A double copy for asymptotic symmetries in the self-dual sector

We give a double copy construction for the symmetries of the self-dual sectors of Yang-Mills (YM) and gravity, in the light-cone formulation. We find an infinite set of double copy constructible symmetries. We focus on two families which correspond to the residual diffeomorphisms on the gravitational side. For the first one, we find novel non-perturbative double copy rules in the bulk. The second family has a more striking structure, as a non-perturbative gravitational symmetry is obtained from a perturbatively defined 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 large gauge YM symmetries are double copied to holomorphic supertranslations. We also identify the single copy of superrotations with certain non-gauge YM transformations that to our knowledge have not been previously presented in the literature.

A Self-duality conditions
Having a DC prescription for the local symmetries of the fields (i.e. the gauge transformations on the Yang-Mills (YM) side and diffeomorphisms on the gravity side) is of course a proof of the robustness of the DC dictionaries for the fields, but the possible applications of such a prescription go beyond this. For example, it was shown in [41,45,59], working in the BRST formalism, that one can exploit it to obtain a gauge-mapping algorithm, which gives 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 so far only been understood at linear level -going beyond this, either perturbatively (or nonperturbatively in certain special cases), could prove very powerful in the DC construction of classical solutions and/or Lagrangians.
A natural arena in which to discuss gravitational and gauge symmetries is the asymptotic boundary of spacetime. In the case of (asymptotically) flat spacetimes, this is given by Penrose's null infinity [61]. It has long been known that gauge and gravitational fields exhibit a rich structure at this boundary, with one of the earliest examples being the Bondi-Metzner-Sachs (BMS) asymptotic symmetry group of gravity [62,63]. Whereas many of the implications of these symmetries were known since the 80's thanks to Ashtekar and others [64][65][66], the subject experienced a veritable renaissance in recent years. The trigger was twofold. On the one hand, Barnich and Troessaert proposed an extension of BMS to include so-called superrotations [67]. On the other hand, Strominger discovered a link between future and past null infinity symmetries [68] which manifests as soft theorems in scattering amplitudes [69,70]. The ramifications of these developments are still ongoing, see [71][72][73] 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 the use of DC for soft theorems [74][75][76][77][78][79][80][81], the memory effect [82], and in the context of celestial amplitudes [83][84][85][86][87]. Additionally, a duality transformation relating the Schwarzschild and Taub-NUT solutions was shown to arise, via a DC construction, from an electric-magnetic duality relation [88,89]. Interestingly, this duality can be interpreted, on the gravity side, as the complexification of a particular supertranslation transformation. 2 These promising results are calling for an interpretation from the perspective of asymptotic symmetries.
In this paper we present some progress in both directions outlined above. We will be restricting to the self-dual sector of gravity and YM theory, where the colour-kinematics duality was made manifest at the level of the algebra by Monteiro and O'Connell [90] (see also [11,39,[91][92][93][94][95] for related work and extensions), leading to a simple DC prescription for the construction of perturbative solutions. In light-cone gauge, the self-dual graviton and YM field have a natural description in terms of scalar fields φ and Φ, respectively [96][97][98][99][100][101][102][103][104], and the self-duality conditions can be expressed as equations for these scalars, referred to as self-dual Einstein (SDE) and self-dual YM (SDYM) equations respectively. We will study the symmetries of these equations, with focus on those arising from the residual gauge symmetries of their respective parent theories. After laying down the 'bulk' DC symmetry map (see below for details), we will move on to study symmetries at null infinity. On the gravitational side we will recover the subset of BMS that preserves the self-duality condition at null infinity. On the YM side, the single copy version of these symmetries will turn out to include 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 [97,98,102,103,[105][106][107][108][109], out of which the symmetries described above turn out to represent only two rungs. We will see that the DC applies to all levels of this ladder. The asymptotic analysis of the higher level symmetries could in principle be worked out along the same lines 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 anticipated above, 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 nonabelian gauge symmetry on the YM side. We discover that a repackaging is required in order to 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 diffeomorphism 2 It appears this duality could be the asymptotic version of the Ehlers transform discussed in [47,48].
It would be interesting to make this connection explicitly. We thank Eoin O Colgain for bringing this possibility to our attention.
parameter of gravity, but to a "Hamiltonian" of the diffeomorphism vector field w.r.t. the Poisson bracket inherent in the self-dual theory. Having performed this repackaging, the DC rules become remarkably simple. The second family is also exact on the gravity side, but perturbative and non-local on the YM side, presenting an apparent puzzle. However, we find that the repackaging described above recasts the gravity transformation as a perturbative non-local series. This is then obtainable from its YM counterpart via the same simple rules as the first family, verifying the robustness of our construction. It is interesting to note that this second family is a subset of diffeomorphisms on the gravity side, but its single copy is not a subset of YM gauge transformations. In the asymptotic limit, the first family maps holomorphic YM large gauge symmetries to holomorphic supertranslations, whereas the second one identifies certain (non-gauge and non-local) SDYM transformations to holomorphic superrotations. 3 The paper is structured as follows: In section 2 we introduce our conventions and present a covariant description of self-dual fields in light-cone gauge. The corresponding residual gauge symmetries are discussed in section 3. In section 4 we describe a perturbative approach to general symmetries of the SDYM and SDE equations. In section 5 we construct the DC map between the symmetries of these equations. Starting with a non-perturbative map for the first family of symmetries, we eventually extend its applicability to an infinite number of families. In section 6 we introduce coordinates adapted to null infinity and review gauge and gravity asymptotic symmetries. In section 7 we take the null-infinity limit of the first two families of symmetries described earlier and establish the DC map for asymptotic symmetries. We summarize our findings and discuss possible future avenues in section 8. Some of the technical discussions are given in appendices.

