ ˘ q δβ f αδγ ˘ q δγ f αβδ q ¨ ¨ ¨ ,q p , q “ ¨ ¨ ¨ p q αεδ ¯ f ¯ α ¯ ε ¯ δ f εβγ ¯ q ¯ ε ¯ β ¯ γ ˘ q αβε ¯ f ¯ α ¯ β ¯ ε f εγδ ¯ q ¯ ε ¯ γ ¯ δ ˘ f αεδ ¯ q ¯ α ¯ ε ¯ δ q εβγ ¯ f ¯ ε ¯ β ¯ γ ˘ f αβε ¯ q ¯ α ¯ β ¯ ε q εγδ ¯ f ¯ ε ¯ γ ¯ δ q ¨ ¨ ¨ . (9.30)We see that q p , q “ splits into two separate conditions on the indices in V and ¯ V andthus it implies ˜ q p , q “ . The condition ˜ q p , q “ can, in principle, be non-trivial, butagain colour–kinematics duality as well as the special form of the BRST operator in thetheories that in which we are interested renders ˜ q p , q “ equivalent to q p , q “ .Finally, we have to check that ˜ Q DCBRST ˜ S DCBRST “ , and we consider q BRST S “ : s p , q ` s p , q ` s p , q , (9.31)where s p , q , s p , q , and s p , q are quadratic, cubic, and quartic in the fields. Analogously,we have ˜ q DCBRST ˜ S DCBRST “ : ˜ s p , q ` ˜ s p , q ` ˜ s p , q , and the discussion for ¯ q BRST and ˜¯ q DCBRST issimilar. Schematically, we compute s p , q “ ż d d x ¨ ¨ ¨ p q γα g γβ ¯ g ¯ α ¯ β l q ¨ ¨ ¨ ,s p , q “ ż d d x ¨ ¨ ¨ p g αδ l q δβγ ` f αδγ q δβ ` f αβδ q δβ q ¯ f ¯ α ¯ β ¯ γ ¨ ¨ ¨ ,s p , q “ ż d d x ¨ ¨ ¨ p f αεδ q εβγ ¯ f ¯ α ¯ ε ¯ δ ¯ f ¯ ε ¯ β ¯ γ ` f αβε q εγδ ¯ f ¯ α ¯ β ¯ ε ¯ f ¯ ε ¯ γ ¯ δ q ¨ ¨ ¨ , (9.32)130here we have assumed that q BRST commutes with the differential and bi-differential oper-ators in the action, which is the case in all our theories. We see that s p , q “ and s p , q “ imply ˜ s p , q “ and ˜ s “ , respectively. The relation ˜ s p , q “ can, in principle, lead toadditional conditions. In a theory with colour–kinematics duality, however, the contractionof the kinematic structure constants ¯ f ¯ α ¯ β ¯ γ appears as in the Jacobi identity, and s p , q as wellas ˜ s p , q vanish automatically. Partial BRST Lagrangian double copy.
There are few theories where we expect theBRST Lagrangian double copy to work perfectly. The reason is that in most formulations,colour–kinematics duality will not hold. In Yang–Mills theory, for example, it is not knownif colour–kinematics duality can be made manifest for off-shell fields. Now if colour–kinematics duality fails to hold up to certain terms, say the ideal offunctions of the fields vanishing on-shell as in the case of Yang–Mills theory, then theequation ˜ Q DCBRST ˜ S DCBRST “ will also fail to hold up to the same ideal. Consequently, ˜ Q DCBRST ˜ S DCBRST is a product of factors whose vanishing amounts to the equations of motionpossibly multiplied by other fields and their derivatives.
Let us now make the abstract discussion above concrete by working out the example of theBRST Lagrangian double copy ˜ L DCBRST : “ Kin st b τ p Kin st b τ Scal q , (9.33)where Kin st is given in (9.11) and Scal in (9.3), respectively.
Field content.
From the discussion in Section 7.5, we already know that the double-copied field content of the BRST-extended Hilbert space of Yang–Mills theory agrees withthe field content of the BRST-extended Hilbert space. We shall continue to use the fieldlabels introduced in Table 7.5.In the interactive case of the full homotopy algebras, however, we have infinitely manyadditional auxiliary fields, arising from the infinitely many additional auxiliary fields ofstrictified and colour–kinematics duality preserving Yang–Mills theory. In the previoussection, we made five of the auxiliary fields in Yang–Mills theory explicit, ˜ K aµ , ˜¯ K aµ , G aµνκ , ˜ K aµ , ˜¯ K aµ , (9.34) Recall that we only extended colour–kinematics to the BRST-extended Hilbert space in Section 8.4,but with all fields still on-shell. t µ , ¯ t µ , t µνκ , t µ , ¯ t µ (9.35)in Kin st . In the tensor product (9.33) this gives rise to 40 auxiliary fields involving oneauxiliary kinetic factor and another 25 auxiliary fields involving two auxiliary kinetic factors.Instead of giving these auxiliary fields individual labels, we shall collectively denote themby k Υ k , where k and k denote the first and second kinematic factors. For example, g Υ g : “ g b g b ˆż d d x s x ϕ gg p x q ˙ “ ˜ λ , v Υ v : “ e a b v µ b v ν b ˆż d d x s x ϕ vv µν p x q ˙ “ ˜ h ` ˜ B , t Υ t : “ t µ b t νκλ b ˆż d d x s x ϕ t t µνκλ p x q ˙ . (9.36) Higher products.
Next, we use the twist (9.13a) and (9.13b) to compute the higherproducts µ and µ between the elements of ˜ L DCBRST . The formulas from Section 6.3 withall the appropriate signs included read as µ p x b y b ϕ q : “ p´ q | τ p q p x q|`| τ p q p y q| τ p q p x q b τ p q p y q b ` τ p q p x qp τ p q p y qp ϕ qq ˘ ,µ p x b y b ϕ , x b y b ϕ q : “ : “ p´ q p| y |`| ϕ |q| x |`| ϕ | | y | ˆˆ τ p q p x , x q b τ p q p y , y q b ` τ p q p x , x q ϕ p x q ˘` τ p q p y , y q ϕ p x q ˘ . (9.37)Note that there are no additional signs because our τ p q i are always even. While the com-putation is readily performed, listing the higher products for all 81 fields is not particularlyhelpful. Action.
