A double copy for asymptotic symmetries in the self-dual sector
PPrepared for submission to JHEP
A double copy for asymptotic symmetries in theself-dual sector
Miguel Campiglia a Silvia Nagy b a Facultad de Ciencias, Universidad de la RepúblicaIgúa 4225, Montevideo, Uruguay b Centre for Research in String Theory, School of Physics and Astronomy,Queen Mary University of London, 327 Mile End Road, London E1 4NS, UK
E-mail: [email protected] , [email protected] Abstract:
We give a double copy construction for the symmetries of the self-dual sectorsof Yang-Mills (YM) and gravity, in the light-cone formulation. We find an infinite setof double copy constructible symmetries. We focus on two families which correspond tothe residual diffeomorphisms on the gravitational side. For the first one, we find novelnon-perturbative double copy rules in the bulk. The second family has a more strikingstructure, as a non-perturbative gravitational symmetry is obtained from a perturbativelydefined symmetry on the YM side.At null infinity, we find the YM origin of the subset of extended Bondi-Metzner-Sachs(BMS) symmetries that preserve the self-duality condition. In particular, holomorphic largegauge YM symmetries are double copied to holomorphic supertranslations. We also identifythe single copy of superrotations with certain non-gauge YM transformations that to ourknowledge have not been previously presented in the literature. a r X i v : . [ h e p - t h ] F e b ontents O (Φ ) symmetry 145.2.2 O ( φ ) symmetry from the DC 155.3 Summary 155.4 Relation with the symmetry raising map 165.5 Double copy for an infinite family of symmetries 17 ˜ δ from Λ = O ( r ) , λ = O ( r ) ˜ δ from Λ = O ( r ) , λ = O ( r ) – i – Self-duality conditions 30
A.1 Yang-Mills field 30A.2 Metric field 31
B Perturbative transformations at higher orders 32
B.1 Second order 32B.1.1 First family 32B.1.2 Second family 33B.2 Recursive construction at arbitrary orders 35
C Comparison with convolution DC 36
The idea of gravity as a double copy (DC) has found applications in the study of numerousaspects of the gravitational theory, most notably scattering amplitudes, where much of therecent success has been driven by the identification of a duality between the color andkinematic factors, the so-called Bern-Carrasco-Johannson (BCJ) duality [4–11]. It hasprogressed to the study of solutions, both perturbative [12–18] and exact [19–34]. Anotherimportant extension has been to find manifestations of it beyond gravity theory [35–41]. Anatural line of work following from these successes has been to formalise the correspondenceby constructing double copy dictionaries for the fields themselves and by studying thesymmetries of the theories, both local [42–47] and global [48–52], as well as by giving directconstructions of Lagrangians via the DC [6, 53–58].Having a DC prescription for the local symmetries of the fields (i.e. the gauge transfor-mations on the Yang-Mills (YM) side and diffeomorphisms on the gravity side) is of coursea proof of the robustness of the DC dictionaries for the fields, but the possible applicationsof such a prescription go beyond this. For example, it was shown in [39, 43, 57], working inthe BRST formalism, that one can exploit it to obtain a gauge-mapping algorithm, whichgives a gravity gauge fixing functional output from a YM gauge-fixing functional input.This has been extended to the study of classical solutions in [16]. However, this has sofar only been understood at linear level - going beyond this, either perturbatively (or non-perturbatively in certain special cases), could prove very powerful in the DC constructionof classical solutions and/or Lagrangians.A natural arena in which to discuss gravitational and gauge symmetries is the asymp-totic boundary of spacetime. In the case of (asymptotically) flat spacetimes, this is givenby Penrose’s null infinity [59]. It has long been known that gauge and gravitational fieldsexhibit a rich structure at this boundary, with one of the earliest examples being the Bondi-Metzner-Sachs (BMS) asymptotic symmetry group of gravity [60, 61]. Whereas many ofthe implications of these symmetries were known since the 80’s thanks to Ashtekar and See [1–3] for some comprehensive reviews. – 1 –thers [62–64], the subject experienced a veritable renaissance in recent years. The triggerwas twofold. On the one hand, Barnich and Troessaert proposed an extension of BMSto include so-called superrotations [65]. On the other hand, Strominger discovered a linkbetween future and past null infinity symmetries [66] which manifests as soft theorems inscattering amplitudes [67, 68]. The ramifications of these developments are still ongoing,see [69–71] for reviews.One may then ask whether the asymptotic perspective could shed light into the DC(and viceversa). In fact, there has already been progress in this direction, through theuse of DC for soft theorems [72–79] and in the context of celestial amplitudes [80–83].Additionally, a duality transformation relating the Schwarzschild and Taub-NUT solutionswas shown to arise, via a DC construction, from an electric-magnetic duality relation [84,85]. Interestingly, this duality can be interpreted, on the gravity side, as the complexificationof a particular supertranslation transformation. These promising results are calling for aninterpretation from the perspective of asymptotic symmetries.In this paper we present some progress in both directions outlined above. We will berestricting to the self-dual sector of gravity and YM theory, where the colour-kinematicsduality was made manifest at the level of the algebra by Monteiro and O’Connell [86] (seealso [11, 37, 87–91] for related work and extensions), leading to a simple DC prescription forthe construction of perturbative solutions. In light-cone gauge, the self-dual graviton andYM field have a natural description in terms of scalar fields φ and Φ , respectively [92–100],and the self-duality conditions can be expressed as equations for these scalars, referred to asself-dual Einstein (SDE) and self-dual YM (SDYM) equations respectively. We will studythe symmetries of these equations, with focus on those arising from the residual gaugesymmetries of their respective parent theories. After laying down the ‘bulk’ DC symmetrymap (see below for details), we will move on to study symmetries at null infinity. On thegravitational side we will recover the subset of BMS that preserves the self-duality conditionat null infinity. On the YM side, the single copy version of these symmetries will turn out toinclude large gauge as well as certain non-gauge symmetries. The latter, to our knowledge,have not been discussed before in the asymptotic symmetry literature.The SDYM and SDE equations are known to possess an infinite ladder of symmetries[93, 94, 98, 99, 101–103], out of which the symmetries described above turn out to representonly two rungs. We will see that the DC applies to all levels of this ladder. The asymptoticanalysis of the higher level symmetries could in principle be worked out along the samelines as the ones studied here, but we do not attempt to do so in this work.Let us now describe in more detail our proposed DC symmetry map. As anticipatedabove, we focus on two families of symmetries. The first is exact in both YM and gravity,and constitutes a subset of diffeomorphisms on the gravity side and a subset of the non-abelian gauge symmetry on the YM side. We discover that a repackaging is required in orderto manifest the color-kinematic duality and we find the appropriate way to perform the DC.Crucially, we find that the gauge parameter of YM is not mapped to the diffeomorphismparameter of gravity, but to a "Hamiltonian" of the diffeomorphism vector field w.r.t. thePoisson bracket inherent in the self-dual theory. Having performed this repackaging, the DCrules become remarkably simple. The second family is also exact on the gravity side, but– 2 –erturbative and non-local on the YM side, presenting an apparent puzzle. However, we findthat the repackaging described above recasts the gravity transformation as a perturbativenon-local series. This is then obtainable from its YM counterpart via the same simplerules as the first family, verifying the robustness of our construction. It is interesting tonote that this second family is a subset of diffeomorphisms on the gravity side, but itssingle copy is not a subset of YM gauge transformations. In the asymptotic limit, the firstfamily maps holomorphic YM large gauge symmetries to holomorphic supertranslations,whereas the second one identifies certain (non-gauge and non-local) SDYM transformationsto holomorphic superrotations. The paper is structured as follows: In section 2 we introduce our conventions andpresent a covariant description of self-dual fields in light-cone gauge. The correspondingresidual gauge symmetries are discussed in section 3. In section 4 we describe a perturbativeapproach to general symmetries of the SDYM and SDE equations. In section 5 we constructthe DC map between the symmetries of these equations. Starting with a non-perturbativemap for the first family of symmetries, we eventually extend its applicability to an infinitenumber of families. In section 6 we introduce coordinates adapted to null infinity and reviewgauge and gravity asymptotic symmetries. In section 7 we take the null-infinity limit of thefirst two families of symmetries described earlier and establish the DC map for asymptoticsymmetries. We summarize our findings and discuss possible future avenues in section 8.Some of the technical discussions are given in appendices.
Consider Minkowski spacetime written in light-cone coordinates ( U, V, Z, ¯ Z ) , related toCartesian coordinates X µ by: U = X − X √ , V = X + X √ , Z = X + iX √ , ¯ Z = X − iX √ . (2.1)We will split these light-cone coordinates in the following two sets of d coordinates: x i := ( U, ¯ Z ) , y α := ( V, Z ) . (2.2)The spacetime metric then takes the form ds = 2 η iα dx i dy α = − dU dV + 2 dZd ¯ Z. (2.3)The description of self-dual fields and their symmetries will involve the “area element” ofthese 2d spaces, described by Ω ij dx i dx j = dU d ¯ Z − d ¯ ZdU (2.4) Π αβ dy α dy β = dV dZ − dZdV (2.5) We have here simplified the discussion in the interest of clarity. Asymptotically, the second family splitsinto two subfamilies, see Figure 1 for the complete picture. – 3 –e will treat Ω ij and Π αβ as antisymmetric 4 dimensional tensors with Ω αµ and Π iµ vanishing. By raising indices with the inverse spacetime metric η iα , we can regard thesetwo tensors as (partial) inverses of each other: Ω αi Π jα = δ ji (2.6) Π iα Ω βi = δ βα . (2.7)The self-dual YM and metric fields will be described in terms of scalar fields Φ and φ according to A µ = Π νµ ∂ ν Φ (2.8) h µν = Π ρµ Π σν ∂ ρ ∂ σ φ. (2.9)In the notation of (2.2), the non-zero components take the form A α = Π iα ∂ i Φ (2.10) h αβ = Π iα Π jβ ∂ i ∂ j φ, (2.11)with the scalar fields satisfying (cid:3) Φ = − i Π ij [ ∂ i Φ , ∂ j Φ] (2.12) (cid:3) φ = 12 Π ij Π kl ∂ i ∂ k φ∂ j ∂ l φ (2.13)where (cid:3) ≡ η iα ∂ i ∂ α is the wave operator. We will refer to (2.12) and (2.13) as the self-dualYM (SDYM) and self-dual Einstein (SDE) equation respectively. We are using conventionswhere the coupling constants are absorbed in the field, so that the field strength is F µν = ∂ µ A ν − ∂ ν A µ − i [ A µ , A µ ] , the spacetime metric g µν = η µν + h µν , and there is no explicitcoupling constant in the field equations.In [86], the notion of a kinematic algebra in the self-dual sector was defined as a Poissonbracket algebra. In our conventions, the Poisson bracket (PB) is defined as: { f, g } := Π ij ∂ i f ∂ j g (2.14)One can now recover (2.13) from(2.12) via the replacement rules [86]: − i [ , ] → { , } . Φ → φ. (2.15)Note that our conventions are slightly different from [86], and so are the color to kinematicsreplacement rules. See Appendix A for the derivation of these equations from the self-duality conditions for YM andgravity. – 4 – .1 Symmetries of the self-dual field equations
The self-dual YM and Einstein equations have long been known to possess infinitely manysymmetries [93, 94, 98, 99, 101–103]. These can be constructed recursively as follows. Let δ Φ and δφ be symmetries of the SDYM and SDE equations respectively: (cid:3) δ Φ = − i Π ij [ ∂ i Φ , ∂ j δ Φ] (2.16) (cid:3) δφ = Π ij Π kl ∂ i ∂ k φ∂ j ∂ l δφ (2.17)One can then obtain new symmetries ˜ δ Φ and ˜ δφ , defined implicitly by the condition ∂ i ˜ δ Φ = Ω αi ∂ α δ Φ − i [ ∂ i Φ , δ Φ] (2.18) ∂ i ˜ δφ = Ω αi ∂ α δφ + 12 { ∂ i φ, δφ } (2.19)In each case, one can verify the consistency condition ∂ [ i ∂ j ] ˜ δ = 0 is satisfied. By directcomputation one can then check that ˜ δ Φ and ˜ δφ satisfy (2.16) and (2.17) provided therespective SDYM and SDE equations are satisfied. We will refer to this δ → ˜ δ map as the symmetry raising map on each respective theory. It will turn out to be useful for identifyingthe DC rules for asymptotic symmetries.We conclude the section by noting that the scalar fields also have intrinsic redundancies:The field Φ should be thought of as defined modulo an arbitrary function Ψ( y ) : Φ ∼ Φ + Ψ( y ) (2.20)as this does not affect the YM field (2.10). Similarly, φ is defined modulo φ ∼ φ + ψ (0) ( y ) + x i ψ (1) i ( y ) . (2.21)which preserves (2.11). Note that (2.20) and (2.21) are not the standard gauge symmetriesof the YM and metric fields (to be discussed in the next section). Rather, they are additionalredundancies that arise in the description given by (2.10) and (2.11).
