Renormalization of spin-one asymptotic charges in AdS D

: We study the renormalized action and the renormalized presymplectic potential for Maxwell fields on Anti de Sitter backgrounds of any dimensions. We then use these results to explicitly derive finite boundary charges for angle-dependent asymptotic symmetries. We consider both Poincaré and Bondi coordinates, the former allowing us to control the systematics for arbitrary D , the latter being better suited for a smooth flat limit.


Introduction
The asymptotic symmetries of four-dimensional asymptotically flat gravity and of Yang-Mills theories on Minkowski backgrounds have been related to soft theorems and memory effects, see e.g.[1] for a review.The relation with soft theorems was also extended to gauge theories describing free particles of arbitrary spin and p-forms [2][3][4][5][6], thus presenting itself as a general feature of any gauge theory.This connection motivated a reassessment of the asymptotic symmetries of both gravity and gauge theories in spacetimes of dimension greater than four, since proper generalizations of the Bondi-Metzner-Sachs (BMS) group [7][8][9] and of its counterparts in gauge theories [10][11][12][13][14] seem instrumental in establishing the link with soft theorems, that in their turn take the same form in any spacetime dimension. 1his observation appears however in tension with the option of defining boundary conditions in higher-dimensional spacetimes which, on the one hand, are compatible with radiative solutions while, on the other hand, lead to finite-dimensional asymptotic symmetry groups [16][17][18][19].
While relaxing the boundary conditions so as to allow for a larger class of residual gauge transformations is always possible, selecting bona fide asymptotic symmetries requires a careful analysis of the associated boundary charges, which are the tool allowing one to distinguish between proper and large gauge transformations.Indeed, the boundary charges associated to infinite-dimensional residual symmetries at face value are found to diverge when D > 4.These divergences can however be renormalized with various techniques, see e.g.[44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62], and in this paper we systematically explore two of them in the instructive and yet relatively handy example of Maxwell fields in Anti de Sitter (AdS) backgrounds of any dimensions.These investigations can be relevant for both AdS/CFT and flat holography, as well as to prepare for a corresponding study of de Sitter spacetime, although in the latter case the boundary analysis would require a different interpretation.
In particular, we derive finite boundary charges parameterizing infinite-dimensional asymptotic symmetries in two ways.We first derive them starting from the renormalized bulk action computed via holographic renormalization [46][47][48]52].We then compare the results with those obtained by directly renormalizing the presymplectic potential derived from the naive bulk action [52,57].The advantage of the first procedure is that it can be performed in a gauge invariant way at each step and that it provides a complete action principle, including the appropriate boundary contributions.The second approach requires instead a gauge fixing, but it is somewhat simpler to implement.
We explore and contrast these two complementary renormalization techniques in two different sets of coordinates.We first deal with Poincaré coordinates, which provide an established arena for holography and allow one to get results applying to arbitrary D, but which do not cover the full AdS space in Lorentzian signature and do not seem appropriate for taking a limit in which the AdS radius ℓ tends to infinity.We then move to Bondi coordinates, which are global coordinates and allow for a straightforward flat, ℓ → ∞ limit.We compute renormalized actions and presymplectic potentials in both coordinate systems, and study the diffeomorphism connecting the two.We stress that holographic renormalization is diffeormorphism invariant, which guarantees the agreement between the charges evaluated in Poincaré and in Bondi coordinates.
However, since both Bondi and Poincaré coordinates represent standard choices in the literature, we opt to present them in parallel, in order to stress their similarities and differences for practical computations.Our renormalized charges are not conserved, and we explicitly discuss the associated flux across the spacetime boundary [42,58].Compared to previous analyses of holographic renormalization and boundary charges of Maxwell fields in AdS D [41,43,45], we provide a general expression for the renormalized action in any D in Poincaré cordinates and, thanks to our analysis in Bondi coordinates, we set up a framework that allows one to recover the surface charges at null infinity [57], rather than spatial infinity, in a smooth flat limit.Let us also mention that our solution spaces include logarithms of the radial coordinate in odd spacetime dimensions.
A natural extension of the current work will be to apply the same strategy and the tools here developed to any linear gauge theory.On the one hand, this is expected to allow one to generalize the boundary conditions employed in [63] to derive conserved asymptotic charges for higher-spin fields in AdS.In addition, the option to get charges at null infinity via the ℓ → ∞ limit computed in Bondi coordinates should provide an alternative derivation of the boundary charges associated to the infinite-dimensional higher-spin asymptotic symmetries of [35], that played a crucial role in establishing the link with soft theorems.
In order to display our strategy in a simpler setting, we begin in section 2 by evaluating the renormalized action and presymplectic potential for a massless scalar field.In section 3 we then move to Maxwell fields on AdS backgrounds, in Poincaré coordinates.This allows us to obtain results applying to generic spacetime dimensions, while in section 4 we recompute from scratch the renormalized action and presymplectic potential in Bondi coordinates in a number of examples.In section 5 we eventually discuss how one can derive the charges in Bondi coordinates by starting from the more general results obtained in Poincaré coordinates and we also discuss how to take the flat limit.A couple of appendices recall some useful facts about the geometry of AdS and the key features of the covariant phase space formalism that we use to renormalize the presymplectic potential.