The factorisation (9.33) induces the following cyclic structure: x x b y b ϕ , x b y b ϕ y : “ : “ p´ q | x | Kin p| y | Kin `| ϕ | Scal q`| x | Kin | ϕ | Scal x x , x y x y , y y x ϕ , ϕ y . (9.38)Together with the formulas for the super homotopy Maurer–Cartan action (4.28), we cancompute the (gauge-fixed) BRST action corresponding to the L -algebra ˜ L DCBRST . Again,listing all the terms would not provide much insight, but we stress that we obtain all theexpected terms, in particular the lowest terms of the Fierz–Pauli version of the N “ supergravity action as well as the evident terms involving ghosts.132 ouble copy of the BRST operator. Let us now also consider the double copy ofthe BRST operator to a BRST operator ˜ Q DCBRST . For our purposes, the double copy of thelinearised part without considering the auxiliary fields will be sufficient. We start fromYang–Mills theory with the factors V : “ g and ¯ V : “ Kin in (9.16) and the usual BRSTrelations in terms of coordinate functions on ˜ L YMBRST , ˜ A aµ Q YM , linBRST ÞÝÝÝÝÝÑ δ ab B µ ˜ c b , ˜ b a Q YM , linBRST ÞÝÝÝÝÝÑ δ ab ´ ? ´ ξ ? ξ ? l ˜ c b , ˜ c a Q YM , linBRST ÞÝÝÝÝÝÑ , ˜¯ c a Q YM , linBRST ÞÝÝÝÝÝÑ δ ab ˆc l ξ ˜ b b ´ ´ ? ´ ξξ B µ ˜ A bµ ˙ . (9.39)We thus have q αβ “ δ αβ , and the non-vanishing components of ¯ q ¯ α ¯ β are given by ¯ q ¯ α ¯ β “ $’’’’’’’&’’’’’’’% B µ for ¯ α “ g ˚ , ¯ β “ v ˚ µ ´? ´ ξ ? ξ ? l for ¯ α “ g ˚ , ¯ β “ n ˚ b l ξ for ¯ α “ n ˚ , ¯ β “ a ˚ ´ ´? ´ ξξ B µ for ¯ α “ v ˚ µ , ¯ β “ a ˚ . (9.40)After the double copy, we have V : “ Kin “ : ¯ V and, correspondingly, q αβ “ ¯ q αβ . Thelinearisation of the double-copied BRST operator is then non-trivial on a field containinga factor of v µ or a and we have in the anti-symmetrised sector ˜ λ ˜ Q DC , linBRST ÞÝÝÝÝÝÑ , ˜Λ µ ˜ Q DC , linBRST ÞÝÝÝÝÝÑ B µ ˜ λ , ˜ γ ˜ Q DC , linBRST ÞÝÝÝÝÝÑ ´ ? ´ ξ ? ξ ? l ˜ λ , ˜ B µν ˜ Q DC , linBRST ÞÝÝÝÝÝÑ B µ ˜Λ ν ´ B ν ˜Λ µ , ˜ α µ ˜ Q DC , linBRST ÞÝÝÝÝÝÑ ´ ? ´ ξ ? ξ ? l ˜Λ µ ´ B µ ˜ γ , ˜ ε ˜ Q DC , linBRST ÞÝÝÝÝÝÑ c l ξ ˜ γ ´ ´ ? ´ ξξ B µ ˜Λ µ , ˜¯Λ µ ˜ Q DC , linBRST ÞÝÝÝÝÝÑ B µ ˜ ε ` c l ξ ˜ α µ ´ ´ ? ´ ξξ B µ ˜ B µν , ˜¯ γ ˜ Q DC , linBRST ÞÝÝÝÝÝÑ ´ ? ´ ξ ? ξ ? l ˜ ε ` ´ ? ´ ξξ B µ ˜ α µ , ˜¯ λ ˜ Q DC , linBRST ÞÝÝÝÝÝÑ c l ξ ˜¯ γ ´ ´ ? ´ ξξ B µ ˜¯Λ µ , (9.41a)133nd in the symmetrised sector ˜ X µ ˜ Q DC , linBRST ÞÝÝÝÝÝÑ , ˜ β ˜ Q DC , linBRST ÞÝÝÝÝÝÑ , ˜ h µν ˜ Q DC , linBRST ÞÝÝÝÝÝÑ B µ ˜ X ν ` B ν ˜ X µ , ˜ (cid:36) µ ˜ Q DC , linBRST ÞÝÝÝÝÝÑ ´ ´ ? ´ ξ ? ξ ? l ˜ X µ ´ B µ ˜ β , ˜ π ˜ Q DC , linBRST ÞÝÝÝÝÝÑ ´ ? ´ ξ ? ξ ? l ˜ β , ˜ δ ˜ Q DC , linBRST ÞÝÝÝÝÝÑ c l ξ ˜ β ´ ´ ? ´ ξξ B µ ˜ X µ , ˜¯ X µ ˜ Q DC , linBRST ÞÝÝÝÝÝÑ ´B µ ˜ δ ´ c l ξ ˜ (cid:36) µ ´ ´ ? ´ ξξ B ν ˜ h νµ , ˜¯ β ˜ Q DC , linBRST ÞÝÝÝÝÝÑ ´ ´ ? ´ ξ ? ξ ? l ˜ δ ` ´ ? ´ ξξ B µ ˜ (cid:36) µ ` c l ξ ˜ π . (9.41b)This BRST operator is related to the usual linearised BRST operator for N “ supergrav-ity, (5.20) and (5.27), λ Q N “ , linBRST ÞÝÝÝÝÝÝÑ , ϕ Q N “ , linBRST ÞÝÝÝÝÝÝÑ , Λ µ Q N “ , linBRST ÞÝÝÝÝÝÝÑ B µ λ , X µ Q N “ , linBRST ÞÝÝÝÝÝÝÑ ,γ Q N “ , linBRST ÞÝÝÝÝÝÝÑ , β Q N “ , linBRST ÞÝÝÝÝÝÝÑ ,B µν Q N “ , linBRST ÞÝÝÝÝÝÝÑ B µ Λ ν ´ B ν Λ µ , h µν Q N “ , linBRST ÞÝÝÝÝÝÝÑ B µ X ν ` B ν X µ ,α µ Q N “ , linBRST ÞÝÝÝÝÝÝÑ , (cid:36) µ Q N “ , linBRST ÞÝÝÝÝÝÝÑ ,ε Q N “ , linBRST ÞÝÝÝÝÝÝÑ γ , δ Q N “ , linBRST ÞÝÝÝÝÝÝÑ β , ¯Λ µ Q N “ , linBRST ÞÝÝÝÝÝÝÑ α µ , ¯ X µ Q N “ , linBRST ÞÝÝÝÝÝÝÑ (cid:36) µ , ¯ γ Q N “ , linBRST ÞÝÝÝÝÝÝÑ , ¯ β Q N “ , linBRST ÞÝÝÝÝÝÝÑ π , ¯ λ Q N “ , linBRST ÞÝÝÝÝÝÝÑ ¯ γ , π Q N “ , linBRST ÞÝÝÝÝÝÝÑ . (9.41c)by the field redefinitions (7.22) and (7.27), respectively. N “ supergravity Let us complete the argument by showing that the double copied action ˜ S DCBRST we con-structed in Section 9.4 is fully perturbatively quantum equivalent to the suitably gauge134xed version of the BV action of N “ supergravity, S N “ , defined in Section 5.5. Forthis, we have to show that up to a field redefinition, both theories have the same tree-levelcorrelation functions. A crucial point in our discussion will be the BRST Lagrangian doublecopy formalism developed in the previous section.In the following, we shall speak of ‘auxiliary fields connected to a field φ ’ by which wemean all auxiliary fields which appear together with φ in Feynman diagrams containingonly propagators of auxiliary fields. Put differently, an auxiliary field ψ connected to a field φ can have an interaction vertex with φ or interact with an auxiliary field that propagatesto an auxiliary field that non-trivially interacts with φ , etc.: . . .ψ φ , . . . . . .ψ φ , . . . . . . . . .ψ φ , . . . , (9.42)where a dashed line denotes an auxiliary field. We also use the terms physical and unphys-ical fields/interaction terms/scattering amplitudes. The unphysical fields are all ghosts,anti-ghosts, and Nakanishi–Lautrup fields as well as auxiliary fields connected to these.Physical fields are the remaining fields, consisting of the metric, the Kalb–Ramond fieldand the dilaton as well as a number of auxiliary fields. Physical interaction vertices arethose consisting exclusively of physical fields. Physical scattering amplitudes are those withphysical fields as external labels. Physical tree-level scattering amplitudes.