In this section we describe the gauge transformations on the YM and gravity fields thatpreserve the form (2.10), (2.11). It is interesting to note that these fields satisfy the followinggauge conditions ∂ µ A µ = 0 , ∂ µ h µν = 0 , η µν h µν = 0 . (3.1)Thus, the symmetries that we will be discussing can be thought of as a subset of the residualgauge symmetries of conditions (3.1). See e.g. Eq. (15) of [97] for the YM case. The attentive reader might notice that whereas (2.20) is a symmetry of the field equation (2.12), thisdoes not appear to be the case for (2.21) and (2.13). The reason for this is that the actual self-dual conditionon the metric, which is of course invariant under (2.21), takes the form of a differential operator acting on(2.13) - see subsection A.2. – 5 – .1 YM field
We start with the YM field. Given a gauge parameter Λ , the corresponding gauge trans-formation is δ Λ A µ = ∂ µ Λ + i [Λ , A µ ] . (3.2)If we want to preserve the condition A i = 0 , we must have ∂ i Λ = 0 = ⇒ Λ = Λ( y ) (3.3)We now ask if a gauge transformation with such gauge parameter is compatible with (2.10).That is, we ask whether there is a δ Λ Φ such that δ Λ A α = Π iα ∂ i δ Λ Φ . (3.4)By substituting (2.10) in (3.2), the LHS of (3.4) can be written as δ Λ A α = ∂ α Λ + i [Λ , Π iα ∂ i Φ] (3.5) = ∂ α Λ + i Π iα ∂ i [Λ , Φ] , (3.6)where to get the last line we used (3.3). The second term in (3.6) is of the required form(3.4). We find it convenient to use the notation δ Λ Φ = δ (0)Λ Φ + δ (1)Λ Φ (3.7)where δ (0)Λ Φ is independent of Φ and δ (1)Λ Φ is linear in Φ . We emphasize that this is an exactsymmetry and that we are not doing a perturbative expansion at this stage. Eq. (3.6) thentells us that δ (1)Λ Φ = i [Λ , Φ] . (3.8)It now remains to see whether the first term in (3.6) can also be written in the form (3.4).That is, we would like to find δ (0)Λ Φ such that Π iα ∂ i δ (0)Λ Φ = ∂ α Λ . (3.9)Using (2.6), this equation can equivalently be written as: ∂ i δ (0)Λ Φ = Ω αi ∂ α Λ . (3.10)Since the RHS is independent of x i , we can integrate it directly to obtain δ (0)Λ Φ = Ω αi x i ∂ α Λ . (3.11)Summarizing: The gauge symmetries compatible with (2.10) are parametrized by gaugeparameters Λ( y ) , and act on Φ according to δ Λ Φ = Ω αi x i ∂ α Λ + i [Λ , Φ] . (3.12)Additionally, one could also consider symmetries of the scalar Φ which do not arise as YMgauge transformations, but are intrinsic symmetries of the equation (2.12). As we will showin section 5, one of these families of transformations can be obtained as the single copy ofa gravitational diffeomorphism symmetry. We ignore an integration ‘constant’ Ψ( y ) due to the redundancy (2.20). – 6 – .2 Metric field In the gravitational case, the spacetime metric is given by g µν = η µν + h µν . (3.13)Let us emphasize this is not just a first order metric but the full-non linear (self-dual) metric.In particular, the inverse metric is exactly given by g µν = η µν − h µν since η ρσ h µρ h σν = 0 .Except for the metric tensor g , we will be raising and lowering indices with η .Given a vector field ξ µ , the gauge transformation on h µν is defined from the lie derivativeon g µν : δ ξ h µν := L ξ η µν + L ξ h µν (3.14)We will be looking for vector fields that are independent of h , so we can study each term in(3.14) separately. We start by imposing the condition that the first term in (3.14) preserves h iµ = 0 : L ξ η iµ = 2 ∂ ( i ξ µ ) = 0 , (3.15)where ξ µ ≡ η µν ξ ν , with ξ ν the infinitesimal diffeomorphism vector field. Setting µ = i, α we obtain ∂ ( i ξ j ) = 0 = ⇒ ξ i = a ij ( y ) x j − c i ( y ) with a ( ij ) = 0 , (3.16)and ∂ i ξ α + ∂ α ξ i = 0 = ⇒ ∂ i ξ α = − ∂ α a ij x j + ∂ α c i (3.17)An equation of the form ∂ i ξ α = v iα can be solved only if ∂ [ i v j ] α = 0 . This implies ∂ α a ij = 0 . (3.18)A constant a ij represents a rigid rotation in the x i plane and is hence part of the globalPoincare symmetries. Since we are interested in local symmetries, we set a ij = 0 . Integrat-ing (3.17) we are led to ξ α = ∂ α c i ( y ) x i + b α ( y ) (3.19)with b α ( y ) an integration “constant”. To continue, we need to (i) determine if the secondterm in (3.14) is compatible with h iµ = 0 and (2.11) and (ii) determine if ∂ ( α ξ β ) can bewritten in a form compatible with (2.11). To simplify the analysis, we will discuss separatelythe cases c i = 0 and b α = 0 . The general case can be obtained as a linear combination ofthese two simple cases. Setting c i = 0 , we have ξ i = 0 , ξ α = b α ( y ) . (3.20)and so the resulting vector field is given by ξ i = η iα b α ( y ) , ξ α = 0 . (3.21) Recall we raise and lower indices with η . For the vector field (3.21) this happens to be equivalent toraising and lowering with g , but this will not be always the case. – 7 –et us now discuss the second term in (3.14) L ξ h µν = ξ ρ ∂ ρ h µν + 2 ∂ ( µ ξ ρ h ν ) ρ . (3.22)From (3.21) and h iµ = 0 we find that the second term in (3.22) vanishes. The first termclearly respects h iµ = 0 . Furthermore, using Eq. (2.11) it can be written as ξ k ∂ k h αβ = ξ k ∂ k Π iα Π jβ ∂ i ∂ j φ (3.23) = Π iα Π jβ ∂ i ∂ j ( ξ k ∂ k φ ) (3.24)where in the second line we used the fact that ∂ i ξ k = 0 . Eq (3.24) gives the part of δ ξ φ that is linear in φ : δ (1) ξ φ = ξ k ∂ k φ. (3.25)It remains to see whether the first term in (3.14) is compatible with (2.11). Thus, we askif there is a δ (0) ξ φ such that Π iα Π jβ ∂ i ∂ j δ (0) ξ φ = 2 ∂ ( α ξ β ) . (3.26)Using (2.6) and (3.20) this equation can be written as: ∂ i ∂ j δ (0) ξ φ = 2Ω αi Ω βj ∂ ( α b β ) . (3.27)Since b β is independent of the x i variables, the equation can be directly integrated, leadingto δ (0) ξ φ = Ω αi Ω βj x i x j ∂ α b β . (3.28)The family of vector fields (3.21) then acts on the scalar field according to δ ξ φ = Ω αi Ω βj x i x j ∂ α b β + η iα b α ∂ i φ. (3.29)At this stage we notice that δ ξ φ does not appear to be a symmetry of the SDE equation(2.13). To see what has gone wrong, we need to recall (see Appendix A) that the SDEequation for φ (2.13) is a sufficient but not necessary condition to produce a self-dualmetric. The actual self-duality condition takes the form of a 2nd order differential operatoracting on the SDE equation (2.13), see Eqs. (A.21) and (A.23). Of course the self-dualitycondition is invariant under the transformation (3.29), since it arises from an infinitesimaldiffeomorphism. In order to have a symmetry of the SDE equation (2.13), one can checkthat it is sufficient to restrict b α ( y ) to be a total derivative, b α ( y ) = ∂ α b ( y ) . (3.30)We will refer to the resulting symmetry transformation as δ b , with: δ b φ = Ω αi Ω βj x i x j ∂ α ∂ β b + η iα ∂ α b∂ i φ. (3.31)We will later see that (3.30) does not actually represent any restriction on the asymptoticvector fields at null infinity. As in the YM case, we do not include the ‘integration constants’ that are part of the ambiguity in thedefinition of φ (2.21). – 8 – .2.2 Second family of symmetries Setting b α = 0 we have ξ i = − c i ( y ) , ξ α = ∂ α c j ( y ) x j . (3.32)We now start by demanding the first term in (3.14) to be compatible with (2.11), as thiswill impose restrictions on c i : ∂ ( α ξ β ) = 2 ∂ α ∂ β c k ( y ) x k = Π iα Π jβ ∂ i ∂ j δ (0) ξ φ (3.33)for some δ (0) ξ φ . Multiplying the equation with Ω ’s as before, the last equality is equivalentto ∂ i ∂ j δ (0) ξ φ = 2Ω αi Ω βj ∂ α ∂ β c k ( y ) x k =: c ijk x k . (3.34)For this to be integrable, we need the tensor c ijk to be symmetric under the exchange ofany two indices. The expression as given is already symmetric in ( ij ) . One can show that,in order to make it symmetric in ( jk ) (and hence in ( ik ) ) one needs to take c k ( y ) = Ω γk ∂ γ c ( y ) (3.35)for some function c ( y ) . The resulting expression can then be integrated, leading to δ (0) ξ φ = 13 Ω αi Ω βj Ω γk x i x j x k ∂ α ∂ β ∂ γ c. (3.36)We finally discuss the second term in (3.14) for the vector field ξ µ : ξ i = η iα Ω βj x j ∂ α ∂ β c, ξ α = − Ω αβ ∂ β c. (3.37)One can show that such a vector field preserves the condition h iµ = 0 as well as the form(2.11). To simplest way to show this is to verify the vector field leaves the tensor Π νµ invariant L ξ Π νµ = 0 , (3.38)from which it follows that L ξ h iµ = 0 and L ξ h αβ = 14 Π iα Π jβ ∂ i ∂ j δ (1) ξ φ (3.39)where δ (1) ξ φ = L ξ φ = ξ i ∂ i φ + ξ α ∂ α φ. (3.40)To summarize, the second family of vector fields acts on φ as: δ ξ φ = δ (0) ξ φ + δ (1) ξ φ , with δ (0) ξ φ and δ (1) ξ φ given by (3.36) and (3.40) respectively.We emphasize that both families are exact, non-perturbative symmetries of the fieldequations (2.13). As we will discuss in subsection 5.4, it turns out that the second familycan be obtained from the first one by the symmetry raising map given in Eq. (2.19).– 9 – Perturbative approach to symmetries
In this section we present a perturbative approach to symmetries of the SDYM and SDEequations. This will offer an alternative perspective on the symmetries described before.The perturbative approach will be useful when constructing the DC symmetry map insection 5. We will present expressions up to first order in the fields. The all-order expansionsare given recursively in subsection B.2.
The field equations (2.12) may be cast as a tower of field equations associated to a pertur-bative expansion Φ = Φ (0) + Φ (1) + · · · (4.1)such that (cid:3) Φ (0) = 0 (4.2) (cid:3) Φ (1) = − i Π ij [ ∂ i Φ (0) , ∂ j Φ (0) ] (4.3) · · · (4.4)We now look at symmetries δ Φ of the field equations (2.12) in the above perturbativescheme. That is, we look for δ Φ = δ Φ (0) + δ Φ (1) + · · · (4.5)such that (cid:3) δ Φ (0) = 0 (4.6) (cid:3) δ Φ (1) = − i Π ij [ ∂ i Φ (0) , ∂ j δ Φ (0) ] (4.7) · · · (4.8)We may think of these equations as determining δ Φ in terms of a “seed” δ Φ (0) : Given δ Φ (0) satisfying (4.6), δ Φ (1) is defined implicitly by (4.7), and similarly for the higher orderterms. Generically this would lead to a δ Φ (1) that depends non-locally on Φ (0) due to theneed to invert the wave operator. The case of gauge symmetries discussed previously is anexception. From (3.11) we see that in this case the seed is given by δ Φ (0) = Ω βj x j ∂ β Λ( y ) , (4.9)which satisfies (4.6): (cid:3) δ Φ (0) = 2 η iα ∂ i ∂ α δ Φ (0) = 2Ω αβ ∂ α ∂ β Λ = 0 . (4.10)If we now substitute (4.9) in the RHS of (4.7) we find − i Π ij [ ∂ i Φ (0) , ∂ j δ Φ (0) ] = − iη iβ [ ∂ i Φ (0) , ∂ β Λ] (4.11) = − i (cid:3) [Φ (0) , Λ] (4.12) Recall we have absorbed the coupling constant in the definition of the field; the order in the expansioncounts the number of Φ (0) fields. – 10 –here in the first equality we used Π ij Ω βj = η iβ and in the second equality we used (4.2)and the fact that Λ is independent of x i . From (4.7) we then conclude that δ Φ (1) = − i [Φ (0) , Λ] , (4.13)consistent with the result from the previous section. One could continue and obtain thehigher order terms, which in this case will resum to the exact symmetry (3.12). See Ap-pendix B for further discussion on higher order terms. Proceeding analogously for the “metric” field φ , we have φ = φ (0) + φ (1) + · · · , (4.14) (cid:3) φ (0) = 0 (4.15) (cid:3) φ (1) = 12 Π ij Π kl ∂ i ∂ k φ (0) ∂ j ∂ l φ (0) (4.16) · · · (4.17)A symmetry δφ = δφ (0) + δφ (1) + · · · of the field equations will then satisfy (cid:3) δφ (0) = 0 (4.18) (cid:3) δφ (1) = Π ij Π kl ∂ i ∂ k φ (0) ∂ j ∂ l δφ (0) (4.19) · · · (4.20)We now discuss how the residual gauge symmetries found in the previous section fit intothis description. For the first family of vector fields described earlier, Eq. (3.31) gives the following candidatefor the “seed” δφ (0) : δφ (0) = Ω αi Ω βj x i x j ∂ α ∂ β b, (4.21)which satisfies (cid:3) δφ (0) = 0 . If we now substitute (4.21) in the RHS of (4.19) we obtain Π ij Π kl ∂ i ∂ k φ (0) ∂ j ∂ l δφ (0) = 2 η kβ η iα ∂ i ∂ k φ (0) ∂ α ∂ β b (4.22) = (cid:3) η iα ∂ i φ (0) ∂ α b (4.23)where to get the last equality we used (4.18) and the fact that b is independent of x i .Comparing with (4.19) we conclude δφ (1) = η iα ∂ i φ (0) ∂ α b, (4.24)compatible with the expected result from (3.31).