Self-dual fields
Consider Minkowski spacetime written in light-cone coordinates (U, V, Z,Z), related to Cartesian coordinates X µ by: We will split these light-cone coordinates in the following two sets of 2d coordinates: The spacetime metric then takes the form The description of self-dual fields and their symmetries will involve the "area element" of these 2d spaces, described by We 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 these two tensors as (partial) inverses of each other: The self-dual YM and metric fields will be described in terms of scalar fields Φ and φ according to (2.9) In the notation of (2.2), the non-zero components take the form with the scalar fields satisfying 4 where ≡ 2η iα ∂ i ∂ α is the wave operator. We will refer to (2.12) and (2.13) as the self-dual YM (SDYM) and self-dual Einstein (SDE) equation respectively. We are using conventions where the coupling constants are absorbed in the field, so that the field strength is , the spacetime metric g µν = η µν + h µν , and there is no explicit coupling constant in the field equations.
In [90], the notion of a kinematic algebra in the self-dual sector was defined as a Poisson bracket algebra. In our conventions, the Poisson bracket (PB) is defined as: One can now recover (2.13) from(2.12) via the replacement rules [90]: Note that our conventions are slightly different from [90], and so are the color to kinematics replacement rules.

Symmetries of the self-dual field equations
The self-dual YM and Einstein equations have long been known to possess infinitely many symmetries [97,98,102,103,[105][106][107]. These can be constructed recursively as follows.
Let δΦ and δφ be symmetries of the SDYM and SDE equations respectively: One can then obtain new symmetriesδΦ andδφ, defined implicitly by the condition 5 In each case, one can verify the consistency condition ∂ [i ∂ j]δ = 0 is satisfied. By direct computation one can then check thatδΦ andδφ satisfy (2.16) and (2.17) provided the respective 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 identifying the 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): as this does not affect the YM field (2.10). Similarly, φ is defined modulo which preserves (2.11). 6 Note that (2.20) and (2.21) are not the standard gauge symmetries of the YM and metric fields (to be discussed in the next section). Rather, they are additional redundancies that arise in the description given by (2.10) and (2.11).

Residual gauge symmetries
In this section we describe the gauge transformations on the YM and gravity fields that preserve the form (2.10), (2.11). It is interesting to note that these fields satisfy the following gauge conditions Thus, the symmetries that we will be discussing can be thought of as a subset of the residual gauge symmetries of conditions (3.1).

YM field
We start with the YM field. Given a gauge parameter Λ, the corresponding gauge transformation is If we want to preserve the condition A i = 0, we must have 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 By substituting (2.10) in (3.2), the LHS of (3.4) can be written as 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 We emphasize that this is an exact symmetry and that we are not doing a perturbative expansion at this stage. Eq. (3.6) then tells us that δ 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 Using (2.6), this equation can equivalently be written as: Since the RHS is independent of x i , we can integrate it directly to obtain 7 Summarizing: The gauge symmetries compatible with (2.10) are parametrized by gauge parameters Λ(y), and act on Φ according to Additionally, one could also consider symmetries of the scalar Φ which do not arise as YM gauge transformations, but are intrinsic symmetries of the equation (2.12). As we will show in section 5, one of these families of transformations can be obtained as the single copy of a gravitational diffeomorphism symmetry.

Metric field
In the gravitational case, the spacetime metric is given by 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 derivative on 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: where ξ µ ≡ η µν ξ ν , with ξ ν the infinitesimal diffeomorphism vector field. Setting µ = i, α we obtain and A constant a ij represents a rigid rotation in the x i plane and is hence part of the global Poincare symmetries. Since we are interested in local symmetries, we set a ij = 0. Integrating (3.17) we are led to with b α (y) an integration "constant". To continue, we need to (i) determine if the second term in (3.14) is compatible with h iµ = 0 and (2.11) and (ii) determine if 2∂ (α ξ β) can be written in a form compatible with (2.11). To simplify the analysis, we will discuss separately the cases c i = 0 and b α = 0. The general case can be obtained as a linear combination of these two simple cases.

First family of symmetries
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 8 Let us now discuss the second term in (3.14) From (3.21) and h iµ = 0 we find that the second term in (3.22) vanishes. The first term clearly respects h iµ = 0. Furthermore, using Eq. (2.11) it can be written as 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 It remains to see whether the first term in (3.14) is compatible with (2.11). Thus, we ask if there is a δ Using (2.6) and (3.20) this equation can be written as: Since b β is independent of the x i variables, the equation can be directly integrated, leading to 9 δ (0) The family of vector fields (3.21) then acts on the scalar field according to 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 SDE equation for φ (2.13) is a sufficient but not necessary condition to produce a self-dual metric. The actual self-duality condition takes the form of a 2nd order differential operator acting on the SDE equation (2.13), see Eqs. (A.21) and (A.23). Of course the self-duality condition is invariant under the transformation (3.29), since it arises from an infinitesimal diffeomorphism. In order to have a symmetry of the SDE equation (2.13), one can check that it is sufficient to restrict b α (y) to be a total derivative, We will refer to the resulting symmetry transformation as δ b , with: We will later see that (3.30) does not actually represent any restriction on the asymptotic vector fields at null infinity.