We first note that the auxiliary fields inthe double copied action ˜ S DCBRST can be integrated out, after which the field content andthe kinematic terms in both actions fully agree, up to the field redefinitions we discussed inSection 7. Implementing these field redefinitions on S N “ , we obtain the action S N “ , .Moreover, the physical tree-level scattering amplitudes computed from the interactionvertices of the action ˜ S DCBRST are by design precisely those arising in the usual double copyprescription for the construction of N “ supergravity tree amplitudes from a factorisationof Yang–Mills amplitudes. The tree-level double copy has been demonstrated to hold,cf. Observation 8.12, and therefore the physical tree-level scattering amplitudes of ˜ S DCBRST and S N “ , agree.If we put all unphysical fields to zero, the resulting theories ˜ S DCBRST , phys and S N “ , phys are classically equivalent by Observation 8.9. In the homotopy algebraic picture, this cor-responds to a restriction L N “ , phys and ˜ L DCphys to two quasi-isomorphic L -subalgebras.In order to improve this restricted classical equivalence to a full perturbative quantumequivalence, we need to adjust and modify the actions or, equivalently, the corresponding135 -algebras. We shall do this now in a sequence of steps, expanding the discussion in [9]. Auxiliary fields of ghost number zero.
The reformulation of the tree-level scatteringamplitudes of N “ supergravity used in the double copy defines a local strictificationof the physical part of the action S N “ to the action S N “ , by promoting all cubicinteraction vertices to cubic interaction terms. This is fully analogous to the strictificationimplied by the manifestly colour–kinematics-dual form of the Yang–Mills action explainedin Section 8.3.By construction, the actions S N “ , and ˜ S DCBRST , phys have the same field content, thesame kinematical terms for the physical and auxiliary fields and identical tree-level scatter-ing amplitudes for the physical fields.Let us now consider the tree-level scattering amplitudes which have auxiliary fields ofghost number zero on their external legs. Such amplitudes are fully determined by the(iterated) collinear limits of physical tree-level scattering amplitudes. Because, again, thephysical tree-level scattering amplitudes of S N “ , and ˜ S DCBRST , phys agree, the tree-levelscattering amplitudes with physical and auxiliary fields of ghost number zero on externallegs agree.By Observation 8.9, we then have a local field redefinition of S N “ , to S N “ , suchthat both actions agree after all fields except for physical ones and auxiliary fields of ghostnumber zero are put to zero. If we integrated out all auxiliary fields in both actions, theresulting actions would agree in their purely physical parts. Nakanishi–Lautrup fields.
In the next step, we deal with the difference between ˜ S DCBRST and S N “ , proportional to any of the Nakanishi–Lautrup fields ( ˜¯ β, ˜ (cid:36) µ , ˜ π, ˜ γ, ˜ α µ , ˜¯ γ ); weshall come to the ghost field β later. After integrating out all auxiliary fields, this differencecan be compensated by Observation 8.8. That is, we can modify the gauge-fixing fermionand perform a field redefinition of the Nakanishi–Lautrup fields such that this difference isremoved. We note that neither of these two processes modifies the physical parts of theaction and both preserve quantum equivalence. We can thus replace all terms in S N “ , containing Nakanishi–Lautrup fields by the terms in ˜ S DCBRST containing Nakanishi–Lautrupfields as well as auxiliary fields connected to Nakanishi–Lautrup fields. We call the resultingaction S N “ , .Recall that there is a ghost number ´ field ¯ λ which is paired with the Nakanishi–Lautrup-type field γ in the gauge fixing fermion (5.23), allowing us to absorb any termproportional to γ in a different gauge choice. This is not the case for the correspondingNakanishi–Lautrup-type field in the gravity sector, γ . Any discrepancy proportional to136 between S N “ , and ˜ S DCBRST (again, after integrating out all the auxiliary fields) shouldinstead be absorbed by shifting the gauge fixing fermion Ψ from (5.35) by a term δP , where γP is the discrepancy. This will generate the desired corrections. This will also lead to newghost terms, which we will treat separately in the next step. Ghost sector.
Let us now examine the ghost interactions. We know that the action S N “ , comes with a BRST operators Q N “ , which satisfies p Q N “ , q “ and Q N “ , S N “ , “ . (9.43)From our discussion in the previous section, we know that the double-copied BRST operator ˜ Q DCBRST satisfies p ˜ Q DCBRST q P I and ˜ Q DCBRST ˜ S DCBRST P I , (9.44)where I is the ideal of polynomials in the fields and their derivatives which vanishes foron-shell fields. We also know from the discussion around (9.41) that the linearisations ofthe BRST operators satisfy ˜ Q DC , linBRST “ Q N “ , linBRST , . (9.45)After integrating out all auxiliary fields, these BRST operators link the physical tree-levelscattering amplitudes to tree-level scattering amplitudes containing ghosts by the on-shellWard identities, cf. Observation 8.2.At the level of the BRST operators ˜ Q DC , linBRST and Q N “ , linBRST , the situation is more involved,but we still end up with similar on-shell Ward identities. In particular, recall that eachauxiliary field introduced in Yang–Mills theory is double copied to a quartet of auxiliaryfields of ghost numbers ´ , , and ` , which are related by double copied (linearised) BRSTtransformations that relates the quartet of states in the BRST-extended Hilbert space ofYang–Mills theory.Therefore, the tree-level scattering amplitudes for the BRST-extended Hilbert spacesof S N “ , and ˜ S DCBRST are fully determined via on-shell Ward identities by the tree-levelscattering amplitudes of the physical and auxiliary fields of ghost number zero. We concludethat all these tree-level scattering amplitudes agree between both theories.