– 11 – .2.2 Second family of symmetries For the second family of symmetries, we take δφ (0) to be given by (3.36) δφ (0) = 13 Ω αi Ω βj Ω γk x i x j x k ∂ α ∂ β ∂ γ c, (4.25)It can be easily checked that the above satisfies (4.18). Substituting in the RHS of (4.19)one finds Π ij Π kl ∂ i ∂ k φ (0) ∂ j ∂ l δφ (0) = − η iβ η lγ Ω αm x m ∂ i ∂ l φ (0) ∂ α ∂ β ∂ γ c. (4.26)Since it is not immediately obvious how to “pull out” a d’Alambertian from such an expres-sion, let us take guidance from the previous section. From (3.40) and (3.37) we expect tohave δφ (1) = η iα Ω βj x j ∂ i φ (0) ∂ α ∂ β c − Ω αβ ∂ α φ (0) ∂ β c. (4.27)Indeed, one can check by direct computation that the d’Alambertian of (4.27) coincideswith (4.26), upon using (4.15). According to [86], the color to kinematics map for the self-dual sector consists in replacingYM matrix commutators by certain Poisson brackets on the x i variables, as reviewed in(2.15): Φ → φ, − i [ , ] → { , } . (5.1)This will be our starting point in understanding the double copy for symmetries. In thefollowing section, we will make the color-kinematics duality manifest at the level of thesymmetry transformations. A key ingredient in this will be to recast the gravitationalsymmetry in a form that is analogous to the YM one. We will achieve this by expressingthe gravitational symmetry parameter in terms of a Hamiltonian λ (w.r.t. the Poissonbracket) associated to the diffeomorphism vector field. This will allow us to supplementthe rules (5.1) with Λ → λ (5.2)where Λ is the YM symmetry parameter and λ is the Hamiltonian describing the gravita-tional symmetry. We will start by comparing the gauge symmetries found in the YM case (3.12) with thefirst family of symmetries found in the gravitational case (3.31): δ Λ Φ = Ω αi x i ∂ α Λ + i [Λ , Φ] (5.3) δ b φ = Ω αi Ω βj x i x j ∂ α ∂ β b + η iα ∂ α b∂ i φ. (5.4)Let us first focus on the terms that are linear in the scalar fields. According to (5.1), theYM term linear in Φ should be mapped to − i [Λ , Φ] → { λ, φ } (5.5)– 12 –or some λ . In order to reproduce the linear term in δ b φ we take λ = 2Ω αi x i ∂ α b, (5.6)which can be thought of as the “Hamiltonian” of the vector field ξ i = η iα ∂ α b with respectto the Poisson bracket { , } .We now notice that the φ -independent piece of δ b φ can be written in terms of λ , leadingto an expression that has the same form as the Φ -independent piece of δ Λ Φ . Thus, we canexpress the total δ b φ in terms of λ as: δ b φ = Ω αi x i ∂ α λ − { λ, φ } . (5.7)Comparing the expression above with (5.3), we find that the tranformation rules map intoeach other under Φ → φ, − i [ , ] → { , } , Λ → λ (5.8)with an apparent mismatch of for the inhomogeneous terms. However we note thatthis is resolved by expressing the transformation rules on the YM and gravitational fieldsthemselves, as: δ Λ A α = ∂ α Λ + i Π iα ∂ i [Λ , Φ] δ λ h αβ =Π i ( α ∂ i (cid:16) ∂ β ) λ − Π jβ ) ∂ j { λ, φ } (cid:17) (5.9)If we insist on working at the level of the scalar fields, then the replacement rules need tobe augmented by the inclusion of a multiplicative factor for the inhomogeneous term: r = deg (Λ) + 1 deg ( λ ) + 1 (5.10)where deg ( f ) counts the power of x i in a function f which is homogeneous in the x i variables.This numerical factor dissapears in the replacement rules for A α = Π iα ∂ i Φ → h αβ = Π iα Π jβ ∂ i ∂ j φ, (5.11)due to the action of the derivatives w.r.t x i on the inhomogeneous term.Finally, we note that deg ( λ ) = deg (Λ) + 1 (5.12)and moreover Λ( y ) → λ = 2Ω αi x i ∂ α b ( y ) . (5.13) Having understood how the first family of gravitational symmetries can be obtained fromthe YM ones, we now focus on the second family of gravitational symmetries: δ c φ = 13 Ω αi Ω βj Ω γk x i x j x k ∂ α ∂ β ∂ γ c + ξ i ∂ i φ + ξ α ∂ α φ (5.14)where ξ i = η iα Ω βj x j ∂ α ∂ β c, ξ α = − Ω αβ ∂ β c. (5.15)– 13 –e first note that the vector field ξ i can again be written in terms of a ‘Hamiltonian’ ˜ λ = Ω αi Ω βj x i x j ∂ α ∂ β c (5.16)so that ξ i ∂ i φ = { φ, ˜ λ } . Using (5.16), we can rewrite (5.14) as δ c φ = 13 Ω αi x i ∂ α ˜ λ − { ˜ λ, φ } − Ω αβ ∂ α φ∂ β c. (5.17)This brings δ c φ into a form that resembles the one obtained before for δ b φ in (5.7), withthe exception of the final term. To understand this extra term we will use the perturbativedescription of symmetries given in section 4, as described below.We would like to find a symmetry of the SDYM theory such that, upon the appropri-ate color to kinematics replacement rules, one recovers the second family of gravitationalsymmetries (5.17). Guided by (5.13), we find that the natural choice for the Φ -independentpiece of this YM symmetry is δ (0)˜Λ Φ = 12 Ω αi x i ∂ α ˜Λ (5.18)with ˜Λ of the form ˜Λ = Ω αi x i ∂ α B ( y ) (5.19)for some function B ( y ) . This generalizes the Φ -independent part of the symmetry (5.3) tothe case where Λ is linear in x . Restricting to the inhomogeneous terms in the transfor-mations for the YM field and the graviton, δ (0)˜Λ A α = ∂ α ˜Λ δ (0)˜ λ h αβ =Π i ( α ∂ i ∂ β ) ˜ λ (5.20)we find that they are related by the same rules (5.8) as the first family. Alternatively, onecan work directly at the level of the scalar fields by taking into acount the multiplicativefactor (5.10), which in this case is r = 2 / , using deg ( ˜Λ) = 1 and deg (˜ λ ) = 2 .The idea now is to use the perturbative method of section 4 to obtain the O (Φ ) sym-metry transformation generated by (5.18). We will then discuss how the color to kinematicrules applied to this term reproduce the two O ( φ ) terms in (5.17). O (Φ ) symmetry Following the perturbative method of section 4, we would like to solve for δ ˜Λ Φ (1) in Eq.(4.7), (cid:3) δ ˜Λ Φ (1) = − i Π ij [ ∂ i Φ (0) , ∂ j δ ˜Λ Φ (0) ] (5.21)given the ‘seed’ δ ˜Λ Φ (0) = 12 Ω αi x i ∂ α ˜Λ . (5.22) We emphasize here that δ ˜Λ is not defined directly on A µ , but through its action on the scalar Φ . Inparticular, using (2.8), we find δ (0)˜Λ A i ≡ Π µi ∂ µ δ (0)˜Λ Φ = 0 , due the fact that Π µi vanishes identically. Notethat δ (0)˜Λ A i (cid:54) = ∂ i ˜Λ , so it cannot be interpreted as a gauge transformation, even at the lowest order. – 14 –e proceed in a similar fashion as in the analysis between Eqs. (4.9) and (4.12) where weverified the perturbative method against the residual YM gauge symmetry. Noting that ∂ j δ ˜Λ Φ (0) = Ω αj ∂ α ˜Λ , the RHS of (5.21) can be written − i Π ij [ ∂ i Φ (0) , ∂ j δ ˜Λ Φ (0) ] = − iη iα [ ∂ i Φ (0) , ∂ α ˜Λ] (5.23) = − i (cid:3) [Φ (0) , ˜Λ] + 2 iη iα [ ∂ α Φ (0) , ∂ i ˜Λ] (5.24)where in the last line we used that (cid:3) Φ (0) = 0 = (cid:3) ˜Λ . Unlike for the residual YM symmetry,there is now a reminder term when one ‘pulls out’ the wave operator. We then concludethat δ ˜Λ Φ (1) = − i [Φ (0) , ˜Λ] + 2 i (cid:3) − η iα [ ∂ α Φ (0) , ∂ i ˜Λ] . (5.25) O ( φ ) symmetry from the DC If we apply the color to kinematic map − i [ , ] → { , } , Φ → φ, ˜Λ → ˜ λ (5.26)to expression (5.25) one finds δ ˜Λ Φ (1) → { φ (0) , ˜ λ } − (cid:3) − η iα { ∂ α φ (0) , ∂ i ˜ λ } (5.27)The first term corresponds to the second term in (5.17). For the second term we have − (cid:3) − η iα { ∂ α φ (0) , ∂ i ˜ λ } = − (cid:3) − η iα Π jk ∂ j ∂ α φ (0) ∂ k ∂ i ˜ λ (5.28) = − (cid:3) − η iα Π jk ∂ j ∂ α φ (0) Ω βi Ω γk ∂ β ∂ γ c (5.29) = − (cid:3) − Ω αβ η jγ ∂ j ∂ α φ (0) ∂ β ∂ γ c (5.30) = − (cid:3) − Ω αβ η jγ ∂ j ∂ γ ( ∂ α φ (0) ∂ β c ) (5.31) = − Ω αβ ∂ α φ (0) ∂ β c, (5.32)where we used the definition of the Poisson bracket, the expression for ˜ λ as given in (5.16),and the fact that (cid:3) φ (0) vanishes. Expression (5.32) reproduces, in the perturbative setting,the last term in (5.17). This establishes δ ˜Λ Φ (1) as the inverse DC map of δ c φ (1) . In fact, asshown in Appendix B the DC map can be extended to arbitrary order in the perturbativeexpansion. Let us summarize our findings. By associating certain “Hamiltonians” to the infinitesimalresidual diffeomorphisms of the SDE equations, we managed to obtain simple DC rulesmapping SDYM to SDE symmetries. In order to compactly write these rules for the scalarfields, let us introduce the notation S := Ω αi x i ∂ α . (5.33)The first family of symmetries take the form δ Λ Φ = S (Λ) + i [Λ , Φ] (5.34) δ λ φ = S ( λ ) − { λ, φ } , (5.35)– 15 –here on the YM side we have a gauge transformation with parameter Λ independent of x i ,and on the gravitational side we have an infinitesimal diffeomorphism with ‘Hamiltonian’ λ that is linear in x i . The two are related by the DC map (5.8), supplemented by themultiplicative coefficient r in the inhomogeneous term (see discussion around (5.10) fordetails): Φ → φ, − i [ , ] → { , } , Λ → λ, S → r S, r = deg (Λ) + 1 deg ( λ ) + 1 . (5.36)The second family can be written as δ ˜Λ Φ = S ( ˜Λ) − i [Φ , ˜Λ] + 2 i (cid:3) − η iα [ ∂ α Φ , ∂ i ˜Λ] + O (Φ ) (5.37) δ ˜ λ φ = S (˜ λ ) + { φ, ˜ λ } − (cid:3) − η iα { ∂ α φ, ∂ i ˜ λ } + O ( φ ) (5.38)The two are related by exactly the same DC rules (5.36) as for the first family. Althoughhere we have presented the argument up to linear order in the field, the map can in factbe extended to arbitrary order, see Appendix B. Let us emphasize that the original grav-itational symmetry, as given in (5.17), is non-perturbative. In (5.38), the higher orderterms arise as a result of parametrizing this symmetry in terms of the Hamiltonian ˜ λ . Thisrequires the use of the field equations, which introduce higher order terms.The second family displays two interesting features. Firstly, it maps a perturbativeexpression on the YM side to a non-perturbative one on the gravity side, as describedabove. Secondly, while on the gravity side the symmetry is a diffeomorphism, on the YMside we have a non-gauge symmetry of the field equation (2.12). In subsection 2.1 we described the existence of symmetry raising maps on YM and gravityself-dual theories that, given a symmetry δ , produce a new symmetry ˜ δ according to Eqs.(2.18), (2.19).Since, as described in section 4 the symmetries we are studying are fully determinedby their inhomogenous piece, it is natural to restrict these equations to the inhomogenousparts of the symmetries: ∂ i ˜ δ (0) Φ = Ω αi ∂ α δ Φ (0) (5.39) ∂ i ˜ δ (0) φ = Ω αi ∂ α δφ (0) . (5.40)It is now straightforward to check that, if we take δ Φ (0) = δ Λ Φ (0) and δφ (0) = δ λ φ (0) , theresulting ˜ δ (0) are given by δ (0)˜Λ Φ and δ (0)˜ λ φ with ˜Λ = δ (0)Λ Φ = S (Λ) (5.41) ˜ λ = δ (0) λ φ = S ( λ ) . (5.42)Since the symmetries are fully determined by their inhomogenous piece, the above showsthat the map δ → ˜ δ takes the full δ Λ Φ into the full δ ˜Λ Φ (and similarly for δ λ φ ). We can– 16 –ummarize the above structure by the diagram δ Λ Φ DC −−→ δ λ φ ↓ ↓ δ ˜Λ Φ DC −−→ δ ˜ λ φ, (5.43)where the horizontal arrows represent the DC map between each family of symmetries, andthe vertical arrows the δ → ˜ δ map on each theory.Said in different words, we have verified that the symmetry raising map commuteswith the DC symmetry map. We will use this result in the next section to obtain the DCsymmetry map at null infinity for the second family of symmetries. We found that the second family of symmetries arises from applying the raising map onthe first family. It is clear that one can continue this process indefinitely and constructan infinite tower of symmetries on both sides. A natural question is whether the DC willcontinue to hold at each level. In other words, we want to check whether the diagram (5.43)extends indefinitely: δ Φ DC −−→ δ φ ↓ ↓ δ Φ DC −−→ δ φ ↓ ↓ ... ... δ n Φ DC −−→ δ n φ ... ... (5.44)where δ n denotes the symmetry of the n -th family. The first two lines of the diagramcorrespond to the original diagram (5.43). It is easy to show by recursion that the DCholds at every level. We have explicitly demonstrated this for δ and δ in subsection 5.1,subsection 5.2, and Appendix B. Assuming that the DC holds at level n − , it immediatelyfollows from the definition of the n -th level symmetry (2.18) and (2.19): ∂ i δ n Φ = Ω αi ∂ α δ n − Φ − i [ ∂ i Φ , δ n − Φ] (5.45) ∂ i δ n φ = Ω αi ∂ α δ n − φ + 12 { ∂ i φ, δ n − φ } (5.46)that δ n φ is obtained from δ n Φ via the DC rules (5.36). The focus in this paper is on the firsttwo families, since they are the ones corresponding to asymptotic diffeomorphisms on thegravitational side. It is however instructive to display the general form of the inhomogenous n -th transformations. Consider first the YM case, starting with the ‘zeroth’ n = 0 family: δ Φ = Λ( y ) (5.47)Notice that these are just the redundancies described in (2.20). It is then easy to solve(5.45) at O (Φ ) . One finds δ (0) n Φ = 1 n ! S n (Λ) . (5.48)– 17 –s we did for the second family in (5.41), we can introduce a gauge-like parameter Λ n = δ (0) n − Φ = 1( n − S n − (Λ) (5.49)at n -th level, such that δ (0) n Φ = 1 n S (Λ n ) = 1 deg (Λ n ) + 1 S (Λ n ) . (5.50)We refer to Λ n as a gauge-like parameter since δ (0) n A α = Π iα ∂ i δ (0) n Φ = ∂ α Λ n . (5.51)Note however that δ (0) n A i = 0 (cid:54) = ∂ i Λ n (see footnote 10 for n = 2 ). Let us now discuss thegravitational case. Recall that in this case there are two kind of redundancies (2.21). Inorder to be consistent with the diagram (5.44), one is lead to start the recursion at n = − with, δ − φ = 2 b ( y ) . (5.52)Solving (5.46) at O ( φ ) . One finds δ (0) n φ = 1( n + 1)! S n +1 (2 b ) = 1( n + 1)! S n ( λ ) , λ := S (2 b ) , (5.53)where we expressed the result in terms of the first family Hamiltonian λ . The n = − , levels describe the two redundancies in (2.21). As we did for the second family (5.42), wecan define a ‘Hamiltonian’ parameter λ n = δ (0) n − φ = 1 n ! S n − ( λ ) (5.54)at n -th level, such that δ (0) n φ = 1 n + 1 S ( λ n ) = 1 deg ( λ n ) + 1 S ( λ n ) . (5.55)As in the first two families, with this definition of λ n one has δ (0) n h αβ = Π i ( α ∂ i ∂ β ) λ n , (5.56)corresponding with the DC version of (5.51). Comparing (5.50) with (5.55) we verify theseare indeed compatible with (5.36). The DC map can be extended to arbitrary order in thefields, following subsection B.2. This extends the applicability of the rules (5.36) to all n levels, Φ → φ, − i [ , ] → { , } , Λ n → λ n , S → r S, r = deg (Λ n ) + 1 deg ( λ n ) + 1 . (5.57)– 18 – omment: It may naively appear that there exists a different DC prescription, one thatmaps the n -th YM family to the ( n − -th gravity one. Upon relabeling the gravitysubscripts, this would yield