Poincaré coordinates
As a warm up to illustrate the renormalization techniques, let us begin by discussing the case of a free massless scalar field.In this section, we employ Poincaré coordinates: where η ab is the Minkowski metric and a, b ∈ {0, . . ., D − 2} (see also appendix A.2).2

Solution space
We consider the action of a free massless scalar field, whose equation of motion where Considering an asymptotic expansion of the field in the radial zcoordinate of the form where we are excluding solutions that explode at the boundary z → 0, one finds the recursion relations These, in their turn, determine the asymptotic solution space, where ϕ (0) and ϕ (D−1) are arbitrary functions of the boundary coordinates x a , while the coefficients that will be relevant in the following read where the last equation is trivial for even D. Therefore, the asymptotic radial expansion is, in even D and in odd D As noticed in [46][47][48]64] for (massive) scalars in any D, the on-shell action typically diverges due to contributions localized at the boundary (z → 0), and one has to implement holographic renormalization to set up the variational principle.

Holographic renormalization
We start by computing the regularized on-shell action, where we defined in order to mimic the role played by the field strength in the spin-one case.Since the boundary is now located at z − ϵ = 0, its normal is given by (2.12) By Stokes' theorem we can then write the regulated on-shell action as (≈ stands for on-shell equality) where the global sign reflects the fact that z = ϵ is the lower limit of the integration.To leading order for small z, the field Φ is z-independent, while the "field-strength" component F z scales like z.Therefore, the regulated action scales like ϵ 3−D and, whenever D ≥ 4, it has divergent contributions.We will now see in a few examples how can one subtract those divergences.
Let us consider D = 6.In this case the free functions in the asymptotic expansion (2.5) are ϕ (0) and ϕ (5) .The regulated on-shell action (2.13) reads where Ψ diverges like 1/ϵ 3 to leading order as ϵ → 0. Our strategy will be to find a counterterm action S (ϵ) ct such that and Ψ is finite as ϵ → 0. Since the additional Φ appearing in the action is finite at the boundary, achieving this will be enough to produce a finite subtracted action.With this objective in mind, we thus proceed by writing where denotes the coefficient of z k in the expansion of F z (z, x a ).Following the above strategy, since Φ remains finite as ϵ → 0 and thus cannot introduce additional divergences,we will eventually choose to not expand it.This has the advantage of simplifying the intermediate calculations, and will make it possible to obtain results holding for generic D below.A similar choice in the spin-one case will allow us to express all counterterms directly in terms of the field strength.
More explicitly, expressing the ) in terms of ϕ (0) by using the equations of motion, expanding also the Φ appearing in (2.16) and multiplying its expansion with the one in the round parenthesis, we obtain the following expression for the divergent terms expressed in terms of ϕ (k) , which shows the dependence on the counterterm from the boundary data.However, inverting the expansion (2.8), we can express back the regulated action in terms of the bulk field, finding Let us now note that the operation of expanding Φ in terms of its asymptotic expansion coefficients ϕ (k) and then inverting this expansion to re-express the resulting sum in terms of Φ acts as the identity on Φ, i.e. gives back Φ itself (regardless whether or not one first multiplies this expansion with that of F z in (2.16)).This fact can be checked explicitly using (2.17) for the present example, but holds in any dimension.Therefore, while needed in order to obtain the explicit expression (2.17) of S (ϵ) reg in terms of ϕ (k) , explicitly expanding Φ is not necessary in order to arrive at the form (2.18) for S (ϵ) reg in terms of the bulk field, which is what we need to the goal of computing the presymplectic potential.In the following, we will restrict our attention to expressions in terms of the bulk fields, so that we will not need to expand the overall Φ appearing in the regulated action, which considerably simplifies the intermediate steps especially as D increases.We can thus define the counterterm action as follows in the present case, and write the subtracted action as (2.20) Here and in the following, the indices a, b, . . .are lowered and raised respectively via the metric η ab and its inverse η ab .In (2.20), we have assumed F a vanishes on the boundary of the z = ϵ surface (which allows one to freely integrate by parts w.r.t.x a ); alternatively we could add the extra "corner term" S (ϵ) which has the effect of canceling such integrations by parts, and clearly does not spoil the variational principle.By varying it on-shell and by injecting the asymptotic solution, we obtain thus leading to In arbitrary even D ≥ 4, by expressing the boundary contributions encoded into the in terms of bulk field Φ, gathering the divergent pieces and assuming the same falloff for F a (equivalently, adding a suitable corner term), one obtains the counterterm action where the coefficients are fixed recursively by Let us remark that the expression (2.23) for the counterterm Lagrangian can be rewritten in the following way, where γ ab = η ab /ϵ 2 is the induced metric on the