Full quantum equivalence.
For full quantum equivalence, it remains to show that thereis a local field redefinition that links the action S N “ , to ˜ S DCBRST . Both actions fully agreein their kinematic terms and their interaction vertices that contain exclusively fields ofghost number zero. Moreover, they have identical tree-level scattering amplitudes on theirBRST-extended (i.e. full) Hilbert spaces. We can therefore invoke Observation 8.9 one137nal time in order to provide us with a local field redefinition that shifts the discrepanciesbetween both actions to interaction vertices of arbitrarily high degree. This renders theactions fully quantum equivalent from the perspective of perturbative quantum field theory.
AppendicesA. Definitions and conventions for homotopy algebras
The homotopy algebras that appear naturally in the context of field theories, namely A -, C -, and L -algebras are homotopy versions of associative, commutative and Lie algebras.In particular, associativity and the Jacobi identity only hold up to coherent homotopies. In the following, we list relevant definitions and our conventions. For more details on L -algebras and some of the calculations detailed in this appendix, see e.g. [177, 178];our conventions match the ones in these references. Other helpful references with originalresults listed in this section are [295, 196, 195]. A unifying description of all the homotopyalgebras and their cyclic structures listed below is given by operads, but we refrain fromintroducing this additional layer of abstraction. A.1. A -algebras A -algebras. An A -algebra or strong homotopy associative algebra is a graded vectorspace A “ À i P Z A i together with higher products which are i -linear maps m i : A ˆ¨ ¨ ¨ˆ A Ñ A of degree ´ i that satisfy the homotopy associativity relation ÿ i ` i ` i “ i p´ q i i ` i m i ` i ` p id b i b m i b id b i q “ (A.1)for all and i P N ` . The lowest identities read as m p m p (cid:96) qq “ , m p m p (cid:96) , (cid:96) qq “ m p m p (cid:96) q , (cid:96) q ` p´ q | (cid:96) | A m p (cid:96) , m p (cid:96) qq , m p m p (cid:96) , (cid:96) , (cid:96) qq ` m p m p (cid:96) q , (cid:96) , (cid:96) q ` p´ q | (cid:96) | A m p (cid:96) , m p (cid:96) q , (cid:96) q `` p´ q | (cid:96) | A `| (cid:96) | A m p (cid:96) , (cid:96) , m p (cid:96) qq “ m p m p (cid:96) , (cid:96) q , (cid:96) q ´ m p (cid:96) , m p (cid:96) , (cid:96) qq , ... (A.2) But graded commutativity (in the case of C -algebras) and graded anti-symmetry (in the case of L -algebras) are not relaxed. (cid:96) , . . . , (cid:96) i P A . We thus see that the unary product m is a differential and a derivationfor the binary product m . Furthermore, the ternary product m captures the failure of thebinary product m to be associative. Cyclic A -algebras. A cyclic A -algebra p A , x´ , ´y A q is an A -algebra A equippedwith a non-degenerate graded-symmetric bilinear form x´ , ´y A : A ˆ A Ñ R such that x (cid:96) , m i p (cid:96) , . . . , (cid:96) i ` qy A “ p´ q i ` i p| (cid:96) | A `| (cid:96) i ` | A q`| (cid:96) i ` | A ř ij “ | (cid:96) j | A x (cid:96) i ` , m i p (cid:96) , . . . , (cid:96) i qy A (A.3)for all (cid:96) i P A . When it is clear from the context, we shall suppress the subscript A on theinner products. Homotopy Maurer–Cartan theory.
Each A -algebra comes with a homotopyMaurer–Cartan theory, where the gauge potential is an element a P A whose curvature f P A is defined as f : “ m p a q ` m p a, a q ` ¨ ¨ ¨ “ ÿ i ě m i p a, . . . , a q (A.4)and satisfies the Bianchi identity ÿ i ě i ÿ j “ p´ q i ` j m i ` p a, . . . , a looomooon j , f, a, . . . , a looomooon i ´ j q “ . (A.5)If the homotopy Maurer–Cartan equation f “ (A.6)holds, we say that a is a homotopy Maurer–Cartan element . Provided A is cyclic withpairing of degree ´ , homotopy Maurer–Cartan elements are the stationary points of the homotopy Maurer–Cartan action S hMC r a s : “ ÿ i ě i ` x a, m i p a, . . . , a qy A . (A.7)Infinitesimal gauge transformations are mediated by elements c P A and are given by δ c a : “ ÿ i ě i ÿ j “ p´ q i ` j m i ` p a, . . . , a looomooon j , c , a, . . . , a looomooon i ´ j q . (A.8)One may check that the action (A.7) is invariant under the transformations (A.8), and thecurvature (A.4) transforms as δ c f “ ÿ i ě i ÿ j “ i ´ j ÿ k “ p´ q k m i ` p a, . . . , a looomooon j , f, a, . . . , a looomooon i ´ j , c , a, . . . , a looomooon i ´ j ´ k q . (A.9)To verify these statements, one makes use of (A.1).139 .2. C -algebras C -algebras. A C -algebra or strong homotopy commutative algebra is an A -algebra C “ À i P Z C i where the higher products m i , in addition to (A.1), also satisfy the homotopycommutativity relations ÿ Sh p i ; i q χ p σ ; (cid:96) , . . . (cid:96) i q m i p (cid:96) σ p q , . . . , (cid:96) σ p i q , (cid:96) σ p i ` q , . . . , (cid:96) σ p i q q “ (A.10)for all ă i ă i , where Sh p i ; i q is the set of unshuffles , i.e. permutations of p , . . . , i q preserving the relative order of the first i as well as the relative order of the last i “ i ´ i elements. The Koszul sign χ p σ ; (cid:96) , . . . , (cid:96) i q for total graded anti-symmetrisation is definedby (cid:96) ^ . . . ^ (cid:96) i “ χ p σ ; (cid:96) , . . . , (cid:96) i q (cid:96) σ p q ^ . . . ^ (cid:96) σ p i q . (A.11)The lowest four homotopy commutativity relations are m p (cid:96) , (cid:96) q ´ p´ q | (cid:96) | C | (cid:96) | C m p (cid:96) , (cid:96) q “ , m p (cid:96) , (cid:96) , (cid:96) q ´ p´ q | (cid:96) | C | (cid:96) | C m p (cid:96) , (cid:96) , (cid:96) q ` p´ q p| (cid:96) | C `| (cid:96) | C q| (cid:96) | C m p (cid:96) , (cid:96) , (cid:96) q “ , m p (cid:96) , (cid:96) , (cid:96) , (cid:96) q ´ p´ q | (cid:96) | C | (cid:96) | C m p (cid:96) , (cid:96) , (cid:96) , (cid:96) q``p´ q | (cid:96) | C p| (cid:96) | C `| (cid:96) | C q m p (cid:96) , (cid:96) , (cid:96) , (cid:96) q ´ p´ q | (cid:96) | C p| (cid:96) | C `| (cid:96) | C `| (cid:96) | C q m p (cid:96) , (cid:96) , (cid:96) , (cid:96) q “ , m p (cid:96) , (cid:96) , (cid:96) , (cid:96) q ´ p´ q | (cid:96) | C | (cid:96) | C m p (cid:96) , (cid:96) , (cid:96) , (cid:96) q``p´ q | (cid:96) | C p| (cid:96) | C `| (cid:96) | C q m p (cid:96) , (cid:96) , (cid:96) , (cid:96) q ` p´ q p| (cid:96) | C `| (cid:96) | C q| (cid:96) | C m p (cid:96) , (cid:96) , (cid:96) , (cid:96) q´´p´ q p| (cid:96) | C `| (cid:96) | C q| (cid:96) | C `| (cid:96) | C | (cid:96) | C m p (cid:96) , (cid:96) , (cid:96) , (cid:96) q``p´ q p| (cid:96) | C `| (cid:96) | C q| (cid:96) | C `p| (cid:96) | C `| (cid:96) | C q| (cid:96) | C m p (cid:96) , (cid:96) , (cid:96) , (cid:96) q “ (A.12)and we see that the product m is indeed graded commutative. Cyclic C -algebras. A cyclic C -algebra is a cyclic A -algebra satisfying the homotopycommutativity relations. A.3. L -algebras L -algebras. An L -algebra or strong homotopy Lie algebra is a graded vector space L “ À i P Z L i together with higher products which are graded anti-symmetric i -linear maps µ i : L ˆ ¨ ¨ ¨ ˆ L Ñ L of degree ´ i that satisfy the homotopy Jacobi identities ÿ i ` i “ i ÿ σ P Sh p i ; i q p´ q i χ p σ ; (cid:96) , . . . , (cid:96) i q µ i ` p µ i p (cid:96) σ p q , . . . , (cid:96) σ p i q q , (cid:96) σ p i ` q , . . . , (cid:96) σ p i q q “ . (A.13)140or all (cid:96) , . . . , (cid:96) i P L and i P N ` ; see below (A.10) for the definitions of the unshuffles Sh p i ; i q and of the Koszul sign χ p σ ; (cid:96) , . . . , (cid:96) i q . The lowest homotopy Jacobi identities,slightly rewritten, read as µ p µ p (cid:96) qq “ ,µ p µ p (cid:96) , (cid:96) qq “ µ p µ p (cid:96) q , (cid:96) q ` p´ q | (cid:96) | L µ p (cid:96) , µ p (cid:96) qq ,µ p µ p (cid:96) , (cid:96) q , (cid:96) q ` p´ q | (cid:96) | L | (cid:96) | L µ p (cid:96) , µ p (cid:96) , (cid:96) qq ´ µ p (cid:96) , µ p (cid:96) , (cid:96) qq ““ µ p µ p (cid:96) , (cid:96) , (cid:96) qq ` µ p µ p (cid:96) q , (cid:96) , (cid:96) q ` p´ q | (cid:96) | L µ p (cid:96) , µ p (cid:96) q , (cid:96) q `` p´ q | (cid:96) | L `| (cid:96) | L µ p (cid:96) , (cid:96) , µ p (cid:96) qq , ... (A.14)and we can interpret them as follows. The unary product µ is a differential and a derivationwith respect to the binary product µ . In addition, the ternary product µ captures thefailure of the binary product µ to satisfy the standard Jacobi identity.We note that any A -algebra yields an L -algebra with higher products obtained fromtotal anti-symmetrisation, µ i p (cid:96) , . . . , (cid:96) i q “ ÿ σ P S i χ p σ ; (cid:96) , . . . , (cid:96) i q m i p (cid:96) σ p q , . . . , (cid:96) σ p i q q . (A.15)In particular, the Lie algebra arising from the commutator on any matrix algebra is an L -algebra. Likewise, the anti-symmetrisation of a C -algebra is an L -algebra with µ i “ for i ě due to the homotopy commutativity relations (A.10).We call an L -algebra nilpotent , if all nested higher products vanish, i.e. µ i p µ j p´ , . . . , ´q , . . . , ´q “ for all i, j ě . (A.16) Cyclic L -algebras. A cyclic L -algebra p L , x´ , ´y L q is an L -algebra L equipped witha non-degenerate graded-symmetric bilinear form x´ , ´y L : L ˆ L Ñ R such that x (cid:96) , µ i p (cid:96) , . . . , (cid:96) i ` qy L “ p´ q i ` i p| (cid:96) | L `| (cid:96) i ` | L q`| (cid:96) i ` | L ř ij “ | (cid:96) j | L x (cid:96) i ` , µ i p (cid:96) , . . . , (cid:96) i qy L (A.17)for all (cid:96) i P L . As before, when it is clear from the context, we shall suppress the subscript L on the inner products. Homotopy Maurer–Cartan theory.
Similar to A -algebras, any L -algebra p L , µ i q comes with its homotopy Maurer–Cartan theory. In particular, a gauge potential is anelement a P L , and its curvature is f : “ µ p a q ` µ p a, a q ` ¨ ¨ ¨ “ ÿ i ě i ! µ i p a, . . . , a q P L . (A.18)141he Bianchi identity reads here as ÿ i ě i ! µ i ` p a, . . . , a, f q “ . (A.19) Homotopy Maurer–Cartan elements , i.e. gauge potentials with vanishing curvature f “ ,are the stationary points of the homotopy Maurer–Cartan action S hMC r a s : “ ÿ i ě p i ` q ! x a, µ i p a, . . . , a qy L (A.20)provided L comes with a cyclic pairing x´ , ´y L of degree ´ . Similarly to (A.8), infinites-imal gauge transformations are of the form δ c a : “ ÿ i ě i ! µ i ` p a, . . . , a, c q (A.21)and are parametrised by elements c P L . The action is invariant under such transforma-tions, and the curvature behaves as δ c f “ ÿ i ě i ! µ i ` p a, . . . , a, f, c q . (A.22)To verify these statements, one makes use of (A.13). Covariant derivative.