identical zeroth order expressions on both sides. However thisprescription has various drawbacks. Firstly, it would map YM gauge symmetries δ A µ = δ Λ A µ = ∂ µ Λ + i [Λ , A µ ] into trivial diffeomorphisms δ h µν = 0 . In particular, the large YMgauge symmetries at null infinity would not have a gravitational counterpart. Secondly,the second family of YM symmetries would be mapped to the first family of gravitationalones. As we shall see, this would imply an additional mismatch in the number of would-beDC related asymptotic symmetries. Finally, our original DC prescription (5.43) is shown inAppendix C to be consistent with the convolution DC [42–44] where they overlap, whereasthe alternative DC would not be. In this section we switch to coordinates adapted to null infinity. Given the light conecoordinates ( U, V, Z, ¯ Z ) introduced in (2.1), a natural choice for Bondi-type coordinates ( r, u, z, ¯ z ) is given by: U = rz ¯ z + u, V = r, Z = rz, ¯ Z = r ¯ z, (6.1)in terms of which the Minkowski line element takes the form ds = − dudr + 2 r dzd ¯ z. (6.2)Future null infinity I is reached by taking r → ∞ while keeping the rest of the variablesfixed. It is thus parametrized by ( u, z, ¯ z ) . Note that we will be working in a flat conformalframe for which the (degenerate) metric at null infinity is dzd ¯ z . Massless fields are captured at null infinity by certain r → ∞ leading components. Fora massless scalar field, this is given by ϕ ( r, u, z, ¯ z ) r →∞ = ϕ I ( u, z, ¯ z ) /r + · · · (6.3)where the dots indicate terms that decay faster than /r . ϕ I ( u, z, ¯ z ) is to be regarded asa field defined intrinsically at null infinity, which contains all the radiative data of ϕ . Wewill also refer to it as the ‘free data’ at I , since it may be regarded as the analogue of theCauchy data for the ‘final time’ hypersurface I , see e.g. [63]. Similarly, gauge and metricfields at null infinity are captured by A z ( r, u, z, ¯ z ) r →∞ = A z ( u, z, ¯ z ) + · · · (6.4) h zz ( r, u, z, ¯ z ) r →∞ = rC zz ( u, z, ¯ z ) + · · · (6.5) This can be contrasted with the often used Bondi frame for which the 2d metric is that of the unitsphere. See e.g. appendix A of [104] for a comparison between the two. – 19 –nd respective z ↔ ¯ z expressions. If we now consider expressions (2.10) and (2.11) in Bondicoordinates (6.1) we find A z = r∂ u Φ , A ¯ z = 0 , (6.6) h zz = r ∂ u φ, h ¯ z ¯ z = 0 , (6.7)from which one easily finds how the gauge and gravity free data is encoded in the scalarfields free data (see (6.3)) A z = ∂ u Φ I , A ¯ z = 0 (6.8) C zz = ∂ u φ I , C ¯ z ¯ z = 0 . (6.9)We thus recover the expected result that self-dual fields are associated with data at nullinfinity that has vanishing antiholomorphic components.In the reminder of the section we will study how the ‘bulk’ symmetries described in theprevious sections manifest at null infinity, with the aim of obtaining DC rules for asymptoticsymmetries. Before getting started, however, it will be useful to briefly review the standarddescription of asymptotic symmetries. In the general (not necessarily self-dual) case, one considers gauge and diffeomorphismsymmetries that are non-trivial at null infinity. In the YM case, these are generated bygauge parameters that asymptote to a non-trivial function on the celestial sphere [67, 105], Λ( r, u, z, ¯ z ) r →∞ = Λ ( z, ¯ z ) + · · · (6.10)where the dots denote terms that go to zero as r → ∞ . The induced symmetry transfor-mation on the null infinity radiative data is then δ Λ A z = ∂ z Λ + i [Λ , A z ] . (6.11)In the gravitational case one finds two classes of vector fields: The Bondi-Metzner-Sachssupertranslations [60, 61], parametrized by functions f ( z, ¯ z ) , ξ f = f ( z, ¯ z ) ∂ u + · · · (6.12)and the Barnich-Troessaert superrotations [65], parametrized by holomorphic 2d vectorfields Y z ( z ) ∂ z + Y ¯ z (¯ z ) ∂ ¯ z , ξ Y = Y z ( z ) ∂ z + u ∂ z Y ( z ) ∂ u − ∂ z Y ( z ) r∂ r + ( z ↔ ¯ z ) + · · · , (6.13)where in both cases the dots denote terms that vanish when r → ∞ . The infinitesimaldiffeomorphisms generated by these vector fields induce the following action on the nullinfinity radiative data: δ f C zz = − ∂ z f + f ∂ u C zz (6.14) δ Y C zz = − u∂ z Y z + ( Y z ∂ z + 2 ∂ z Y z + u ∂ · Y ∂ u − ∂ · Y ) C zz (6.15)– 20 –here ∂ · Y = ∂ z Y z + ∂ ¯ z Y ¯ z . The transformation properties for C ¯ z ¯ z can be obtained bydoing the replacement z → ¯ z in the expressions above.Whereas the above represent the simplest large gauge symmetries, it is possible tofind additional ones by carefully relaxing the standard fall-off conditions (6.4), (6.5). Inparticular, one can make sense of gauge symmetries with O ( r ) gauge parameter [106–109]and non-holomorphic superrotations [110–112]. Interestingly, the O ( r ) gauge symmetrieswill turn out to play a key role in identifying the single copy version of superrotations, seesection 7.Coming back to the self-dual case, we can ask what is the subset of the previouslydiscussed symmetries that preserve the conditions A ¯ z = 0 , C ¯ z ¯ z = 0 (6.16)at null infinity. On the YM side, this condition is preserved provided Λ is holomorphic, Λ = Λ ( z ) . (6.17)On the gravity side, the supertranslation function has to be of the form f = f ( z ) + ¯ zg ( z ) . (6.18)Finally, the antiholomorphic component of the superrotation vector field has to be of theform Y ¯ z = a + a ¯ z + a ¯ z for some constants a , a and a . This corresponds to Lorentzgenerators and is hence part of the global symmetry group. If, as in section 3, we only keeptrack of local symmetries, we are left with vector fields of the form Y z = Y ( z ) , Y ¯ z = 0 . (6.19)Below we will recover these restricted symmetries as the null infinity limit of the residualgauge symmetries discussed in section 3. We now discuss the null infinity limit of the residual gauge symmetries described in section 3.In order to do so, we will chose the functional form of the symmetry parameters that yieldsa finite and non-trivial action at null infinity.Recall that the symmetry parameters can be constructed from functions which dependonly on the y α = ( V, Z ) variables: Λ( y ) for the YM gauge symmetries and b ( y ) and c ( y ) for the first and second family of gravitational symmetries. In terms of Bondi coordinates(6.1), this translates into functions of the variables ( r, z ) . We will consider simple functionalforms, F ( r, z ) = r n F ( z ) , and choose n so as to obtain well defined r → ∞ limits.Let us start with the YM residual gauge symmetry (3.12). In order to have a well-defined action at null infinity, we need the RHS of (3.12) to be O ( r − ) . For the homogeneousterm, this can only be achieved if Λ = O ( r ) . We are thus led to choose Λ as Λ( r, z ) = Λ ( z ) (6.20)– 21 –et us now discuss the inhomogeneous term in (3.12). To do so we note that the differentialoperator S (5.33) in Bondi coordinates takes the form S ≡ Ω αi x i ∂ α = − ¯ zu∂ u − ¯ z ∂ ¯ z + ¯ zr∂ r + ur ∂ z . (6.21)When (6.21) acts on Λ (6.20), only the last term survives, yielding a O ( r − ) result asexpected. From here one can read off the action on the radiative data to be δ Λ Φ I = u∂ z Λ ( z ) + i [Λ ( z ) , Φ I ] . (6.22)The corresponding action on A z can be obtained from the relation (6.8), where one recovers(6.11) for holomorphic Λ .Consider now the first family of gravitational symmetries, (3.29), with b α = ∂ α b as inEq.(3.31). Writing the homogenous term in Bondi coordinates, one easily finds that, inorder for this term to be O ( r − ) one needs to take b = O ( r ) . We then choose b ( r, z ) = − rf ( z ) . (6.23)The inhomogeneous term associated to b turns out to be O ( r − ) due to subtle cancellations.One finally arrives at δ b φ I = − u f (cid:48)(cid:48) ( z ) + f ( z ) ∂ u φ I (6.24)for the induced action on the free data. The corresponding action on C zz can be obtainedfrom the relation (6.9), where one recovers (6.14) for f = f ( z ) . By similar considerations,one can extend the previous analysis to parameters b α that are not necessarily a totalderivative. One finds that, at null infinity, the only finite and non-trivial transformationthat can be obtained is still of the form (6.24). Thus, the condition b α = ∂ α b does notimply a restriction as far as the asymptotic symmetries are concerned.Let us finally discuss the second family of gravitational symmetries introduced in sub-subsection 3.2.2. In this case one finds there are two possible choices of c ( y ) , either O ( r ) or O ( r ) , that yield a finite and non-trivial null infinity limit: c ( r, z ) = rg ( z ) or c ( r, z ) = − r Y ( z ) / (6.25)For the first choice, one obtains δ c φ I = − u ¯ zg (cid:48)(cid:48) ( z ) + ¯ zg ( z ) ∂ u φ I , (6.26)which, upon translating to C zz , yields a supertranslation (6.14) with f ( z, ¯ z ) = ¯ zg ( z ) . Forthe second choice of c in (6.25) one finds δ c φ I = − u Y (cid:48)(cid:48)(cid:48) ( z ) + ( Y ( z ) ∂ z + u Y (cid:48) ( z ) ∂ u + 12 Y (cid:48) ( z )) φ I , (6.27)which, when translated to C zz via (6.9), reproduces the superrotation action (6.15) withvector field (6.19). – 22 – Color to kinematic symmetry map at null infinity
Having understood how large gauge symmetries emerge from the residual symmetries dis-cussed in section 3, we now study how the DC for ‘bulk’ symmetries described in section 5translates into a DC for asymptotic symmetries at null infinity. This will be achieved bystudying the DC symmetry relations to leading order in a /r expansion. Whereas for thefirst family (Eqs. (5.34) and (5.35)) this is rather straightforward, for the second family(Eqs. (5.37) and (5.38)) the expansion is trickier due to the appearance of the inverse waveoperator. We will circumvent this difficulty by making use of the symmetry raising mapdiscussed in subsection 5.4.To get started, let us express the operators that appear in the description of bulksymmetries in Bondi coordinates, paying special attention to their /r expansion. Givena spacetime function of the form r n F n ( u, z, ¯ z ) , we will regard F n ( u, z, ¯ z ) as a function of“weight" n at null infinity ( n determines the the transformation properties of F n ( u, z, ¯ z ) under a conformal rescaling). The PB in Bondi coordinates takes the form { f, g } = r − { f, g } I (7.1)with { f, g } I = ( ∂ ¯ z f ∂ u g − ∂ u f ∂ ¯ z g ) . (7.2)We will regard { , } I as a PB defined intrinsically at null infinity (with weight − ). Considernow the differential operator S = Ω αi x i ∂ α (7.3)that features in the inhomogeneous part of the symmetries. Its form in Bondi coordinateswas given in (6.21). When acting on a function of the form r n F n ( u, z, ¯ z ) , it can be writtenas: S ( r n F n ) = r n S ( F n ) + r n − S − ( F n ) (7.4)where S = − ¯ zu∂ u + n ¯ z − ¯ z ∂ ¯ z (7.5) S − = u∂ z . (7.6) S and S − will be regarded as differential operators defined intrinsically at null infinity.Note that S depends on the weight n of the function being acted upon.We now proceed to describe the DC for the first and second family of asymptoticsymmetries. The YM symmetry (5.34) at null infinity was described in Eq. (6.22) for
Λ = Λ ( z ) . Usingthe notation introduced above, we write it as δ Λ Φ I = S − (Λ ) + i [Λ , Φ I ] . (7.7) Notice that this n has nothing to do with the n of subsection 5.5, which labeled the symmetry families. – 23 –or the gravity symmetry (5.35), we need to compute the ‘Hamiltonian’ λ (5.6) for thechoice of b ( y ) given in (6.23). This gives λ = 2 S ( b = − rf ( z )) = rλ + λ (7.8)with λ = − S ( f ) = − zf ( z ) (7.9) λ = − S − ( f ) = − uf (cid:48) ( z ) . (7.10)The homogenous term in (5.35) then has the expected order in r , since a λ with an O ( r ) piece will compensate for the r − factor in the PB (7.1). Naively, it appears that theinhomogenous term could have O ( r ) and O ( r ) terms, thus spoiling the required O ( r − ) fall-offs. However, such coefficients turn out to be zero. In the end one finds a well definedaction on the radiative data, given by δφ I = 12 S − ( λ ) − { λ , φ I } I . (7.11)We emphasize that (7.11) is just a rewriting of (6.24), but one that makes the DC structuremanifest. Comparing (7.7) to (7.11), we find that the DC rules at null infinity have the samestructure as those in the bulk, as expected. The new ingredient is that, in the homogeneousterm, the weight of the symmetry parameter increases by 1 (this is to compensate for thefactor of /r in the PB (7.1)). For the second family of symmetries, the DC relation in the bulk was made manifest byexpressing the symmetries perturbatively, Eqs. (5.37) and (5.38). Unfortunately, suchexpressions are difficult to expand in /r due to the appearance of the inverse wave operator.However, the discussion given in subsection 5.4 offers an alternative route to obtaining theDC symmetry map: the second family of symmetries for each theory can be obtained fromthe first family via the “symmetry raising map”, see the vertical arrows of (5.43). Thus,rather than going directly from δ ˜Λ Φ to δ ˜ λ φ via the perturbative bulk DC map (bottomhorizontal arrow of (5.43)), one can try to get there via the first family of symmetries. Thisis the strategy we will follow in this section to describe the second family of asymptoticsymmetries and their DC