surface z = ϵ and □ γ = γ ab ∂ a ∂ b .This makes the diffeomorphism invariance of the procedure manifest, since no explicit (coordinatedependent) dependence on ϵ appears in (2.25).The on-shell variation of the subtracted action takes the form and yields the following variation of the renormalized on-shell action depending only on the subleading free function since 1) .In the odd-dimensional case there appear new finite terms in the counterterm action when expressed in terms of bulk fields, although, in the present renormalization scheme, we systematically drop them.For concreteness, let us first illustrate this phenomenon for the case D = 7.The asymptotic expansion of the scalar field is given by (2.9) where the arbitrary orders are ϕ (0) and ϕ (6) .As explained below (2.14), if we don't radially expand Φ, the regulated on-shell action is and re-expressing it in terms of the bulk fields leads to As anticipated, in this last step additional contributions of the type ϵ 4 ℓ 4 □ 2 would appear inside the round parenthesis, but we systematically drop them as they correspond to finite local functionals of ϕ (0) , and, following [46][47][48]64], we recall that the result after the subtraction of the divergent parts is only defined up to the addition of such scheme-dependent finite terms.An additional source of (harmless) ambiguities of this type would come from first expanding the overall Φ in (2.28), dropping the resulting finite pieces depending on ϕ (k) and then re-expressing the divergent contributions in terms of the bulk field Φ.The counterterm action, with the inclusion of the corner term, thus takes the following form: and, in the limit ϵ → 0, the on-shell variation of the subtracted action S x δΦ F (5)  z . (2.31) Similarly to the even-dimensional case, we can generalize the procedure to any odd dimension, for D > 4, by means of the following counterterm ) which, as usual, includes a corner-term contribution.The last two equations involve the recursive coefficients c (k) N (2.24) as well as a logarithmic coefficient (2.33) Let us rewrite (2.32) also in a manner analogous to (2.25), where all the dependence on ϵ enters via the cutoff surface and the induced metric γ ab thereon, except for the log ϵ term.When one approaches the boundary, it leads to the same expression for the variation of the renormalized on-shell action as in (2.27).
Symplectic renormalization Using the covariant phase space formalism (see appendix B) one can renormalize the scalar theory directly at the level of the symplectic structure [52,57].As we shall see, an advantage of this set-up is that we will obtain a radial renormalization equation providing the systematics for the introduction of the additional terms.
In order to see this explicitly, following [57], let us write the variation of the action (2.2) in the form where the presymplectic potential is Splitting Θ µ into radial and boundary components and factoring out an overall z −(D−2) for convenience, we define and for the off-shell Lagrangian.Thus, we can split the total divergence in (2.35) into radial and boundary divergences and obtain the asymptotic renormalization equation: The radial equation (2.40) implies that the symplectic potential can be made finite on-shell by subtracting counterterms which are either total variations or total boundary derivatives, as allowed by the ambiguity equation (B.7).Indeed, assuming an asymptotic radial expansion of the form and substituting in (2.40) one finds and similar equations involving the coefficients of the log terms.The above equation shows in particular that the orders n < D − 2 of Θ z (n) , i.e. the ones that come with divergent prefactors in Θ z , are fixed on-shell to be total derivatives plus total variations, while Θ z (D−2) , which gives the finite order of Θ z , is left undetermined.The remaining terms do not contribute in the asymptotic limit z → 0. Let us illustrate the procedure for the two cases of even and odd dimensions.
For even D ≥ 4, we can write where the summation includes all diverging terms, given on-shell by (see (2.7)) while the finite order is According to the radial equation (2.42), we can cancel the divergent orders adding terms of the form ∂ a Θ a (2n) and δ L (2n) , with and The counterterm is therefore while the renormalized presymplectic potential is By following similar steps, we can deal with the case of odd D > 4. The asymptotic expansion of the radial component of the presymplectic potential is where the diverging orders are the terms in the sum and the log term.They are given on-shell respectively by (2.44) and while the finite order is given by 2n) . (2.52) where Therefore, the renormalized presymplectic potential is Upon further adding the following corner term the above renormalized potential can be expressed in terms of the arbitrary coefficients as where we denoted the n-th harmonic number by H n One can see that the last term of (2.57) is a combination of symplectic ambiguities (B.7).Indeed, it can be cancelled through the addition of and At the end, we obtain the same result (2.49) as for the even-dimensional case.Note that the results obtained for the presymplectic potential from the two renormalization procedures lead to the exact same expression in even dimensions, see (2.27) and (2.49), while they differ by local terms in ϕ (0) in odd dimensions, see (2.27) and (2.57).We will encounter a similar situation for the case of spin one.