Second family of symmetries
Setting b α = 0 we have We now start by demanding the first term in (3.14) to be compatible with (2.11), as this will impose restrictions on c i : Multiplying the equation with Ω's as before, the last equality is equivalent to For this to be integrable, we need the tensor c ijk to be symmetric under the exchange of any 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 for some function c(y). The resulting expression can then be integrated, leading to We finally discuss the second term in (3.14) for the vector field ξ µ : 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 ν from which it follows that L ξ h iµ = 0 and To summarize, the second family of vector fields acts on φ as: ξ φ given by (3.36) and (3.40) respectively. We emphasize that both families are exact, non-perturbative symmetries of the field equations (2.13). As we will discuss in subsection 5.4, it turns out that the second family can be obtained from the first one by the symmetry raising map given in Eq. (2.19).

Perturbative approach to symmetries
In this section we present a perturbative approach to symmetries of the SDYM and SDE equations. This will offer an alternative perspective on the symmetries described before. The perturbative approach will be useful when constructing the DC symmetry map in section 5. We will present expressions up to first order in the fields. The all-order expansions are given recursively in subsection B.2.

YM field
The field equations (2.12) may be cast as a tower of field equations associated to a perturbative expansion 10 such that We now look at symmetries δΦ of the field equations (2.12) in the above perturbative scheme. That is, we look for such that · · · (4.8) We may think of these equations as determining δΦ in terms of a "seed" δΦ (0) : Given δΦ (0) satisfying (4.6), δΦ (1) is defined implicitly by (4.7), and similarly for the higher order terms. Generically this would lead to a δΦ (1) that depends non-locally on Φ (0) due to the need to invert the wave operator. The case of gauge symmetries discussed previously is an exception. From (3.11) we see that in this case the seed is given by which satisfies (4.6): If we now substitute (4.9) in the RHS of (4.7) we find where in the first equality we used Π ij Ω β j = 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 (4.13) consistent with the result from the previous section. One could continue and obtain the higher order terms, which in this case will resum to the exact symmetry (3.12). See Appendix B for further discussion on higher order terms.

First family of symmetries
For the first family of vector fields described earlier, Eq. (3.31) gives the following candidate for the "seed" δφ (0) : which satisfies δφ (0) = 0. If we now substitute (4.21) in the RHS of (4.19) we obtain 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 compatible with the expected result from (3.31).

Second family of symmetries
For the second family of symmetries, we take δφ (0) to be given by (3.36) It can be easily checked that the above satisfies (4.18). Substituting in the RHS of (4.19) one finds Since it is not immediately obvious how to "pull out" a d'Alambertian from such an expression, let us take guidance from the previous section. From (3.40) and (3.37) we expect to have Indeed, one can check by direct computation that the d'Alambertian of (4.27) coincides with (4.26), upon using (4.15).

Color to kinematics map for symmetries
According to [90], the color to kinematics map for the self-dual sector consists in replacing YM matrix commutators by certain Poisson brackets on the x i variables, as reviewed in This will be our starting point in understanding the double copy for symmetries. In the following section, we will make the color-kinematics duality manifest at the level of the symmetry transformations. A key ingredient in this will be to recast the gravitational symmetry in a form that is analogous to the YM one. We will achieve this by expressing the gravitational symmetry parameter in terms of a Hamiltonian λ (w.r.t. the Poisson bracket) associated to the diffeomorphism vector field. This will allow us to supplement the rules (5.1) with where Λ is the YM symmetry parameter and λ is the Hamiltonian describing the gravitational symmetry.

First family of symmetries
We will start by comparing the gauge symmetries found in the YM case (3.12) with the first family of symmetries found in the gravitational case (3.31): Let us first focus on the terms that are linear in the scalar fields. According to (5.1), the YM term linear in Φ should be mapped to for some λ. In order to reproduce the linear term in δ b φ we take which can be thought of as the "Hamiltonian" of the vector field ξ i = η iα ∂ α b with respect to the Poisson bracket 1 2 {, }. We now notice that the φ-independent piece of δ b φ can be written in terms of λ, leading to an expression that has the same form as the Φ-independent piece of δ Λ Φ. Thus, we can express the total δ b φ in terms of λ as: Comparing the expression above with (5.3), we find that the tranformation rules map into each other under with an apparent mismatch of 1 2 for the inhomogeneous terms. However we note that this is resolved by expressing the transformation rules on the YM and gravitational fields themselves, as: If we insist on working at the level of the scalar fields, then the replacement rules need to be augmented by the inclusion of a multiplicative factor for the inhomogeneous term: 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 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

Second family of symmetries
Having understood how the first family of gravitational symmetries can be obtained from the YM ones, we now focus on the second family of gravitational symmetries: where We first note that the vector field ξ i can again be written in terms of a 'Hamiltonian' Using (5.16), we can rewrite (5.14) as This brings δ c φ into a form that resembles the one obtained before for δ b φ in (5.7), with the exception of the final term. To understand this extra term we will use the perturbative description of symmetries given in section 4, as described below.
We would like to find a symmetry of the SDYM theory such that, upon the appropriate color to kinematics replacement rules, one recovers the second family of gravitational symmetries (5.17). Guided by (5.13), we find that the natural choice for the Φ-independent piece of this YM symmetry is for some function B(y). This generalizes the Φ-independent part of the symmetry (5.3) to the case where Λ is linear in x . Restricting to the inhomogeneous terms in the transformations for the YM field and the graviton, we find that they are related by the same rules (5.8) as the first family. 11 Alternatively, one can work directly at the level of the scalar fields by taking into acount the multiplicative factor (5.10), which in this case is r = 2/3, 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 1 ) symmetry transformation generated by (5.18). We will then discuss how the color to kinematic rules applied to this term reproduce the two O(φ 1 ) terms in (5.17).