relation.Let us start with a general discussion that applies to both theories; for concretenesswe only display the equations corresponding to the YM case. Since the first family ofsymmetries can be written as a sum of O (Φ ) and O (Φ ) terms, δ Φ = δ (0) Φ + δ (1) Φ , theraising map (2.18) produces a symmetry ˜ δ that is at most O (Φ ) : ˜ δ Φ = ˜ δ (0) Φ+˜ δ (1) Φ+˜ δ (2) Φ .Collecting powers of Φ in (2.18), one finds each term should satisfy ∂ i ˜ δ (0) Φ = Ω αi ∂ α δ (0) Φ , (7.12) ∂ i ˜ δ (1) Φ = Ω αi ∂ α δ (1) Φ − i [ ∂ i Φ , δ (0) Φ] , (7.13) ∂ i ˜ δ (2) Φ = − i [ ∂ i Φ , δ (1) Φ] . (7.14)– 24 –q. (7.12) was already discussed in subsection 5.4, where it was found that ˜ δ (0) Φ = 12 S (Λ) = 12 S ( ˜Λ) , with ˜Λ = δ (0)Λ Φ = S (Λ) . (7.15)Similarly, if we start with a gravitational first family symmetry, so that δ (0) φ = S ( λ ) (with λ = 2 S ( b ) ) one finds ˜ δ (0) φ = 16 S ( λ ) = 13 S (˜ λ ) , with ˜ λ = δ (0) λ φ = 12 S ( λ ) . (7.16)The numerical factors in (7.15) and (7.16) arise as particular cases of the general coefficientsexplained in subsection 5.5. We will later expand these expressions in /r in order to obtain ˜ δ (0) Φ I and ˜ δ (0) φ I .To obtain ˜ δ (1) Φ I and ˜ δ (2) Φ I , we will solve Eqs. (7.13) and (7.14) asymptotically asfollows. First, we note that the x i derivatives expressed in Bondi coordinates, ∂ U = ∂ u (7.17) ∂ ¯ Z = − z∂ u + r − ∂ ¯ z (7.18)are not independent in the r → ∞ limit. Thus, to leading order in /r it is enough toconsider only one component of Eqs. (7.13) and (7.14). Choosing i = U and writing themin Bondi coordinates one obtains ∂ u ˜ δ (1) Φ = ( − ¯ z∂ u + r − ∂ z ) δ (1) Φ − i [ ∂ u Φ , δ (0) Φ] (7.19) ∂ u ˜ δ (2) Φ = − i [ ∂ u Φ , δ (1) Φ] (7.20)In order to have a well defined action at null infinity, we need the RHS of these equationsto be O ( r − ) . The resulting O ( r − ) factors will then give expressions for ∂ u ˜ δ (1) Φ I and ∂ u ˜ δ (2) Φ I , from which the final answer can be obtained by a single integral in u .Similar considerations apply to the gravitational case, in which case ˜ δ (1) φ I , ˜ δ (2) φ I areto be determined from the O ( r − ) part of the equations ∂ u ˜ δ (1) φ = ( − ¯ z∂ u + r − ∂ z ) δ (1) φ + { ∂ u φ, δ (0) φ } (7.21) ∂ u ˜ δ (2) φ = { ∂ u φ, δ (1) φ } . (7.22)In order to perform the required /r expansion, we need to specify the asymptoticbehavior of the symmetry parameters Λ and λ . We now describe two possibilities thatyield a well defined action of ˜ δ at null infinity. ˜ δ from Λ = O ( r ) , λ = O ( r ) We start by considering symmetry parameters as those discussed in subsection 7.1,
Λ = Λ ( z ) (7.23) λ = rλ + λ (7.24)– 25 –ith λ as in Eqs. (7.8), (7.9), (7.10). The corresponding ˜Λ and ˜ λ defined in Eqs. (7.15)and (7.16) take the form ˜Λ = r − ˜Λ − , ˜ λ = r − ˜ λ − (7.25)with ˜Λ − = S − (Λ ) = u Λ (cid:48) ( z ) (7.26) ˜ λ − = S − ( λ ) = − u f (cid:48)(cid:48) ( z ) . (7.27)One then obtains ˜ δ (0) Φ I = 12 S ( ˜Λ − ) = − u ¯ z Λ (cid:48) ( z ) (7.28) ˜ δ (0) φ I = 13 S (˜ λ − ) = u ¯ zf (cid:48)(cid:48) ( z ) (7.29)for the inhomogeneous part of the symmetry transformations. The linear part of the sym-metries can be obtained from Eqs. (7.19), (7.21), taking into account the form of δ (0) and δ (1) given in subsection 7.1. One finds that the only terms contributing to order O ( r − ) are those coming from the ¯ z∂ u term hitting the commutator/PB. This allows for a trivialintegration in u , leading to ˜ δ (1) Φ I = − i ¯ z [Λ , Φ I ] , (7.30) ˜ δ (1) φ I = 12 ¯ z { λ , φ I } I . (7.31)Finally, one finds that the RHS of Eqs. (7.20) and (7.22) decay faster than O ( r − ) andhence ˜ δ (2) Φ I = 0 , ˜ δ (2) φ I = 0 . (7.32)The gravitational symmetry ˜ δφ I can be seen to coincide with a supertranslation associatedto ˜ f ( z, ¯ z ) = − ¯ zf ( z ) (Eq. (6.26) with g ( z ) → f ( z ) ). Thus, the symmetry raising map takesa supertranslation with f = f ( z ) into a supertranslation with ˜ f = − ¯ zf ( z ) . ˜ δ from Λ = O ( r ) , λ = O ( r ) We now consider symmetry parameters Λ and λ of one order higher in r . Although thesegenerate transformations that violate the r → ∞ fall-offs of the YM and gravity fields, thecorresponding ˜ δ symmetries will turn out to preserve such fall-offs.On the YM side, we consider a O ( r ) gauge transformation Λ = r Λ ( z ) + Λ (7.33)where Λ = u∂ z ∂ ¯ z Λ ( z ) = 0 (as obtained from the condition (cid:3) Λ = 0 ). We explicitly keepthis vanishing term in (7.33) since it will get mapped into a non-trivial term under the DC.The corresponding ˜Λ is: ˜Λ = S (Λ) = rS (Λ) + S (Λ) (7.34)with ˜Λ = S (Λ) = S (Λ ) = ¯ z Λ ( z ) , ˜Λ = S (Λ) = S − (Λ ) = u Λ (cid:48) ( z ) . (7.35) Since S (Λ) has a non-trivial O ( r ) part, there is no need in this case to explicitly include the vanishing Λ contribution. – 26 –ubstituting this expansion in (7.15) one finds ˜ δ (0) Φ I = 12 S − ( ˜Λ ) = u (cid:48)(cid:48) ( z ) . (7.36)To obtain ˜ δ (1) Φ I , we consider Eq. (7.19) for δ (0)Λ Φ = S (Λ) and δ (1)Λ Φ = i [Λ , Φ] . Interestingly,the potentially divergent terms cancel out, and one arrives at a finite equation for the nullinfinity free data Φ I : ∂ u ˜ δ (1) Φ I = − i ¯ z∂ u [Λ , Φ I ] + i∂ z [Λ , Φ I ] − i [ ∂ u Φ I , S − (Λ )] . (7.37)Similarly, from Eq. (7.20) one obtains ∂ u ˜ δ (2) Φ I = [ ∂ u Φ I , [Λ , Φ I ]] . (7.38)On the gravitational side, we now consider a symmetry parameter b that is O ( r ) b = − r Y ( z ) / . (7.39)The corresponding λ = 2 S ( b ) takes the form λ = r λ + rλ (7.40)with λ = − zY ( z ) , λ = − uY (cid:48) ( z ) . (7.41) S ( λ ) = 2˜ λ is then given by S ( λ ) = r S ( λ ) + rS ( λ ) + S ( λ ) (7.42)with S ( λ ) = S ( λ ) = − z Y ( z ) , (7.43) S ( λ ) = S − ( λ ) = − u ¯ zY (cid:48) ( z ) , (7.44) S ( λ ) = S − ( λ ) = − u Y (cid:48)(cid:48) ( z ) , (7.45)where we used that S ( λ ) = 0 . To obtain ˜ δ (0) φ I , we substitute the above expansion in(7.16). It turns out that all potentially divergent terms cancel out and one is left with a O ( r − ) term, yielding ˜ δ (0) φ I = 13 S − (˜ λ ) = − u Y (cid:48)(cid:48)(cid:48) ( z ) . (7.46)This precisely reproduces the inhomogenous part of the superrotation action (6.27).To obtain ˜ δ (1) φ I , we consider Eq. (7.21) for δ (0) λ φ = S ( λ ) and δ (1) λ φ = − { λ, φ } , with λ , S ( λ ) as given by (7.40), (7.42). Once again the divergent terms cancel out, and oneobtains a well defined equation at null infinity: ∂ u ˜ δ (1) φ I = 12 ¯ z∂ u { λ , φ I } I − ∂ z { λ , φ I } I + 14 { ∂ u φ I , S − ( λ ) } I . (7.47) The structure of ˜Λ in this case is formally identical to the λ of the first gravitational symmetry (7.8),with Λ = − f . The same cancellations occur when computing S (˜Λ) , so that only the O ( r − ) part survives. – 27 –his is the DC version of (7.37), with the rule described in subsection 7.1 for the increasein weight when going from Λ to λ , and the multiplicative r factor associated to the operator S , see Equation 5.36. Finally, the RHS of (7.20) is found to be O ( r − ) and hence ˜ δ (2) φ I = 0 . (7.48)It may be puzzling that the non-zero quadratic term (7.38) trivializes after the DC. Fromthe perspective of null infinity, this happens because the DC takes the weight commutator [ , ] into the weight − Poisson bracket { , } I . Since in both cases one should have overallweight minus one, one would need to map Λ into a weight 3 λ to get a non-trivial DCresult. However there is no λ in the gravitational symmetry under consideration.We conclude by noting that if one expands (7.47) by explicitly writing the PB (7.2)and the parameters (7.41), (7.44), one recovers, as expected, the total u -derivative of thehomogenous part of the superrotation action (6.27). Thus, the symmetry raising map takesa “divergent supertranslation” defined by (7.39) into a superrotation. In Figure 1 we summarize the different symmetries at null infinity, together with the func-tional form of their parameters. The diagram may be thought of as an augmented versionof (5.43), with a new layer due to the two possibilities that arise in the second family ofsymmetries. In the first line we have holomorphic non-abelian large gauge transformations(6.22) and holomorphic supertranslations (6.24). In the second line we have the non-gaugeYM transformation (7.28), (7.30) and the second type of allowed supertranslations (6.26).The third line displays the non-gauge YM transformation defined by Eqs. (7.36), (7.37),(7.38) and the superrotations (6.27).The horizontal arrows represent the double copy relations between the different sym-metries. The replacement rules are as in the bulk, with the following clarification: thecommutator is replaced by the PB at null infinity (7.2), which has weight − . To com-pensate for this, YM parameters of weight n are replaced with Hamiltonians of weight n + 1 .Vertical arrows describe the symmetry raising map, see subsection 5.4. Diagonal arrowsdescribe the increase in the power of r associated to the parameter of the second familyof symmetries, see e.g. Eq. (6.25) for the gravitational case. To better understand thesearrows, one can imagine completing the diagram by raising the power of r in the parametersof the first family of YM and gravity symmetries, thus completing the cube. We can thenrecover the bottom two symmetries by applying the symmetry raising map to these newvertices, see subsubsection 7.2.2 for details. We chose not to display them in the diagramsince they do not preserve the r → ∞ field fall-offs. See the discussion section for further comments on this point. – 28 –
LGT, Λ = Λ(𝑧) ST , 𝑓 = 𝑓(𝑧) √ST , Λ̃ 𝑇 = 𝑧̅Λ T (𝑧) Λ̃ ST , 𝑓̃ = 𝑧̅𝑔(𝑧) √SR, Λ̃ 𝑅 = 𝑢Λ R′ (𝑧) SR, 𝑌 𝑧 = 𝑌(𝑧) Figure 1 . Summary of the asymptotic symmetries and their various relations. LGT stands fornon-abelian large gauge transformations, ST and SR for supertranslations and superrotations. Hor-izontal lines represent the DC relations. Vertical lines represent the symmetry raising map betweenthe first and second family of symmetries. Diagonal arrows represent an increase in the power of r of symmetry parameters, see main text for details. In this paper we presented the double copy construction of a subset of asymptotic symme-tries. Our starting point was the study of residual symmetries of the self-dual sectors of YMand gravity, in the light-cone formulation. We identified two families of symmetries. Forthe first one, we found novel non-pertubative double copy rules in the bulk, leading, in theasymptotic regime, to the mapping of a holomorphic YM symmetry to a holomorphic su-pertranslation. The second family has a more striking structure, in the sense that an exactsubset of diffeomorphism symmetries is obtained as the double copy of a perturbatively-defined and non-local transformation in the YM self-dual sector. At null infinity, we identifythe YM origin of a subset of superrotations with a novel symmetry transformation, whichto our knowledge has not been previously presented in the literature.An important open problem is how to extend our DC symmetry prescription beyond theself-dual context. Promising avenues for extending the Monteiro and O’Connell formulationhave been given in e.g. [87, 88]. Another possibility would be to make use of the techniquesdeveloped in [113], where the full BMS algebra is obtained from the residual symmetries ofthe gravitational theory in the light-cone gauge. It would also be interesting to approachthe problem via the formalism developed in [114].The self-dual theories are known to possess infinitely many symmetries. An infinitetower of such symmetries can be obtained by iterating the ‘symmetry raising map’ reviewedin Eqs. (2.18), (2.19). From this perspective, the two families of symmetries described aboverepresent just the first two rungs of an infinite ladder. Whereas we have demonstrated that– 29 –C relations in the bulk are obeyed at each level, we have not yet explored the realizationof these higher level symmetries at null infinity. It is tempting to speculate that these wouldbe related, in the scattering amplitudes context, to the remarkable simplicity of MaximallyHelicity Violating (MHV) amplitudes. This expectation is justified by existing explanationsof MHV formulae based on the integrability properties of the self-dual equations, see e.g.[100, 115, 116].It is worthwhile to point out that, asymptotically, there is potentially yet anotherhierarchy if one allows for gauge transformations that diverge as some power of r [106, 117–121]. There appears to be an interesting interplay between this hierarchy and the onedescribed in the previous paragraph. Indeed, we were lead to consider O ( r ) divergentgauge symmetries in order to fully implement the raising map between the first two familiesof asymptotic symmetries. It may be that the higher level families are related to gaugetransformations diverging with higher powers of r . We would like to thank Alok Laddha for illuminating discussions. MC acknowledges supportfrom PEDECIBA and from ANII grant FCE-1-2019-1-155865. SN is supported by STFCgrant ST/T000686/1. This research was supported by the Munich Institute for Astro-and Particle Physics (MIAPP) which is funded by the Deutsche Forschungsgemeinschaft(DFG, German Research Foundation) under Germany ’s Excellence Strategy – EXC-2094– 390783311.