Given an L -algebra p L , µ i q , consider ϕ P L k for some k P Z andrequire that under infinitesimal gauge transformations, ϕ transforms adjointly, that is, δ c ϕ : “ ÿ i ě i ! µ i ` p a, . . . , a, ϕ, c q (A.23)for c P L . We then define the covariant derivative ∇ : L k Ñ L k ` by ∇ ϕ : “ µ p ϕ q ` µ p a, ϕ q ` ¨ ¨ ¨ “ ÿ i ě i ! µ i ` p a, . . . , a, ϕ q (A.24)for a P L . Using (A.13), one can show that under infinitesimal gauge transforma-tions (A.21) and (A.23), ∇ ϕ transforms as δ c p ∇ ϕ q “ ÿ i ě i ! µ i ` p a, . . . , a, ∇ ϕ, c q ` ÿ i ě i ! µ i ` p a, . . . , a, f, ϕ, c q , (A.25)where f is the curvature (A.18) of a . Thus, for homotopy Maurer–Cartan elements a , thecovariant derivative transforms adjointly as well. Using (A.13) again, we obtain in addition ∇ ϕ “ ÿ i ě i ! µ i ` p a, . . . , a, f, ϕ q . (A.26) It will always transform adjointly when µ i “ for all i ą , that is, for differential graded Lie algebrasalso known as strict L -algebras, cf. Appendix A.4. urved morphisms of L -algebras. Morphisms between Lie algebras are maps pre-serving the Lie bracket. In the context of L -algebras, this notion generalises and one ob-tains what is known as a curved morphism (of L -algebras) . Specifically, a curved morphism φ : p L , µ i q Ñ p ˜ L , ˜ µ i q between two L -algebras p L , µ i q and p ˜ L , ˜ µ i q is a collection of i -lineartotally graded anti-symmetric maps φ i : L ˆ ¨ ¨ ¨ ˆ L Ñ ˜ L of degree ´ i such that ÿ i ` i “ i ÿ σ P Sh p i ; i q p´ q i χ p σ ; (cid:96) , . . . , (cid:96) i q φ i ` p µ i p (cid:96) σ p q , . . . , (cid:96) σ p i q q , (cid:96) σ p i ` q , . . . , (cid:96) σ p i q q ““ ÿ j ě j ! ÿ k `¨¨¨` k j “ i ÿ σ P Sh p k ,...,k j ´ ; i q χ p σ ; (cid:96) , . . . , (cid:96) i q ζ p σ ; (cid:96) , . . . , (cid:96) i q ˆˆ ˜ µ j ´ φ k ` (cid:96) σ p q , . . . , (cid:96) σ p k q ˘ , . . . , φ k j ` (cid:96) σ p k `¨¨¨` k j ´ ` q , . . . , (cid:96) σ p i q ˘¯ (A.27a)for i P N ` Y t u with χ p σ ; (cid:96) , . . . , (cid:96) i q the Koszul sign (A.11) and ζ p σ ; (cid:96) , . . . , (cid:96) i q given by ζ p σ ; (cid:96) , . . . , (cid:96) i q : “ p´ q ř ď m ă n ď j k m k n ` ř j ´ m “ k m p j ´ m q` ř jm “ p ´ k m q ř k `¨¨¨` km ´ k “ | (cid:96) σ p k q | L . (A.27b)Note that φ : R Ñ ˜ L is the constant map. Explicitly, the lowest expressions of (A.27)read as “ ÿ i ě i ! ˜ µ i p φ , . . . , φ q ,φ p µ p (cid:96) qq “ ˜ µ p φ p (cid:96) qq ` ÿ i ě i ! ˜ µ i ` p φ , . . . , φ , φ p (cid:96) qq ,φ p µ p (cid:96) , (cid:96) qq ´ φ p µ p (cid:96) q , (cid:96) q ` p´ q | (cid:96) | L | (cid:96) | L φ p µ p (cid:96) q , (cid:96) q ““ ˜ µ p φ p (cid:96) , (cid:96) qq ` ˜ µ p φ p (cid:96) q , φ p (cid:96) qq `` ÿ i ě i ! ˜ µ i ` p φ , . . . , φ , φ p (cid:96) , (cid:96) qq ` ÿ i ě i ! ˜ µ i ` p φ , . . . , φ , φ p (cid:96) q , φ p (cid:96) qq , ... (A.28)It is easily seen that this definition reduces to the standard definition of a Lie algebramorphism in the context of Lie algebras. Note that a curved morphism is simply called an (uncurved) morphism (of L -algebras) whenever φ “ , and this notion of morphisms isusually used in the literature when discussing L -algebras. As we will see below, we shallneed the more general notion of curved morphisms to reinterpret gauge transformations asmorphisms of L -algebras.Evidently, the first equation of (A.28) implies that φ is necessarily a homotopy Maurer–Cartan element of ˜ L . For such φ , we now set ˜ µ φ i p ˜ (cid:96) , . . . , ˜ (cid:96) i q : “ ÿ j ě j ! ˜ µ i ` j p φ , . . . , φ , ˜ (cid:96) , . . . , ˜ (cid:96) i q (A.29)143or all ˜ (cid:96) , . . . , ˜ (cid:96) i P ˜ L and i P N ` . By virtue of (A.26), we immediately have that ˜ µ φ isa differential. In fact, one can show that p ˜ L , ˜ µ φ i q forms an L -algebra, that is, the ˜ µ φ i satisfy the homotopy Jacobi identities (A.13) thus defining another L -structure on ˜ L .From (A.27) we may then conclude that any curved morphism between two L -algebras p L , µ i q and p ˜ L , ˜ µ i q can be viewed as an uncurved morphism between p L , µ i q and p ˜ L , ˜ µ φ i q . Maurer–Cartan elements and curved morphisms.