O(Φ 1 ) symmetry
Following the perturbative method of section 4, we would like to solve for δΛΦ (1) given the 'seed' 11 We emphasize here that δΛ is not defined directly on Aµ, but through its action on the scalar Φ. In particular, using (2.8), we find δ Λ Φ = 0, due the fact that Π µ i vanishes identically. Note that δ (0) Λ Ai = ∂iΛ, so it cannot be interpreted as a gauge transformation, even at the lowest order.
We proceed in a similar fashion as in the analysis between Eqs. (4.9) and (4.12) where we verified the perturbative method against the residual YM gauge symmetry. Noting that ∂ j δΛΦ (0) = Ω α j ∂ αΛ , the RHS of (5.21) can be written where in the last line we used that Φ (0) = 0 = Λ . Unlike for the residual YM symmetry, there is now a reminder term when one 'pulls out' the wave operator. We then conclude that If we apply the color to kinematic map to expression (5.25) one finds The first term corresponds to the second term in (5.17). For the second term we have where we used the definition of the Poisson bracket, the expression forλ as given in (5.16), and the fact that φ (0) vanishes. Expression (5.32) reproduces, in the perturbative setting, the last term in (5.17). This establishes δΛΦ (1) as the inverse DC map of δ c φ (1) . In fact, as shown in Appendix B the DC map can be extended to arbitrary order in the perturbative expansion.

Summary
Let us summarize our findings. By associating certain "Hamiltonians" to the infinitesimal residual diffeomorphisms of the SDE equations, we managed to obtain simple DC rules mapping SDYM to SDE symmetries. In order to compactly write these rules for the scalar fields, let us introduce the notation The first family of symmetries take the form where 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 the multiplicative coefficient r in the inhomogeneous term (see discussion around (5.10) for details): The second family can be written as The two are related by exactly the same DC rules (5.36) as for the first family. Although here we have presented the argument up to linear order in the field, the map can in fact be extended to arbitrary order, see Appendix B. Let us emphasize that the original gravitational symmetry, as given in (5.17), is non-perturbative. In (5.38), the higher order terms arise as a result of parametrizing this symmetry in terms of the Hamiltonianλ. This requires the use of the field equations, which introduce higher order terms. The second family displays two interesting features. Firstly, it maps a perturbative expression on the YM side to a non-perturbative one on the gravity side, as described above. Secondly, while on the gravity side the symmetry is a diffeomorphism, on the YM side we have a non-gauge symmetry of the field equation (2.12).

Relation with the symmetry raising map
In subsection 2.1 we described the existence of symmetry raising maps on YM and gravity self-dual theories that, given a symmetry δ, produce a new symmetryδ according to Eqs. (2.18), (2.19).
Since, as described in section 4 the symmetries we are studying are fully determined by their inhomogenous piece, it is natural to restrict these equations to the inhomogenous parts of the symmetries: It is now straightforward to check that, if we take δΦ (0) = δ Λ Φ (0) and δφ (0) = δ λ φ (0) , the resultingδ (0) are given by δ Since the symmetries are fully determined by their inhomogenous piece, the above shows that the map δ →δ takes the full δ Λ Φ into the full δΛΦ (and similarly for δ λ φ). We can summarize the above structure by the diagram where the horizontal arrows represent the DC map between each family of symmetries, and the vertical arrows the δ →δ map on each theory. Said in different words, we have verified that the symmetry raising map commutes with the DC symmetry map. We will use this result in the next section to obtain the DC symmetry map at null infinity for the second family of symmetries.

Double copy for an infinite family of symmetries
We found that the second family of symmetries arises from applying the raising map on the first family. It is clear that one can continue this process indefinitely and construct an infinite tower of symmetries on both sides. A natural question is whether the DC will continue to hold at each level. In other words, we want to check whether the diagram (5.43) extends indefinitely: where δ n denotes the symmetry of the n-th family. The first two lines of the diagram correspond to the original diagram (5.43). It is easy to show by recursion that the DC holds at every level. We have explicitly demonstrated this for δ 1 and δ 2 in subsection 5.1, subsection 5.2, and Appendix B. Assuming that the DC holds at level n − 1, it immediately follows from the definition of the n-th level symmetry (2.18) and (2.19): that δ n φ is obtained from δ n Φ via the DC rules (5.36). The focus in this paper is on the first two families, since they are the ones corresponding to asymptotic diffeomorphisms on the gravitational 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: Notice that these are just the redundancies described in (2.20). It is then easy to solve (5.45) at O(Φ 0 ). One finds As we did for the second family in (5.41), we can introduce a gauge-like parameter at n-th level, such that We refer to Λ n as a gauge-like parameter since Note however that δ (0) n A i = 0 = ∂ i Λ n (see footnote 11 for n = 2). Let us now discuss the gravitational case. Recall that in this case there are two kind of redundancies (2.21). In order to be consistent with the diagram (5.44), one is lead to start the recursion at n = −1 with, δ −1 φ = 2b(y).
where we expressed the result in terms of the first family Hamiltonian λ. The n = −1, 0 levels describe the two redundancies in (2.21). As we did for the second family (5.42), we can define a 'Hamiltonian' parameter at n-th level, such that As in the first two families, with this definition of λ n one has Comment: It may naively appear that there exists a different DC prescription, one that maps the n-th YM family to the (n − 1)-th gravity one. Upon relabeling the gravity subscripts, this would yield identical zeroth order expressions on both sides. However this prescription has various drawbacks. Firstly, it would map YM gauge symmetries In particular, the large YM gauge 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 gravitational ones. As we shall see, this would imply an additional mismatch in the number of would-be DC related asymptotic symmetries. Finally, our original DC prescription (5.43) is shown in Appendix C to be consistent with the convolution DC [44][45][46] where they overlap, whereas the alternative DC would not be.
6 Large gauge symmetries at null infinity

Asymptotic fields at null infinity
In this section we switch to coordinates adapted to null infinity. Given the light cone coordinates (U, V, Z,Z) introduced in (2.1), a natural choice for Bondi-type coordinates (r, u, z,z) is given by: Future null infinity I is reached by taking r → ∞ while keeping the rest of the variables fixed. It is thus parametrized by (u, z,z). Note that we will be working in a flat conformal frame for which the (degenerate) metric at null infinity is 2dzdz. 12 Massless fields are captured at null infinity by certain r → ∞ leading components. For a 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 1/r. ϕ I (u, z,z) is to be regarded as a field defined intrinsically at null infinity, which contains all the radiative data of ϕ. We will also refer to it as the 'free data' at I, since it may be regarded as the analogue of the Cauchy data for the 'final time' hypersurface I, see e.g. [65]. Similarly, gauge and metric fields at null infinity are captured by h zz (r, u, z,z) r→∞ = rC zz (u, z,z) + · · · (6.5) and respective z ↔z expressions. If we now consider expressions (2.10) and (2.11) in Bondi coordinates (6.1) we find from which one easily finds how the gauge and gravity free data is encoded in the scalar fields free data (see (6.3)) A z = ∂ u Φ I , Az = 0 (6.8) We thus recover the expected result that self-dual fields are associated with data at null infinity that has vanishing antiholomorphic components.
In the reminder of the section we will study how the 'bulk' symmetries described in the previous sections manifest at null infinity, with the aim of obtaining DC rules for asymptotic symmetries. Before getting started, however, it will be useful to briefly review the standard description of asymptotic symmetries.

Review of large gauge symmetries at null infinity
In the general (not necessarily self-dual) case, one considers gauge and diffeomorphism symmetries that are non-trivial at null infinity. In the YM case, these are generated by gauge parameters that asymptote to a non-trivial function on the celestial sphere [69,111], where the dots denote terms that go to zero as r → ∞. The induced symmetry transformation on the null infinity radiative data is then In the gravitational case one finds two classes of vector fields: The Bondi-Metzner-Sachs supertranslations [62,63], parametrized by functions f (z,z), ξ f = f (z,z)∂ u + · · · (6.12) and the Barnich-Troessaert superrotations [67], parametrized by holomorphic 2d vector fields Y z (z)∂ z + Yz(z)∂z, where in both cases the dots denote terms that vanish when r → ∞. The infinitesimal diffeomorphisms generated by these vector fields induce the following action on the null infinity radiative data: where ∂ · Y = ∂ z Y z + ∂zYz. The transformation properties for Czz can be obtained by doing the replacement z →z in the expressions above. Whereas the above represent the simplest large gauge symmetries, it is possible to find additional ones by carefully relaxing the standard fall-off conditions (6.4), (6.5). In particular, one can make sense of gauge symmetries with O(r) gauge parameter [112][113][114][115] and non-holomorphic superrotations [116][117][118]. Interestingly, the O(r) gauge symmetries will turn out to play a key role in identifying the single copy version of superrotations, see section 7.
Coming back to the self-dual case, we can ask what is the subset of the previously discussed symmetries that preserve the conditions Az = 0, Czz = 0 (6.16) at null infinity. On the YM side, this condition is preserved provided Λ 0 is holomorphic, On the gravity side, the supertranslation function has to be of the form f = f (z) +zg(z). Below we will recover these restricted symmetries as the null infinity limit of the residual gauge symmetries discussed in section 3.

Residual gauge symmetries at null infinity
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 yields a finite and non-trivial action at null infinity.
Recall that the symmetry parameters can be constructed from functions which depend only 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 functional forms, 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 welldefined action at null infinity, we need the RHS of (3.12) to be O(r −1 ). For the homogeneous term, this can only be achieved if Λ = O(r 0 ). We are thus led to choose Λ as Λ(r, z) = Λ 0 (z) (6.20) Let us now discuss the inhomogeneous term in (3.12). To do so we note that the differential operator S (5.33) in Bondi coordinates takes the form When (6.21) acts on Λ (6.20), only the last term survives, yielding a O(r −1 ) result as expected. From here one can read off the action on the radiative data to be The corresponding action on A z can be obtained from the relation (6.8), where one recovers (6.11) for holomorphic Λ 0 . Consider now the first family of gravitational symmetries, (3.29), with b α = ∂ α b as in Eq. (3.31). Writing the homogenous term in Bondi coordinates, one easily finds that, in order for this term to be O(r −1 ) 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 −1 ) due to subtle cancellations. One finally arrives at for the induced action on the free data. The corresponding action on C zz can be obtained from 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 total derivative. One finds that, at null infinity, the only finite and non-trivial transformation that can be obtained is still of the form (6.24). Thus, the condition b α = ∂ α b does not imply a restriction as far as the asymptotic symmetries are concerned. Let us finally discuss the second family of gravitational symmetries introduced in subsubsection 3.2.2. In this case one finds there are two possible choices of c(y), either O(r) or O(r 2 ), that yield a finite and non-trivial null infinity limit: c(r, z) = rg(z) or c(r, z) = −r 2 Y (z)/2 (6.25) For the first choice, one obtains which, upon translating to C zz , yields a supertranslation (6.14) with f (z,z) =zg(z). For the second choice of c in (6.25) one finds 27) which, when translated to C zz via (6.9), reproduces the superrotation action (6.15) with vector field (6.19).