A Self-duality conditions
A.1 Yang-Mills field
Given the field strength F µν = ∂ µ A ν − ∂ ν A µ − i [ A µ , A µ ] , (A.1)its dual is defined as ˜ F µν := 12 (cid:15) ρσµν F ρσ . (A.2)We say the field strength is self-dual if ˜ F µν = iF µν . (A.3)In the notation of ( x i , y α ) coordinates, the independent components of the volume form (cid:15) µνρσ are (cid:15) αβij = i Π αβ Ω ij . (A.4)The field strength of the YM field (2.8) is given by F ij = 0 (A.5) F iα = Π jα ∂ i ∂ j Φ (A.6) F αβ = 2Π i [ α ∂ β ] ∂ i Φ − i Π iα Π jβ [ ∂ i Φ , ∂ j Φ] (A.7)– 30 –t is easy to see that ˜ F ij = 0 . Using (2.6) one can verify that ˜ F iα = iF iα . Let us then focuson F αβ . Since it is antisymmetric in the 2d indices ( α, β ) , it can be written as multiple of Π αβ : F αβ = f Π αβ with f = −
12 Ω αβ F αβ (A.8)The spacetime dual is then found to be given by ˜ F αβ = − iF αβ . Thus, the only way tosatisfy the selfdual condition (A.3) is that this component vanishes: F αβ = 0 ⇐⇒ Ω αβ F αβ = 0 . (A.9)From (A.7) and using (2.6) one finds Ω αβ F αβ = − ( (cid:3) Φ + i Π ij [ ∂ i Φ , ∂ j Φ]) (A.10)so that indeed it vanishes provided (2.12) is satisfied.
A.2 Metric field
For spacetime metrics, the dual of the curvature tensor is defined as: ˜ R σµνρ := 12 (cid:15) ηλµν R σηλρ , (A.11)and we say the metric is self-dual if ˜ R σµνρ = iR σµνρ . (A.12)We now study this condition for the spacetime metric g µν = η µν + h µν (A.13)with h µν given by (2.9). Note that we are not requiring h to be a perturbation, and regard(A.13) as a full, non-linear metric. The form of η and h imply the inverse metric is givenexactly by g µν = η µν − h µν . (A.14)One can also check that the determinant of g coincides with that of η . In particular, thevolume form is still given by (A.4). The non-zero Christoffel symbols are: Γ jiα = 12 η βj ∂ i h αβ , Γ iαβ = 12 η γi ( ∂ α h βγ + ∂ β h αγ − ∂ γ h αβ ) + 12 h ij ∂ j h αβ , Γ γαβ = − η γi ∂ i h αβ (A.15)These expressions lead to the following independent components of the curvature tensor: R lijk = 0 (A.16) R αijk = 0 (A.17) R kiαj = − η βk ∂ i ∂ j h αβ (A.18) R δαβγ = 12 η δi ∂ i ∂ α h βγ + 14 η δi η (cid:15)j ∂ j h αγ ∂ i h β(cid:15) − ( α ↔ β ) (A.19) R iαβγ = ∂ β Γ iαγ + Γ δαγ Γ iδβ + Γ jαγ Γ ijβ − ( α ↔ β ) (A.20)– 31 –e now consider the dual curvature tensor (A.12). Here it is important to note that theraising of indices of the volume form must be realized with the full inverse metric (A.14).We will however continue to raise and lower indices on Π and Ω with the flat metric η .It is easy to check that ˜ R lijk = ˜ R αijk = 0 , as well as ˜ R kiαj = iR kiαj . We now discussthe self-dual condition for the components (A.19). Since the expression is antisymmetric inthe 2d α, β indices, we can contract with Ω αβ . After some algebra, one finds Ω αβ ( ˜ R δαβγ − iR δαβγ ) = − i Π mγ η δn ∂ m ∂ n E φ (A.21)with E φ = (cid:3) φ −
12 Π ij Π kl ∂ i ∂ k φ∂ j ∂ l φ (A.22)where E φ = 0 is the SDE equation (2.13). Finally, the self-dual condition for (A.20),contracted with Ω αβ , gives Ω αβ ( ˜ R iαβγ − iR iαβγ ) = i Π ij ∂ j ∂ γ E φ − iη iα Π jγ ∂ j ∂ α E φ + i Π jk Π lγ Π im ∂ j ∂ l E φ ∂ k ∂ m φ. (A.23) B Perturbative transformations at higher orders
In a perturbative setting, the double copy rules (5.36) can be written as: Φ ( i ) → φ ( i ) , − i [ , ] → { , } , Λ → λ, r = deg (Λ) + 1 deg ( λ ) + 1 (B.1)where Φ =Φ (0) + Φ (1) + · · · φ = φ (0) + φ (1) + · · · (B.2) B.1 Second orderB.1.1 First family
The first family of symmetries was shown to double copy non-perturbatively in subsec-tion 5.1. However, as a warm-up, we demonstrate the perturbative construction to secondorder in perturbation theory, before proceeding to the second family. Working to linearorder, the first family of transformations acts on the YM scalar as (see (4.9) and (4.13) ): δ Λ Φ (0) =Ω βj x j ∂ β Λ ,δ Λ Φ (1) = − i [Φ (0) , Λ] (B.3)We will now treat these as seeds and use the e.o.m. at second order in Φ : (cid:3) Φ (2) = − i Π ij [ ∂ i Φ (0) , ∂ j Φ (1) ] (B.4)to derive δ Λ Φ (2) . We have (cid:3) δ Λ Φ (2) = − i Π ij [ ∂ i δ Λ Φ (0) , ∂ j Φ (1) ] − i Π ij [ ∂ i Φ (0) , ∂ j δ Λ Φ (1) ]= − i Π ij [Ω βi ∂ β Λ , ∂ j Φ (1) ] − ij [ ∂ i Φ (0) , [ ∂ j Φ (0) , Λ]]= i (cid:3) [Λ , Φ (1) ] − i [Λ , (cid:3) Φ (1) ] + Π ij [Λ , [ ∂ i Φ (0) , ∂ j Φ (0) ]] (B.5)– 32 –here we used the Jacobi identity for the commutator and the fact that Λ = Λ( y ) . Then,using the eom for Φ (1) , we are left with δ Λ Φ (2) = − i [Φ (1) , Λ] (B.6)Next, we look at the gravity transformation. Our seeds can be obtained by perturbing(5.7): δ λ φ (0) = Ω αi x i ∂ α λδ λ φ (1) = { φ (0) , λ } (B.7)We will now use the e.o.m. at second order in φ , (cid:3) φ (2) = Π ij Π kl ∂ i ∂ k φ (0) ∂ j ∂ l φ (1) (B.8)to derive δ λ φ (2) . We have (cid:3) δ λ φ (2) = Π ij Π kl ∂ i ∂ k δ λ φ (0) ∂ j ∂ l φ (1) + Π ij Π kl ∂ i ∂ k φ (0) ∂ j ∂ l δ λ φ (1) =Π ij Π kl ∂ i (Ω αk ∂ α λ ) ∂ j ∂ l φ (1) + Π ij Π kl ∂ i ∂ k φ (0) ∂ j { ∂ l φ (0) , λ } = − η lα Π ij ∂ i ∂ α λ∂ j ∂ l φ (1) − Π kl { λ, { ∂ k φ (0) , ∂ l φ (0) }} = − (cid:3) { ˜ λ, φ (1) } + { ˜ λ, (cid:3) φ (1) } − Π kl { ˜ λ, { ∂ k φ (0) , ∂ l φ (0) }} (B.9)where we used the Jacobi identity for the Poisson bracket and the fact that λ is linear in x . Then, using the e.o.m. for φ (1) we get δ λ φ (2) = { φ (1) , λ } (B.10)and, comparing with (B.6), we see that it is obtained from δ Λ Φ (2) via the double copy rules(B.1). B.1.2 Second family
Here we extend the construction of the second family of symmetries and their double copyto second order in perturbation theory. Working to linear order, the second family oftransformations acts on the YM scalar as (see (5.22) and (5.25)) δ ˜Λ Φ (0) = 12 Ω αi x i ∂ α ˜Λ ,δ ˜Λ Φ (1) = − i [Φ (0) , ˜Λ] + 2 i (cid:3) − η iα [ ∂ α Φ (0) , ∂ i ˜Λ] (B.11)We will now treat these as seeds and use the e.o.m. at second order in Φ : (cid:3) Φ (2) = − i Π ij [ ∂ i Φ (0) , ∂ j Φ (1) ] (B.12)– 33 –o derive δ ˜Λ Φ (2) . We have (cid:3) δ ˜Λ Φ (2) = − i Π ij [ ∂ i δ ˜Λ Φ (0) , ∂ j Φ (1) ] − i Π ij [ ∂ i Φ (0) , ∂ j δ ˜Λ Φ (1) ]= − i Π ij [Ω αi ∂ α ˜Λ , ∂ j Φ (1) ] − ij [ ∂ i Φ (0) , [ ∂ j Φ (0) , ˜Λ]] − ij [ ∂ i Φ (0) , [Φ (0) , ∂ j ˜Λ]]+ 4Π ij [ ∂ i Φ (0) , ∂ j (cid:3) − η kα [ ∂ α Φ (0) , ∂ k ˜Λ]]=2 iη iα [ ∂ α ˜Λ , ∂ i Φ (1) ] + Π ij [ ˜Λ , [ ∂ i Φ (0) , ∂ j Φ (0) ]] − ij [ ∂ i Φ (0) , [Φ (0) , ∂ j ˜Λ]]+ 4Π ij [ ∂ i Φ (0) , ∂ j (cid:3) − η kα [ ∂ α Φ (0) , ∂ k ˜Λ]]= i (cid:3) [ ˜Λ , Φ (1) ] − iη iα [ ∂ i ˜Λ , ∂ α Φ (1) ] − i [ ˜Λ , (cid:3) Φ (1) ] + Π ij [ ˜Λ , [ ∂ i Φ (0) , ∂ j Φ (0) ]] − ij [ ∂ i Φ (0) , [Φ (0) , ∂ j ˜Λ]] + 4Π ij [ ∂ i Φ (0) , ∂ j (cid:3) − η kα [ ∂ α Φ (0) , ∂ k ˜Λ]] (B.13)where we used the Jacobi identity to get to the third line and (cid:3) ˜Λ = 0 to get to the fourthline. Then, using the eom for Φ (1) , we are left with δ ˜Λ Φ (2) = − i [Φ (1) , ˜Λ] + 2 i (cid:3) − η iα [ ∂ α Φ (1) , ∂ i ˜Λ] − (cid:3) − Π ij [ ∂ i Φ (0) , [Φ (0) , ∂ j ˜Λ]] + 4 (cid:3) − Π ij [ ∂ i Φ (0) , ∂ j (cid:3) − η kα [ ∂ α Φ (0) , ∂ k ˜Λ]] (B.14)Next, we look at the gravity transformation. Our seeds are (5.38): δ ˜ λ φ (0) = 13 Ω αi x i ∂ α ˜ λδ ˜ λ φ (1) = { φ (0) , ˜ λ } − (cid:3) − η iα { ∂ α φ (0) , ∂ i ˜ λ } (B.15)We will now use the e.o.m. at second order in φ , (cid:3) φ (2) = Π ij Π kl ∂ i ∂ k φ (0) ∂ j ∂ l φ (1) (B.16)to derive δ ˜ λ φ (2) . We have (cid:3) δ ˜ λ φ (2) = Π ij Π kl ∂ i ∂ k δ ˜ λ φ (0) ∂ j ∂ l φ (1) + Π ij Π kl ∂ i ∂ k φ (0) ∂ j ∂ l δ ˜ λ φ (1) =Π ij Π kl ∂ i (cid:16) Ω αk ∂ α ˜ λ (cid:17) ∂ j ∂ l φ (1) + Π ij Π kl ∂ i ∂ k φ (0) ∂ j { ∂ l φ (0) , ˜ λ } + Π ij Π kl ∂ i ∂ k φ (0) ∂ j { φ (0) , ∂ l ˜ λ } − Π ij Π kl ∂ i ∂ k φ (0) ∂ j ∂ l (cid:3) − η iα { ∂ α φ (0) , ∂ i ˜ λ } = − η lα Π ij ∂ i ∂ α ˜ λ∂ j ∂ l φ (1) − Π kl { ˜ λ, { ∂ k φ (0) , ∂ l φ (0) }} + Π ij Π kl ∂ i ∂ k φ (0) ∂ j { φ (0) , ∂ l ˜ λ } − Π ij Π kl ∂ i ∂ k φ (0) ∂ j ∂ l (cid:3) − η iα { ∂ α φ (0) , ∂ i ˜ λ } = − (cid:3) { ˜ λ, φ (1) } + η lα { ∂ l ˜ λ, ∂ α φ (1) } + { ˜ λ, (cid:3) φ (1) } − Π kl { ˜ λ, { ∂ k φ (0) , ∂ l φ (0) }} + Π ij Π kl ∂ i ∂ k φ (0) ∂ j { φ (0) , ∂ l ˜ λ } − Π ij Π kl ∂ i ∂ k φ (0) ∂ j ∂ l (cid:3) − η iα { ∂ α φ (0) , ∂ i ˜ λ } (B.17)where we used the Jacobi identity for the Poisson bracket to get to the third line and (cid:3) ˜ λ = 0 to get to the fourth line. Then, using the e.o.m. for φ (1) , we get δ ˜ λ φ (2) = { φ (1) , ˜ λ } − (cid:3) − η iα { ∂ α φ (1) , ∂ i ˜ λ } + (cid:3) − Π ij { ∂ i φ (0) , { φ (0) , ∂ j ˜ λ }} − (cid:3) − Π ij { ∂ i φ (0) , ∂ j (cid:3) − η iα { ∂ α φ (0) , ∂ i ˜ λ }} (B.18)and comparing with the YM transformation (B.14) we see that it double copies correctlyunder the rules (B.1). – 34 – .2 Recursive construction at arbitrary orders We will demonstrate recursively that the double copy rules (B.1) work at all orders inperturbation theory. The proof applies simultaneously to the first and second family ofsymmetries (and, indeed, to the infinite set of families constructed in subsection 5.5). Startby assuming we have shown that the transformation rules for self-dual YM up to some order ( n − : δ Λ Φ (0) , δ Λ Φ (1) , · · · , δ Λ Φ ( n − (B.19)double copy correctly, under the rules (B.1), into the respective gravity transformations: δ λ φ (0) , δ λ φ (1) , · · · , δ λ φ ( n − (B.20)We now treat the transformations δ ˜Λ Φ (0) , · · · , δ ˜Λ Φ ( n − as seeds, and make use of the YMself-dual equation at order n : (cid:3) Φ ( n ) = − i Π ij (cid:88) m + p = n − [ ∂ i Φ ( m ) , ∂ j Φ ( p ) ] (B.21)to derive δ ˜Λ Φ ( n ) = − i Π ij (cid:3) − (cid:88) m + p = n − [ ∂ i δ ˜Λ Φ ( m ) , ∂ j Φ ( p ) ] − i Π ij (cid:3) − (cid:88) m + p = n − [ ∂ i Φ ( m ) , ∂ j δ ˜Λ Φ ( p ) ] (B.22)Then using the d.c. rules (B.1) and the assumption that the double copy works up to order ( n − we get δ ˜ λ φ ( n ) = Π ij (cid:3) − (cid:88) m + p = n − { ∂ i δ ˜ λ φ ( m ) , ∂ j φ ( p ) } + Π ij (cid:3) − (cid:88) m + p = n − { ∂ i φ ( m ) , ∂ j δ ˜ λ φ ( p ) } (B.23)Separately, we can directly derive the gravity transformation by taking δ ˜ λ φ (0) , · · · , δ ˜ λ φ ( n − as seeds, and using the gravity self-dual equation at order n: (cid:3) φ ( n ) = Π ij (cid:88) m + p = n − { ∂ i φ ( m ) , ∂ j φ ( p ) } (B.24)we get (cid:3) δ ˜ λ φ ( n ) = Π ij (cid:88) m + p = n − { ∂ i δ ˜ λ φ ( m ) , ∂ j φ ( p ) } + Π ij (cid:88) m + p = n − { ∂ i φ ( m ) , ∂ j δ ˜ λ φ ( p ) } (B.25)so indeed δ ˜ λ φ ( n ) = Π ij (cid:3) − (cid:88) m + p = n − { ∂ i δ ˜ λ φ ( m ) , ∂ j φ ( p ) } + Π ij (cid:3) − (cid:88) m + p = n − { ∂ i φ ( m ) , ∂ j δ ˜ λ φ ( p ) } (B.26)as needed. As seen in this section, the fact that the DC rules continue to hold at all orderscan be traced back to the fact that the e.o.m. themselves exhibit a DC structure.– 35 – Comparison with convolution DC