Consider a P L and let f P L be its curvature (A.18). We define the image of a gauge potential under a curved morphism φ : p L , µ i q Ñ p ˜ L , ˜ µ i q as ˜ a : “ φ ` φ p a q ` φ p a, a q ` ¨ ¨ ¨ “ ÿ i ě i ! φ i p a, . . . , a q P ˜ L . (A.30)The curvature of ˜ a is then ˜ f “ ÿ i ě i ! ˜ µ i p ˜ a, . . . , ˜ a q “ ÿ i ě i ! φ i ` p a, . . . , a, f q P ˜ L , (A.31)which one can verify using (A.13) and (A.27). Hence, homotopy Maurer–Cartan elementsin L are mapped to homotopy Maurer–Cartan elements in ˜ L .Let us extend the above observation to gauge orbits. Consider gauge transforma-tions (A.21) a ÞÑ a ` δ c a and ˜ a ÞÑ ˜ a ` δ ˜ c ˜ a with the image of the gauge parameter c P L given by ˜ c : “ φ p c q ` φ p a, c q ` ¨ ¨ ¨ “ ÿ i ě i ! φ i ` p a, . . . , a, c q P ˜ L . (A.32)A short calculation involving (A.13) reveals that δ ˜ c ˜ a “ ´ ÿ i ě i ! φ i ` p a, . . . , a, f, c q ` ÿ i ě i ! φ i ` p δ c a, a, . . . , a q . (A.33)This immediately yields ÿ i ě i ! φ i p a ` δ c a, . . . , a ` δ c a q “ ÿ i ě i ! φ i p a, . . . , a q ` ÿ i ě i ! φ i ` p δ c a, a, . . . , a q“ ˜ a ` δ ˜ c ˜ a ` ÿ i ě i ! φ i ` p a, . . . , a, f, c q . (A.34)Consequently, gauge equivalence classes of homotopy Maurer–Cartan elements in L aremapped to gauge equivalence classes of homotopy Maurer–Cartan elements in ˜ L undercurved morphisms. 144 orphisms of cyclic L -algebras. Consider an uncurved morphism between two L -algebras p L , µ i q and p ˜ L , ˜ µ i q , that is, a curved morphism with φ “ . If, in addition, wehave inner products x´ , ´y L on L and x´ , ´y ˜ L on ˜ L , then a morphism of cyclic L -algebrashas to satisfy x (cid:96) , (cid:96) y L “ x φ p (cid:96) q , φ p (cid:96) qy ˜ L (A.35a)for all (cid:96) , P L and for all i ě and (cid:96) , . . . , (cid:96) i P L ÿ i ` i “ ii ,i ě x φ i p (cid:96) , . . . , (cid:96) i q , φ i p (cid:96) i ` , . . . , (cid:96) i qy ˜ L “ . (A.35b)We note that the morphisms of cyclic L -algebras defined here require φ to be injective.More general notions of such morphisms can be defined using Lagrangian correspondences,cf. [296].Suppose now that the inner product x´ , ´y L on L and x´ , ´y ˜ L on ˜ L of degree ´ sothat the homotopy Maurer–Cartan equations, f “ and ˜ f “ , are variational. Then,under a morphism φ : p L , µ i q Ñ p ˜ L , ˜ µ i q , we obtain ÿ i ě p i ` q ! x a, µ i p a, . . . , a qy L “ S hMC r a s“ ˜ S hMC r ˜ a s “ ÿ i ě p i ` q ! x ˜ a, ˜ µ i p ˜ a, . . . , ˜ a qy ˜ L (A.36)by virtue of (A.35) and (A.30). Quasi-isomorphisms of curved L -algebras. Recall that the homotopy Jacobi iden-tities (A.13) (see also (A.14)) imply that µ “ . Hence, we may consider the cohomology H ‚ µ p L q “ à k P Z H kµ p L q with H kµ p L q : “ ker p µ | L k q{ im p µ | L k ´ q . (A.37)A curved morphism of L -algebras φ : p L , µ i q Ñ p ˜ L , ˜ µ i q is called a quasi-isomorphism(of curved L -algebras) whenever φ induces an isomorphism H ‚ µ p L q – H ‚ ˜ µ φ p ˜ L q ; theproducts ˜ µ φ i were defined in (A.29). There is a bijection between the moduli spaces ofgauge equivalence classes of homotopy Maurer–Cartan elements of L and ˜ L . A curvedquasi-isomorphism is called an (uncurved) quasi-isomorphism whenever φ “ . Gauge transformations as curved morphisms.
Let us revisit the infinitesimal gaugetransformations (A.21) and first explain how they arise from partially flat homotopies. Inparticular, set I : “ r , s Ď R and consider the tensor product L Ω : “ Ω ‚ p I q b L “ à k P Z p L Ω q k with p L Ω q k “ C p I q b L k ‘ Ω p I q b L k ´ (A.38)145etween the de Rham complex p Ω ‚ p I q , d q on the interval I and an L -algebra p L , µ i q .Hence, a general element a P p L Ω q is of the form a p t q “ a p t q ` d t b c p t q with a p t q P C p I q b L and c p t q P C p I q b L . Its curvature f P p L Ω q is then f p t q “ f p t q ` d t b B a p t qB t ´ ÿ i ě i ! µ i ` p a p t q , . . . , a p t q , c p t qq + , (A.39)where f p t q P C p I q b L is the curvature of a p t q . The requirement of partial flatness of f p t q amounts to saying that f p t q has no components along d t . Thus, B a p t qB t “ ÿ i ě i ! µ i ` p a p t q , . . . , a p t q , c p t qq (A.40)and we recover the gauge transformations (A.21) from δ c a “ B a p t qB t ˇˇˇˇ t “ (A.41)with a “ a p q and c “ c p q . Furthermore, upon solving the ordinary differential equa-tion (A.40), we will obtain finite gauge transformations. Let us now explain how one canunderstand this as a curved morphism that preserves the products µ i .Concretely, we consider (A.30) and (A.32) and make the ansatz a p t q : “ ÿ i ě i ! φ i p t qp a, . . . , a q and c p t q : “ ÿ i ě i ! φ i ` p t qp a, . . . , a, c q . (A.42)Here, we again set a “ a p q and c “ c p q which, in turn, translates to the conditions φ i p q “ for all i ‰ and φ p q “ . Upon substituting the ansatz (A.42) into (A.40) andremembering (A.33), we obtain B a p t qB t “ ÿ i ě i ! B φ i p t qB t p a, . . . , a q“ ´ ÿ i ě i ! φ i ` p t qp a, . . . , a, f, c q ` ÿ i ě i ! φ i ` p t qp δ c a, a, . . . , a q , (A.43)where f is the curvature of a . Thus, solving the ordinary differential equation (A.40) forgauge transformations is equivalent to solving the ordinary differential equation (A.43)for a curved morphism φ i on the L -algebra that preserves the L -algebra structure.Put differently, finite gauge transformations are given by curved morphisms that arise assolutions to (A.43).Let us exemplify these discussions by considering a standard Lie algebra valued one-form gauge potential on Minkowski space M d . Here, a “ A P Ω p M d q b g and c “ c P C p M d q b g for a Lie algebra g . Moreover, in this case it is enough to consider φ p t q and146 p t q and set φ i p t q “ for all i ą . Consequently, the ordinary differential equation (A.43)reduces to B A p t qB t “ B φ p t qB t ` B φ p t qB t p A q “ φ p t qp d c ` r A, c sq (A.44)and is solved by A p t q “ φ p t q ` φ p t qp A q and c p t q “ φ p t qp c q with φ p t q “ t d c ` t r d c, c s ` t rr d c, c s , c s ` ¨ ¨ ¨ “ e ´ tc de tc ,φ p t qp A q “ A ` t r A, c s ` t rr A, c s , c s ` t rrr A, c s , c s , c s ` ¨ ¨ ¨ “ e ´ tc A e tc ,φ p t qp c q “ c (A.45)as a short calculation reveals; recall from (A.28) that φ p t q must be a homotopy Maurer–Cartan element. A.4. Structure theorems
In the following, the term ‘homotopy algebra’ refers to either an A -, C -, or L -algebra.Note that the unary higher product is a differential for any homotopy algebra. We call ahomotopy algebra minimal provided the unary product vanishes. A homotopy algebra iscalled strict if only the unary and binary products are non-vanishing. Moreover, a homotopyalgebra is called linearly contractible if only the unary product is nonvanishing and it hastrivial cohomology. Structure theorems.