Color to kinematic symmetry map at null infinity
Having understood how large gauge symmetries emerge from the residual symmetries discussed in section 3, we now study how the DC for 'bulk' symmetries described in section 5 translates into a DC for asymptotic symmetries at null infinity. This will be achieved by studying the DC symmetry relations to leading order in a 1/r expansion. Whereas for the first 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 wave operator. We will circumvent this difficulty by making use of the symmetry raising map discussed in subsection 5.4. To get started, let us express the operators that appear in the description of bulk symmetries in Bondi coordinates, paying special attention to their 1/r expansion. Given a 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). 13 The PB in Bondi coordinates takes the form We will regard {, } I as a PB defined intrinsically at null infinity (with weight −1). Consider now the differential operator that features in the inhomogeneous part of the symmetries. Its form in Bondi coordinates was given in (6.21). When acting on a function of the form r n F n (u, z,z), it can be written as: S(r n F n ) = r n S 0 (F n ) + r n−1 S −1 (F n ) (7.4) where S 0 = −zu∂ u + nz −z 2 ∂z (7.5) S 0 and S −1 will be regarded as differential operators defined intrinsically at null infinity. Note that S 0 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 asymptotic symmetries.

First family
The YM symmetry (5.34) at null infinity was described in Eq. (6.22) for Λ = Λ 0 (z). Using the notation introduced above, we write it as (7.7) 13 Notice that this n has nothing to do with the n of subsection 5.5, which labeled the symmetry families.
For the gravity symmetry (5.35), we need to compute the 'Hamiltonian' λ (5.6) for the choice of b(y) given in (6.23). This gives 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 −1 factor in the PB (7.1). Naively, it appears that the inhomogenous term could have O(r) and O(r 0 ) terms, thus spoiling the required O(r −1 ) fall-offs. However, such coefficients turn out to be zero. In the end one finds a well defined action on the radiative data, given by We emphasize that (7.11) is just a rewriting of (6.24), but one that makes the DC structure manifest. Comparing (7.7) to (7.11), we find that the DC rules at null infinity have the same structure as those in the bulk, as expected. The new ingredient is that, in the homogeneous term, the weight of the symmetry parameter increases by 1 (this is to compensate for the factor of 1/r in the PB (7.1)).

Second family
For the second family of symmetries, the DC relation in the bulk was made manifest by expressing the symmetries perturbatively, Eqs. (5.37) and (5.38). Unfortunately, such expressions are difficult to expand in 1/r due to the appearance of the inverse wave operator. However, the discussion given in subsection 5.4 offers an alternative route to obtaining the DC symmetry map: the second family of symmetries for each theory can be obtained from the first family via the "symmetry raising map", see the vertical arrows of (5.43). Thus, rather than going directly from δΛΦ to δλφ via the perturbative bulk DC map (bottom horizontal arrow of (5.43)), one can try to get there via the first family of symmetries. This is the strategy we will follow in this section to describe the second family of asymptotic symmetries and their DC relation. Let us start with a general discussion that applies to both theories; for concreteness we only display the equations corresponding to the YM case. Since the first family of symmetries can be written as a sum of O(Φ 0 ) and O(Φ 1 ) terms, δΦ = δ (0) Φ + δ (1) Φ, the raising map (2.18) produces a symmetryδ that is at most O(Φ 2 ):δΦ =δ (0) Φ+δ (1) Φ+δ (2) Φ. Collecting powers of Φ in (2.18), one finds each term should satisfy (7.14) Eq. (7.12) was already discussed in subsection 5.4, where it was found that Similarly, if we start with a gravitational first family symmetry, so that δ (0) φ = 1 2 S(λ) (with λ = 2S(b)) one finds The numerical factors in (7.15) and (7.16) arise as particular cases of the general coefficients explained in subsection 5.5. We will later expand these expressions in 1/r in order to obtaiñ To obtainδ (1) Φ I andδ (2) Φ I , we will solve Eqs. (7.13) and (7.14) asymptotically as follows. First, we note that the x i derivatives expressed in Bondi coordinates, are not independent in the r → ∞ limit. Thus, to leading order in 1/r it is enough to consider only one component of Eqs. (7.13) and (7.14). Choosing i = U and writing them in Bondi coordinates one obtains In order to have a well defined action at null infinity, we need the RHS of these equations to be O(r −1 ). The resulting O(r −1 ) factors will then give expressions for ∂ uδ (1) Φ I and ∂ uδ (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 caseδ (1) φ I ,δ (2) φ I are to be determined from the O(r −1 ) part of the equations In order to perform the required 1/r expansion, we need to specify the asymptotic behavior of the symmetry parameters Λ and λ. We now describe two possibilities that yield a well defined action ofδ at null infinity.
We start by considering symmetry parameters as those discussed in subsection 7.1, with λ as in Eqs. (7.8), (7.9), (7.10). The correspondingΛ andλ defined in Eqs. (7.15) and (7.16) take the formΛ for the inhomogeneous part of the symmetry transformations. The linear part of the symmetries 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 −1 ) are those coming from thez∂ u term hitting the commutator/PB. This allows for a trivial integration in u, leading toδ Finally, one finds that the RHS of Eqs. (7.20) and (7.22) decay faster than O(r −1 ) and henceδ (2) The gravitational symmetryδφ I can be seen to coincide with a supertranslation associated tof (z,z) = −zf (z) (Eq. (6.26) with g(z) → f (z)). Thus, the symmetry raising map takes a supertranslation with f = f (z) into a supertranslation withf = −zf (z).