At zeroth order in the fields there exists an alternative DC construction for gravity and YMsymmetries, based on a convolution dictionary [42–44]. In this appendix we will show thatour DC is consistent with the convolution one where they overlap. Since the latter relateslinearized YM and gravity gauge symmetries, we will compare it with our first family atzeroth order in the fields.For simplicity, we present the comparison in the case of (2 , signature, where Z and ¯ Z are treated as independent real variables. This will allow us to perform Fouriertransforms along Z and ¯ Z independently.In momentum space, the convolution DC relates a linearized gravity diffeomorphism ˜ ξ µ ( p ) to a linearized YM gauge parameter ˜Λ( p ) by ˜ ξ µ ( p ) = Tr [ ˜Λ( p ) ˜ ϕ − ( p ) ˜ A µ ( p )] (C.1)where ϕ is the biadjoint “spectactor” scalar field. Since A i = 0 , we immediately see that ξ i = 0 , (C.2)consistent with (3.20). Next we note that since Λ = Λ( y ) is independent of x i = ( U, ¯ Z ) itfollows that ˜Λ( p ) = δ ( p U ) δ ( p ¯ Z ) ˜Λ( p V , p Z ) . (C.3)Substituting in (C.1), we find that ˜ ξ µ is proportional to δ ( p U ) δ ( p ¯ Z ) . Hence, when trans-formed back to position space, it can only depend on the y variables ξ α = ξ α ( y ) , (C.4)consistent with (3.20). References [1] Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson, and R. Roiban, “The DualityBetween Color and Kinematics and its Applications”, arXiv:1909.01358 [hep-th] .[2] L. Borsten, “Gravity as the square of gauge theory: a review”,
Riv. Nuovo Cim. (2020)no. 3, 97–186.[3] L. Borsten and M. J. Duff, “Gravity as the square of Yang–Mills?”, Phys. Scripta (2015)108012, arXiv:1602.08267 [hep-th] .[4] Z. Bern, J. Carrasco, and H. Johansson, “New Relations for Gauge-Theory Amplitudes”, Phys.Rev.
D78 (2008) 085011, arXiv:0805.3993 [hep-ph] .[5] Z. Bern, J. J. M. Carrasco, and H. Johansson, “Perturbative Quantum Gravity as a DoubleCopy of Gauge Theory”,
Phys.Rev.Lett. (2010) 061602, arXiv:1004.0476 [hep-th] . See e.g. [122] for a recent discussion. Note ˜Λ is the Fourier transform of the gauge parameter Λ . It should not be confused with the parameterof the second family of symmetries, which in the main text is also denoted by ˜Λ . For simplicity we haveassumed that the two gauge sectors are identical. – 36 –
6] Z. Bern, T. Dennen, Y.-t. Huang, and M. Kiermaier, “Gravity as the Square of GaugeTheory”,
Phys.Rev.
D82 (2010) 065003, arXiv:1004.0693 [hep-th] .[7] Z. Bern, J. J. M. Carrasco, W.-M. Chen, H. Johansson, R. Roiban, and M. Zeng, “Five-loopfour-point integrand of N = 8 supergravity as a generalized double copy”, Phys. Rev.
D96 (2017) no. 12, 126012, arXiv:1708.06807 [hep-th] .[8] Z. Bern, S. Davies, T. Dennen, and Y.-t. Huang, “Absence of Three-Loop Four-PointDivergences in N=4 Supergravity”,
Phys. Rev. Lett. (2012) 201301, arXiv:1202.3423[hep-th] .[9] Z. Bern, J. J. Carrasco, W.-M. Chen, A. Edison, H. Johansson, J. Parra-Martinez,R. Roiban, and M. Zeng, “Ultraviolet Properties of N = 8 Supergravity at Five Loops”,
Phys. Rev. D (2018) no. 8, 086021, arXiv:1804.09311 [hep-th] .[10] N. Bjerrum-Bohr, P. H. Damgaard, R. Monteiro, and D. O’Connell, “Algebras forAmplitudes”, JHEP (2012) 061, arXiv:1203.0944 [hep-th] .[11] R. Monteiro and D. O’Connell, “The Kinematic Algebras from the Scattering Equations”,
JHEP (2014) 110, arXiv:1311.1151 [hep-th] .[12] A. Luna, R. Monteiro, I. Nicholson, A. Ochirov, D. O’Connell, N. Westerberg, and C. D.White, “Perturbative spacetimes from Yang-Mills theory”,
JHEP (2017) 069, arXiv:1611.07508 [hep-th] .[13] W. D. Goldberger and A. K. Ridgway, “Radiation and the classical double copy for colorcharges”, Phys. Rev.
D95 (2017) no. 12, 125010, arXiv:1611.03493 [hep-th] .[14] G. L. Cardoso, S. Nagy, and S. Nampuri, “A double copy for N = 2 supergravity: alinearised tale told on-shell”, JHEP (2016) 127, arXiv:1609.05022 [hep-th] .[15] G. Cardoso, S. Nagy, and S. Nampuri, “Multi-centered N = 2 BPS black holes: a doublecopy description”,
JHEP (2017) 037, arXiv:1611.04409 [hep-th] .[16] A. Luna, S. Nagy, and C. White, “The convolutional double copy: a case study with apoint”, JHEP (2020) 062, arXiv:2004.11254 [hep-th] .[17] W. D. Goldberger, J. Li, and S. G. Prabhu, “Spinning particles, axion radiation, and theclassical double copy”, Phys. Rev. D (2018) no. 10, 105018, arXiv:1712.09250[hep-th] .[18] R. Monteiro, D. O’Connell, D. P. Veiga, and M. Sergola, “Classical Solutions and theirDouble Copy in Split Signature”, arXiv:2012.11190 [hep-th] .[19] R. Monteiro, D. O’Connell, and C. D. White, “Black holes and the double copy”, JHEP (2014) 056, arXiv:1410.0239 [hep-th] .[20] A. K. Ridgway and M. B. Wise, “Static Spherically Symmetric Kerr-Schild Metrics andImplications for the Classical Double Copy”,
Phys. Rev.
D94 (2016) no. 4, 044023, arXiv:1512.02243 [hep-th] .[21] C. D. White, “Exact solutions for the biadjoint scalar field”,
Phys. Lett.
B763 (2016)365–369, arXiv:1606.04724 [hep-th] .[22] D. S. Berman, E. Chacón, A. Luna, and C. D. White, “The self-dual classical double copy,and the Eguchi-Hanson instanton”,
JHEP (2019) 107, arXiv:1809.04063 [hep-th] .[23] I. Bah, R. Dempsey, and P. Weck, “Kerr-Schild Double Copy and Complex Worldlines”, JHEP (2020) 180, arXiv:1910.04197 [hep-th] . – 37 –
24] L. Alfonsi, C. D. White, and S. Wikeley, “Topology and Wilson lines: global aspects of thedouble copy”,
JHEP (2020) 091, arXiv:2004.07181 [hep-th] .[25] K. Kim, K. Lee, R. Monteiro, I. Nicholson, and D. Peinador Veiga, “The Classical DoubleCopy of a Point Charge”, JHEP (2020) 046, arXiv:1912.02177 [hep-th] .[26] N. Bahjat-Abbas, R. Stark-Muchão, and C. D. White, “Monopoles, shockwaves and theclassical double copy”, JHEP (2020) 102, arXiv:2001.09918 [hep-th] .[27] E. Lescano and J. A. Rodríguez, “ N = 1 supersymmetric Double Field Theory and thegeneralized Kerr-Schild ansatz”, JHEP (2020) 148, arXiv:2002.07751 [hep-th] .[28] P.-J. De Smet and C. D. White, “Extended solutions for the biadjoint scalar field”, Phys.Lett. B (2017) 163–167, arXiv:1708.01103 [hep-th] .[29] K. Lee, “Kerr-Schild Double Field Theory and Classical Double Copy”,
JHEP (2018)027, arXiv:1807.08443 [hep-th] .[30] C. Keeler, T. Manton, and N. Monga, “From Navier-Stokes to Maxwell via Einstein”, JHEP (2020) 147, arXiv:2005.04242 [hep-th] .[31] R. Alawadhi, D. S. Berman, and B. Spence, “Weyl doubling”, JHEP (2020) 127, arXiv:2007.03264 [hep-th] .[32] H. Godazgar, M. Godazgar, R. Monteiro, D. Peinador Veiga, and C. Pope, “The WeylDouble Copy for Gravitational Waves”, arXiv:2010.02925 [hep-th] .[33] C. D. White, “A Twistorial Foundation for the Classical Double Copy”, arXiv:2012.02479[hep-th] .[34] D. S. Berman, K. Kim, and K. Lee, “The Classical Double Copy for M-theory from aKerr-Schild Ansatz for Exceptional Field Theory”, arXiv:2010.08255 [hep-th] .[35] J. J. M. Carrasco, C. R. Mafra, and O. Schlotterer, “Semi-abelian Z-theory: NLSM+phi3from the open string”, JHEP (2017) 135, arXiv:1612.06446 [hep-th] .[36] J. J. M. Carrasco, C. R. Mafra, and O. Schlotterer, “Abelian Z-theory: NLSM amplitudesand α ’-corrections from the open string”, JHEP (2017) 093, arXiv:1608.02569[hep-th] .[37] C. Cheung and C.-H. Shen, “Symmetry for Flavor-Kinematics Duality from an Action”, Phys. Rev. Lett. (2017) no. 12, 121601, arXiv:1612.00868 [hep-th] .[38] L. Borsten, “ D = 6 , N = (2 , and N = (4 , theories”, Phys. Rev.
D97 (2018) 066014, arXiv:1708.02573 [hep-th] .[39] L. Borsten, I. Jubb, V. Makwana, and S. Nagy, “Gauge × gauge on spheres”,
JHEP (2020) 096, arXiv:1911.12324 [hep-th] .[40] S. Nagy, “Chiral Squaring”, JHEP (2016) 142, arXiv:1412.4750 [hep-th] .[41] F. Cachazo, S. He, and E. Y. Yuan, “Scattering Equations and Matrices: From Einstein ToYang-Mills, DBI and NLSM”, JHEP (2015) 149, arXiv:1412.3479 [hep-th] .[42] A. Anastasiou, L. Borsten, M. J. Duff, L. J. Hughes, and S. Nagy, “Yang-Mills origin ofgravitational symmetries”, Phys.Rev.Lett. (2014) no. 23, 231606, arXiv:1408.4434[hep-th] .[43] A. Anastasiou, L. Borsten, M. J. Duff, S. Nagy, and M. Zoccali, “Gravity as Gauge Theory – 38 – quared: A Ghost Story”,
Phys. Rev. Lett. (2018) no. 21, 211601, arXiv:1807.02486[hep-th] .[44] G. Lopes Cardoso, G. Inverso, S. Nagy, and S. Nampuri, “Comments on the double copyconstruction for gravitational theories”, in . 2018. arXiv:1803.07670 [hep-th] . http://inspirehep.net/record/1663475/files/1803.07670.pdf .[45] R. Alawadhi, D. S. Berman, B. Spence, and D. Peinador Veiga, “S-duality and the doublecopy”, JHEP (2020) 059, arXiv:1911.06797 [hep-th] .[46] A. Banerjee, E. Colgáin, J. Rosabal, and H. Yavartanoo, “Ehlers as EM duality in thedouble copy”, Phys. Rev. D (2020) 126017, arXiv:1912.02597 [hep-th] .[47] L. Borsten, I. Jubb, V. Makwana, and S. Nagy to appear .[48] A. Anastasiou, L. Borsten, M. J. Duff, A. Marrani, S. Nagy, and M. Zoccali, “Are allsupergravity theories Yang-Mills squared?”,
Nucl. Phys.