We now have the following structure theorems:(i) The decomposition theorem : any homotopy algebra is isomorphic to the direct sum ofa minimal and a linearly contractible one; see e.g. [195] for the case of A -algebras.(ii) The minimal model theorem : any homotopy algebra is quasi-isomorphic to a minimalone. This follows directly from the decomposition theorem, see also [194, 195] for thecase of L -algebras.(iii) The strictification theorem : any homotopy algebra is quasi-isomorphic to a strictone [297, 298].We note that strict A -, C -, and L -algebras are simply differential graded associative,differential graded commutative, and differential graded Lie algebras, respectively. We alsonote that mathematicians would probably use the term ‘rectify’ over ‘strictify’; we foundthe latter term more descriptive. We can also consider the more general case φ p t q “ g ´ p t q d g p t q , φ p t qp A q “ g ´ p t q A g p t q , and φ p t qp c q “ g ´ p t q B t g p t q for g P C p I, G q with g p q “ , that is, g solves the ordinary differential equation B t g p t q “ g p t q c p t q ; note that B t g p t q| t “ “ c . emark A.1. We also would like to make a few remarks on the relations between thehomotopy algebras:(i) As we saw above in (A.15) , any A -algebra carries an L -structure by (graded) anti-symmetrisation the higher products.(ii) All higher products of a C -algebra (which is also in particular an A -algebra) exceptfor the differential vanish after anti-symmetrisation. B. Inverses of wave operators
In this paper, we have glossed over some of the finer analytical details as not to hide thesimplicity of our constructions (too much) behind arcane notation. In particular, we mostlyignored the difference between C p M d q and the actual function space F : “ F int ‘ F free “ S p M d q ‘ ker S p l q , (B.1)cf. (4.29). This is unproblematic, but the mathematically minded reader may wonder aboutthe definition of inverses of the operator l which appear throughout our discussion. Below,we give an answer to this point.A first point to note is that only gluons can label scattering amplitudes (we are nottalking about correlation functions) and therefore they are the only relevant object in theminimal model consisting of (several copies of) the kernel of the wave operator. However, wedo want to have gauge symmetries also at the level of free fields, and we therefore allow alsothe ghosts to have free components. This is not a problem for the scattering amplitudes,as gluons and ghosts live in homogeneously differently graded spaces. The anti-ghostsand Nakanishi–Lautrup field are only relevant as interacting fields, and we arrive at an L -algebra with underlying chain complex ˜ A aµ F Ω b g ´ ˜ A ` aµ F Ω b g ˜ b a S p M d q b g ˜ b ` a S p M d q b g ˜ c a F b g ˜¯ c ` S p M d q b g ˜¯ c a S p M d q b g ˜ c ` a F b g ´ l ´ ˜ ξ ? l B µ ´ l ˜ ξ ? l B µ l l (B.2)148here F Ω are one-forms on M d with coefficients in F and l (as well as ? l ) vanishes on ker S p l q b g . In this complex, expressions such as ? l ˜ b a are clearly well-defined.We may feel slightly uncomfortable with this restriction of fields as the tensor productof scalar fields with elements in Kin that we used to construct the Yang–Mills fields doesnot make any distinction between gluons and ghost on the one side and anti-ghosts andNakanishi–Lautrup fields on the other side. To resolve this issue, we can consider a quasi-isomorphic L -algebra with underlying complex ˜ A aµ F Ω b g ´ ˜ A ` aµ F Ω b g ˜ b a S p M d q b g ˜ b ` a S p M d q b g ˜ b a ker S p l q b g ˜ b ` a ker S p l q b g ˜ c a F b g ˜¯ c ` S p M d q b g ˜¯ c a S p M d q b g ˜ c ` a F b g ˜¯ c ` ker S p l q b g ˜¯ c a ker S p l q b g ´ l ´ ˜ ξ ? l B µ ´ l ˜ ξ ? l B µ id l l id (B.3)We note that the identity component is readily implemented with a modification of theYang–Mills twist τ in (7.19). However, the required modification is technically a bit cum-bersome, and had we inserted it, it would have distracted from the main point of the twist.Finally, note that the twists τ governing interactions can be left unmodified and merelyneed to be restricted to the Schwartz parts S p M d q b g for anti-ghosts and Nakanishi–Lautrup fields. This is also evident, as scattering amplitudes will never lead to interactionsinvolving non-propagating anti-ghosts or Nakanishi–Lautrup fields. Acknowledgements
We gratefully acknowledge stimulating conversations with Johannes Brödel, Michael Duff,Jan Gutowski, Henrik Johansson, Silvia Nagy, and Alessandro Torrielli. We also thankthe organisers of and participants at
QCD Meets Gravity VI for interesting questions and149ruitful discussions about our related work [9]. L.B., H.K., and C.S. were supported by theLeverhulme Research Project Grant RPG–2018–329
The Mathematics of M5-Branes . B.J.was supported by the GAČR Grant EXPRO 19–28268X. T.M. was partially supported bythe EPSRC grant EP/N509772. BJ would also like to thank the MPI Bonn for hospitality.
Data Management
No additional research data beyond the data presented and cited in this work are neededto validate the research findings in this work.
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