7.2.2δ from Λ = O(r), λ = O(r 2 )
We now consider symmetry parameters Λ and λ of one order higher in r. Although these generate transformations that violate the r → ∞ fall-offs of the YM and gravity fields, the correspondingδ symmetries will turn out to preserve such fall-offs. On the YM side, we consider a O(r) gauge transformation where Λ 0 = u∂ z ∂zΛ 1 (z) = 0 (as obtained from the condition Λ = 0). We explicitly keep this vanishing term in (7.33) since it will get mapped into a non-trivial term under the DC. The correspondingΛ is: Substituting this expansion in (7.15) one finds 15 To obtainδ (1) Φ I , we consider Eq. (7.19) for δ (0) Interestingly, the potentially divergent terms cancel out, and one arrives at a finite equation for the null infinity free data Φ I : Similarly, from Eq. (7.20) one obtains On the gravitational side, we now consider a symmetry parameter b that is O( The corresponding λ = 2S(b) takes the form S(λ) = 2λ is then given by where we used that S 0 (λ 1 ) = 0. To obtainδ (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 −1 ) term, yieldingδ This precisely reproduces the inhomogenous part of the superrotation action (6.27).
To obtainδ (1) φ I , we consider Eq. (7.21) for δ (0) λ φ = − 1 2 {λ, φ}, with λ, S(λ) as given by (7.40), (7.42). Once again the divergent terms cancel out, and one obtains a well defined equation at null infinity: This is the DC version of (7.37), with the rule described in subsection 7.1 for the increase in 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 −2 ) and hencẽ It may be puzzling that the non-zero quadratic term (7.38) trivializes after the DC. From the perspective of null infinity, this happens because the DC takes the weight 0 commutator [, ] into the weight −1 Poisson bracket {, } I . Since in both cases one should have overall weight minus one, one would need to map Λ 1 into a weight 3 λ 3 to get a non-trivial DC result. However there is no λ 3 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 the homogenous part of the superrotation action (6.27). Thus, the symmetry raising map takes a "divergent supertranslation" defined by (7.39) into a superrotation.

Summary
In Figure 1 we summarize the different symmetries at null infinity, together with the functional form of their parameters. The diagram may be thought of as an augmented version of (5.43), with a new layer due to the two possibilities that arise in the second family of symmetries. 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-gauge YM 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 symmetries. The replacement rules are as in the bulk, with the following clarification: the commutator is replaced by the PB at null infinity (7.2), which has weight −1. To compensate 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 arrows describe the increase in the power of r associated to the parameter of the second family of symmetries, see e.g. Eq. (6.25) for the gravitational case. To better understand these arrows, one can imagine completing the diagram by raising the power of r in the parameters of the first family of YM and gravity symmetries, thus completing the cube. We can then recover the bottom two symmetries by applying the symmetry raising map to these new vertices, see subsubsection 7.2.2 for details. We chose not to display them in the diagram since they do not preserve the r → ∞ field fall-offs. 16 LGT, Λ = Λ( )

Discussion
In this paper we presented the double copy construction of a subset of asymptotic symmetries. Our starting point was the study of residual symmetries of the self-dual sectors of YM and gravity, in the light-cone formulation. We identified two families of symmetries. For the first one, we found novel non-pertubative double copy rules in the bulk, leading, in the asymptotic regime, to the mapping of a holomorphic YM symmetry to a holomorphic supertranslation. The second family has a more striking structure, in the sense that an exact subset of diffeomorphism symmetries is obtained as the double copy of a perturbativelydefined and non-local transformation in the YM self-dual sector. At null infinity, we identify the YM origin of a subset of superrotations with a novel symmetry transformation, which to our knowledge has not been previously presented in the literature.
An important open problem is how to extend our DC symmetry prescription beyond the self-dual context. Promising avenues for extending the Monteiro and O'Connell formulation have been given in e.g. [91,92]. Another possibility would be to make use of the techniques developed in [119], where the full BMS algebra is obtained from the residual symmetries of the gravitational theory in the light-cone gauge. It would also be interesting to approach the problem via the formalism developed in [120]. Alternatively, one could try to extend our analysis to other families of solutions for which the doubly copy is manifest. An obvious first step would be to look at solutions obtained from the Kerr-Schild ansatz [19], which is structurally similar to the self-dual metric ansatz used in this paper.
The self-dual theories are known to possess infinitely many symmetries. An infinite tower of such symmetries can be obtained by iterating the 'symmetry raising map' reviewed in Eqs. (2.18), (2.19). From this perspective, the two families of symmetries described above represent just the first two rungs of an infinite ladder. Whereas we have demonstrated that DC relations in the bulk are obeyed at each level, we have not yet explored the realization of these higher level symmetries at null infinity. It is tempting to speculate that these would be related, in the scattering amplitudes context, to the remarkable simplicity of Maximally Helicity Violating (MHV) amplitudes. This expectation is justified by existing explanations of MHV formulae based on the integrability properties of the self-dual equations, see e.g. [104,121,122].
It is worthwhile to point out that, asymptotically, there is potentially yet another hierarchy if one allows for gauge transformations that diverge as some power of r [112,[123][124][125][126][127]]. There appears to be an interesting interplay between this hierarchy and the one described in the previous paragraph. Indeed, we were lead to consider O(r) divergent gauge symmetries in order to fully implement the raising map between the first two families of asymptotic symmetries. It may be that the higher level families are related to gauge transformations diverging with higher powers of r.