B934 (2018) 606–633, arXiv:1707.03234 [hep-th] .[49] L. Borsten, M. J. Duff, L. J. Hughes, and S. Nagy, “A magic square from Yang-Millssquared”,
Phys.Rev.Lett. (2014) 131601, arXiv:1301.4176 [hep-th] .[50] A. Anastasiou, L. Borsten, M. J. Duff, L. J. Hughes, and S. Nagy, “A magic pyramid ofsupergravities”,
JHEP (2014) 178, arXiv:1312.6523 [hep-th] .[51] A. Anastasiou, L. Borsten, L. J. Hughes, and S. Nagy, “Global symmetries of Yang-Millssquared in various dimensions”,
JHEP (2016) 1601, arXiv:1502.05359 [hep-th] .[52] A. Anastasiou, L. Borsten, M. J. Duff, A. Marrani, S. Nagy, and M. Zoccali, “The MileHigh Magic Pyramid”, in
Nonassociative Mathematics and its Applications ,P. Vojtechovsky, ed., vol. 721 of
Contemporary Mathematics , pp. 1–27. Amer. Math. Soc.,Providence, RI, 2019. arXiv:1711.08476 [hep-th] .[53] M. Tolotti and S. Weinzierl, “Construction of an effective Yang-Mills Lagrangian withmanifest BCJ duality”,
JHEP (2013) 111, arXiv:1306.2975 [hep-th] .[54] J. Plefka, J. Steinhoff, and W. Wormsbecher, “Effective action of dilaton gravity as theclassical double copy of Yang-Mills theory”, Phys. Rev.
D99 (2019) no. 2, 024021, arXiv:1807.09859 [hep-th] .[55] J. Plefka, C. Shi, and T. Wang, “Double copy of massive scalar QCD”,
Phys. Rev. D (2020) no. 6, 066004, arXiv:1911.06785 [hep-th] .[56] L. Borsten, B. Jurčo, H. Kim, T. Macrelli, C. Saemann, and M. Wolf, “BRST-LagrangianDouble Copy of Yang-Mills Theory”, arXiv:2007.13803 [hep-th] .[57] L. Borsten and S. Nagy, “The pure BRST Einstein-Hilbert Lagrangian from thedouble-copy to cubic order”,
JHEP (2020) 093, arXiv:2004.14945 [hep-th] .[58] P. Ferrero and D. Francia, “On the Lagrangian formulation of the double copy to cubicorder”, arXiv:2012.00713 [hep-th] .[59] R. Penrose, “Asymptotic properties of fields and space-times”, Phys. Rev. Lett. (1963)66–68.[60] H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, “Gravitational waves in general – 39 – elativity. 7. Waves from axisymmetric isolated systems”, Proc. Roy. Soc. Lond. A (1962) 21–52.[61] R. K. Sachs, “Gravitational waves in general relativity. 8. Waves in asymptotically flatspace-times”,
Proc. Roy. Soc. Lond. A (1962) 103–126.[62] A. Ashtekar, “Radiative Degrees of Freedom of the Gravitational Field in Exact GeneralRelativity”,
J. Math. Phys. (1981) 2885–2895.[63] A. Ashtekar and M. Streubel, “Symplectic Geometry of Radiative Modes and ConservedQuantities at Null Infinity”, Proc. Roy. Soc. Lond. A (1981) 585–607.[64] A. Ashtekar,
ASYMPTOTIC QUANTIZATION: BASED ON 1984 NAPLES LECTURES .1987.[65] G. Barnich and C. Troessaert, “Symmetries of asymptotically flat 4 dimensional spacetimesat null infinity revisited”,
Phys. Rev. Lett. (2010) 111103, arXiv:0909.2617 [gr-qc] .[66] A. Strominger, “On BMS Invariance of Gravitational Scattering”,
JHEP (2014) 152, arXiv:1312.2229 [hep-th] .[67] A. Strominger, “Asymptotic Symmetries of Yang-Mills Theory”, JHEP (2014) 151, arXiv:1308.0589 [hep-th] .[68] T. He, V. Lysov, P. Mitra, and A. Strominger, “BMS supertranslations and Weinberg’s softgraviton theorem”, JHEP (2015) 151, arXiv:1401.7026 [hep-th] .[69] A. Strominger, “Lectures on the Infrared Structure of Gravity and Gauge Theory”, arXiv:1703.05448 [hep-th] .[70] A. Ashtekar, M. Campiglia, and A. Laddha, “Null infinity, the BMS group and infraredissues”, Gen. Rel. Grav. (2018) no. 11, 140–163, arXiv:1808.07093 [gr-qc] .[71] S. Pasterski, “Implications of Superrotations”, Phys. Rept. (2019) 1–35, arXiv:1905.10052 [hep-th] .[72] N. E. J. Bjerrum-Bohr, P. H. Damgaard, T. Sondergaard, and P. Vanhove, “The MomentumKernel of Gauge and Gravity Theories”,
JHEP (2011) 001, arXiv:1010.3933 [hep-th] .[73] S. Oxburgh and C. D. White, “BCJ duality and the double copy in the soft limit”, JHEP (2013) 127, arXiv:1210.1110 [hep-th] .[74] S. G. Naculich, H. Nastase, and H. J. Schnitzer, “All-loop infrared-divergent behavior ofmost-subleading-color gauge-theory amplitudes”, JHEP (2013) 114, arXiv:1301.2234[hep-th] .[75] S. He, Y.-t. Huang, and C. Wen, “Loop Corrections to Soft Theorems in Gauge Theoriesand Gravity”, JHEP (2014) 115, arXiv:1405.1410 [hep-th] .[76] C. White, “Diagrammatic insights into next-to-soft corrections”, Phys. Lett. B (2014)216–222, arXiv:1406.7184 [hep-th] .[77] A. Sabio Vera and M. A. Vazquez-Mozo, “The Double Copy Structure of Soft Gravitons”,
JHEP (2015) 070, arXiv:1412.3699 [hep-th] .[78] A. Luna, S. Melville, S. Naculich, and C. White, “Next-to-soft corrections to high energyscattering in QCD and gravity”, JHEP (2017) 052, arXiv:1611.02172 [hep-th] .[79] A. P.V. and A. Manu, “Classical double copy from Color Kinematics duality: A proof in thesoft limit”, Phys. Rev. D (2020) no. 4, 046014, arXiv:1907.10021 [hep-th] . – 40 –
80] E. Casali and A. Puhm, “A Double Copy for Celestial Amplitudes”, arXiv:2007.15027[hep-th] .[81] E. Casali and A. Sharma, “Celestial double copy from the worldsheet”, arXiv:2011.10052[hep-th] .[82] N. Kalyanapuram, “Soft S-Matrices, Defects and the Double Copy on the Celestial Sphere”, arXiv:2011.11412 [hep-th] .[83] N. Kalyanapuram, “Gauge and Gravity Amplitudes on the Celestial Sphere”, arXiv:2012.04579 [hep-th] .[84] Y.-T. Huang, U. Kol, and D. O’Connell, “Double copy of electric-magnetic duality”,
Phys.Rev. D (2020) no. 4, 046005, arXiv:1911.06318 [hep-th] .[85] W. T. Emond, Y.-T. Huang, U. Kol, N. Moynihan, and D. O’Connell, “Amplitudes fromCoulomb to Kerr-Taub-NUT”, arXiv:2010.07861 [hep-th] .[86] R. Monteiro and D. O’Connell, “The Kinematic Algebra From the Self-Dual Sector”,
JHEP (2011) 007, arXiv:1105.2565 [hep-th] .[87] C.-H. Fu and K. Krasnov, “Colour-Kinematics duality and the Drinfeld double of the Liealgebra of diffeomorphisms”,
JHEP (2017) 075, arXiv:1603.02033 [hep-th] .[88] G. Chen, H. Johansson, F. Teng, and T. Wang, “On the kinematic algebra for BCJnumerators beyond the MHV sector”, JHEP (2019) 055, arXiv:1906.10683 [hep-th] .[89] G. Elor, K. Farnsworth, M. L. Graesser, and G. Herczeg, “The Newman-Penrose Map andthe Classical Double Copy”, arXiv:2006.08630 [hep-th] .[90] E. Chacón, H. García-Compeán, A. Luna, R. Monteiro, and C. D. White, “New heavenlydouble copies”, arXiv:2008.09603 [hep-th] .[91] R. H. Boels, R. S. Isermann, R. Monteiro, and D. O’Connell, “Colour-Kinematics Dualityfor One-Loop Rational Amplitudes”, JHEP (2013) 107, arXiv:1301.4165 [hep-th] .[92] J. F. Plebanski, “Some solutions of complex Einstein equations”, J. Math. Phys. (1975)2395–2402.[93] M. Prasad, A. Sinha, and L.-L. Wang, “Nonlocal Continuity Equations for Selfdual SU( N ){Yang-Mills} Fields”, Phys. Lett. B (1979) 237–238.[94] L. Dolan, “Kac-moody Algebras and Exact Solvability in Hadronic Physics”, Phys. Rept. (1984) 1.[95] A. Parkes, “A Cubic action for selfdual Yang-Mills”,
Phys. Lett. B (1992) 265–270, arXiv:hep-th/9203074 .[96] G. Chalmers and W. Siegel, “The Selfdual sector of QCD amplitudes”,
Phys. Rev. D (1996) 7628–7633, arXiv:hep-th/9606061 .[97] D. Cangemi, “Selfduality and maximally helicity violating QCD amplitudes”, Int. J. Mod.Phys. A (1997) 1215–1226, arXiv:hep-th/9610021 .[98] A. Popov, M. Bordemann, and H. Romer, “Symmetries, currents and conservation laws ofselfdual gravity”, Phys. Lett. B (1996) 63–74, arXiv:hep-th/9606077 .[99] A. Popov, “Selfdual Yang-Mills: Symmetries and moduli space”,
Rev. Math. Phys. (1999) 1091–1149, arXiv:hep-th/9803183 . – 41 – Prog. Theor. Phys. Suppl. (1996) 1–8.[101] Q.-H. Park, “Selfdual Gravity as a Large N Limit of the Two-dimensional Nonlinear σ Model”,
Phys. Lett. B (1990) 287–290.[102] V. Husain, “Selfdual gravity as a two-dimensional theory and conservation laws”,
Class.Quant. Grav. (1994) 927–938, arXiv:gr-qc/9310003 .[103] V. Husain, “The Affine symmetry of selfdual gravity”, J. Math. Phys. (1995) 6897–6906, arXiv:hep-th/9410072 .[104] D. Kapec and P. Mitra, “A d -Dimensional Stress Tensor for Mink d +2 Gravity”,
JHEP (2018) 186, arXiv:1711.04371 [hep-th] .[105] T. He, P. Mitra, and A. Strominger, “2D Kac-Moody Symmetry of 4D Yang-Mills Theory”, JHEP (2016) 137, arXiv:1503.02663 [hep-th] .[106] M. Campiglia and A. Laddha, “Subleading soft photons and large gauge transformations”, JHEP (2016) 012, arXiv:1605.09677 [hep-th] .[107] A. Laddha and P. Mitra, “Asymptotic Symmetries and Subleading Soft Photon Theorem inEffective Field Theories”, JHEP (2018) 132, arXiv:1709.03850 [hep-th] .[108] T. He and P. Mitra, “Asymptotic symmetries in (d + 2)-dimensional gauge theories”, JHEP (2019) 277, arXiv:1903.03607 [hep-th] .[109] M. Campiglia and J. Peraza to appear .[110] M. Campiglia and A. Laddha, “New symmetries for the Gravitational S-matrix”, JHEP (2015) 076, arXiv:1502.02318 [hep-th] .[111] G. Compère, A. Fiorucci, and R. Ruzziconi, “Superboost transitions, refraction memoryand super-Lorentz charge algebra”, JHEP (2018) 200, arXiv:1810.00377 [hep-th] .[Erratum: JHEP 04, 172 (2020)].[112] M. Campiglia and J. Peraza, “Generalized BMS charge algebra”, Phys. Rev. D (2020)no. 10, 104039, arXiv:2002.06691 [gr-qc] .[113] S. Ananth, L. Brink, and S. Majumdar, “BMS algebra from residual gauge invariance inlight-cone gravity”, arXiv:2101.00019 [hep-th] .[114] A. Ashtekar and M. Varadarajan, “Gravitational dynamics: A novel shift in theHamiltonian paradigm”,
Universe (2021) 13, arXiv:2012.12094 [gr-qc] .[115] A. A. Rosly and K. G. Selivanov, “On amplitudes in selfdual sector of Yang-Mills theory”, Phys. Lett. B (1997) 135–140, arXiv:hep-th/9611101 .[116] L. J. Mason and D. Skinner, “Gravity, Twistors and the MHV Formalism”,
Commun. Math.Phys. (2010) 827–862, arXiv:0808.3907 [hep-th] .[117] M. Campiglia and A. Laddha, “Sub-subleading soft gravitons and large diffeomorphisms”,
JHEP (2017) 036, arXiv:1608.00685 [gr-qc] .[118] M. Campiglia and A. Laddha, “Asymptotic charges in massless QED revisited: A view fromSpatial Infinity”, JHEP (2019) 207, arXiv:1810.04619 [hep-th] .[119] A. Seraj, “Multipole charge conservation and implications on electromagnetic radiation”, JHEP (2017) 080, arXiv:1610.02870 [hep-th] . – 42 – JHEP (2018) 054, arXiv:1711.08806 [hep-th] .[121] G. Compère, “Infinite towers of supertranslation and superrotation memories”, Phys. Rev.Lett. (2019) no. 2, 021101, arXiv:1904.00280 [gr-qc] .[122] A. Atanasov, A. Ball, W. Melton, A.-M. Raclariu, and A. Strominger, “ (2 , Scattering andthe Celestial Torus”, arXiv:2101.09591 [hep-th] ..