Aknowledgements
We would like to thank Alok Laddha for illuminating discussions. MC acknowledges support from PEDECIBA and from ANII grant FCE-1-2019-1-155865. SN is supported by STFC grant ST/T000686/1. This research was supported by the Munich Institute for Astroand Particle Physics (MIAPP) which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany 's Excellence Strategy -EXC-2094 -390783311.

A.1 Yang-Mills field
Given the field strength its dual is defined asF We say the field strength is self-dual ifF In the notation of (x i , y α ) coordinates, the independent components of the volume form µνρσ are The field strength of the YM field (2.8) is given by It is easy to see thatF ij = 0. Using (2.6) one can verify thatF iα = iF iα . Let us then focus on F αβ . Since it is antisymmetric in the 2d indices (α, β), it can be written as multiple of Π αβ : The spacetime dual is then found to be given byF αβ = −iF αβ . Thus, the only way to satisfy the selfdual condition (A.3) is that this component vanishes: From (A.7) and using (2.6) one finds 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: We now study this condition for the spacetime metric 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 given exactly by One can also check that the determinant of g coincides with that of η. In particular, the volume form is still given by (A.4). The non-zero Christoffel symbols are: These expressions lead to the following independent components of the curvature tensor: We now consider the dual curvature tensor (A.12). Here it is important to note that the raising 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 thatR l ijk =R α ijk = 0, as well asR k iαj = iR k iαj . We now discuss the self-dual condition for the components (A. 19). Since the expression is antisymmetric in the 2d α, β indices, we can contract with Ω αβ . After some algebra, one finds where E φ = 0 is the SDE equation (2.13). Finally, the self-dual condition for (A.20), contracted with Ω αβ , gives

B Perturbative transformations at higher orders
In a perturbative setting, the double copy rules (5.36) can be written as: The first family of symmetries was shown to double copy non-perturbatively in subsection 5.1. However, as a warm-up, we demonstrate the perturbative construction to second order in perturbation theory, before proceeding to the second family. Working to linear order, the first family of transformations acts on the YM scalar as (see (4.9) and (4.13) ): We will now treat these as seeds and use the e.o.m. at second order in Φ: to derive δ Λ Φ (2) . We have where 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 Next, we look at the gravity transformation. Our seeds can be obtained by perturbing (5.7): We will now use the e.o.m. at second order in φ, to derive δ λ φ (2) . We have 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) = 1 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 copy to second order in perturbation theory. Working to linear order, the second family of transformations acts on the YM scalar as (see (5.22) and (5.25)) (B.11) We will now treat these as seeds and use the e.o.m. at second order in Φ: to derive δΛΦ (2) . We have (B.13) where we used the Jacobi identity to get to the third line and Λ = 0 to get to the fourth line. Then, using the eom for Φ (1) , we are left with (B.14) Next, we look at the gravity transformation. Our seeds are (5.38): We will now use the e.o.m. at second order in φ, to derive δλφ (2) . We have where we used the Jacobi identity for the Poisson bracket to get to the third line and λ = 0 to get to the fourth line. Then, using the e.o.m. for φ (1) , we get and comparing with the YM transformation (B.14) we see that it double copies correctly under the rules (B.1).

B.2 Recursive construction at arbitrary orders
We will demonstrate recursively that the double copy rules (B.1) work at all orders in perturbation theory. The proof applies simultaneously to the first and second family of symmetries (and, indeed, to the infinite set of families constructed in subsection 5.5). Start by assuming we have shown that the transformation rules for self-dual YM up to some order (n − 1): 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−1) (B.20) We now treat the transformations δΛΦ (0) , · · · , δΛΦ (n−1) as seeds, and make use of the YM self-dual equation at order n: to derive (B.22) Then using the d.c. rules (B.1) and the assumption that the double copy works up to order (n − 1) we get (B.23) Separately, we can directly derive the gravity transformation by taking δλφ (0) , · · · , δλφ (n−1) as seeds, and using the gravity self-dual equation at order n: we get (B.26) as needed. As seen in this section, the fact that the DC rules continue to hold at all orders can be traced back to the fact that the e.o.m. themselves exhibit a DC structure.

C Comparison with convolution DC
At zeroth order in the fields there exists an alternative DC construction for gravity and YM symmetries, based on a convolution dictionary [44][45][46]. In this appendix we will show that our DC is consistent with the convolution one where they overlap. Since the latter relates linearized YM and gravity gauge symmetries, we will compare it with our first family at zeroth order in the fields.
For simplicity, we present the comparison in the case of (2, 2) signature, 17 where Z andZ are treated as independent real variables. This will allow us to perform Fourier transforms along Z andZ independently.
In momentum space, the convolution DC relates a linearized gravity diffeomorphism ξ µ (p) to a linearized YM gauge parameterΛ(p) by 18 where ϕ is the biadjoint "spectactor" scalar field. Since A i = 0, we immediately see that consistent with (3.20). Next we note that since Λ = Λ(y) is independent of x i = (U,Z) it follows thatΛ (p) = δ(p U )δ(pZ)Λ(p V , p Z ).