Bondi coordinates
We now revisit the previous construction in Bondi coordinates, where γ ij is the metric on the unit (D − 2)-dimensional celestial sphere, parameterised by the angular coordinates x i (see also appendix A.3). Furthermore, we focus on the renormalization of the presymplectic potential.The resulting techniques will be instrumental to consider the flat limit of the surface charges in the spin-one case discussed in section 5.
Solution space In Bondi coordinates, in which the AdS metric is given by (2.61), the equation of motion (2.3) for a massless scalar reads where ∆ denotes the Laplacian on the (D − 2)−sphere.We assume a radial expansion of the form which turns the above equations of motion into the following recursive relation: and similarly for the log terms.We obtain that ϕ (0) and ϕ (D−1) are undetermined.In terms of F µ := ∇ µ Φ, the equation of motion takes the form where the round brackets denote symmetrization on the corresponding indices, and where " • " stands for contraction of indices on the sphere, raised and lowered using γ ij and γ ij .

Symplectic renormalization
In Bondi coordinates the presymplectic potential has radial divergences when r → ∞, so that one has to renormalize the symplectic structure along the lines of section 2.1 and appendix B. The components of the presymplectic potential (2.36) and the Lagrangian are and Upon substituting in (2.35) one obtains which, in its turn, when taking into account the radial expansions (2.72) In the following, we shall illustrate along these lines two specific examples in even and odd dimensions, D = 4, 5.
Let us first consider D = 4. First, we compute the asymptotic solution space and obtain that the orders ϕ (0) , ϕ (3) are arbitrary functions of the boundary coordinates functions of u and x i .Next, we inject the corresponding expansions into the radial component of the presymplectic potential: where (2.74b)So we have a divergent order, Θ r (1) to be renormalized.Using the radial renormalization equation (2.72) we have so that Thus, the counterterm to be added to the presymplectic potential is where which implies that the renormalized potential takes the following form: (2.79) In the odd dimensional example of D = 5, the asymptotic solution space is given by where ϕ (0) and ϕ (4) are arbitrary functions of u and x i .It leads to the following radial expansion of the presymplectic potential where the first three terms diverge with r.If we add the following counterterm to the presymplectic potential where one obtains the renormalized potential, whose explicit form is (2.85)

Spin-one fields in Poincaré coordinates
In this section we compute and renormalize asymptotic charges of massless spin-one fields in AdS space of arbitrary dimension, in Poincaré coordinates A.2.

Solution space
Maxwell's equations ∂ µ ( √ −g F µν ) = 0 in Poincaré coordinates read and combining them with the Bianchi identities, To begin with, let us explore the solution space in terms of the field strength.
For even D we assume an expansion in z of the schematic form and get and where the first of (3.4) is the divergence of the second one, for n ̸ = D − 4.
One sees that (3.5) fixes ab for any positive even integer n, Similarly, one can fix F .Thus, for even D, we assume the following structure of the solution as an expansion in z, The free data in this case are encoded in ab , which is an arbitrary antisymmetric tensor, and in F (D−4) az , which by the first equation in (3.4) is divergence-free.These two characteristic orders at which the new boundary data appear correspond to the "source" and to the "VEV" according to the standard terminology.
For odd D > 4, we include logarithmic terms and expand the field strength according to which modifies (3.1) into and so as to lead, together with the Bianchi identities, to the recursion relations ) In this case, we assume ) It turns out that the logarithmic branch can be fixed in terms of the free data specified by the source, in such a way that ab remains unconstrained.To summarize, the structure of the solution space for arbitrary D is as follows: the arbitrary functions are F , the last one being divergence-free, while Note that in particular the order F The falloffs (3.7) and (3.12) for even and odd D, respectively, capture the two expected branches of solutions associated to radiation (or source) and to static (or VEV) contributions.
Let us now discuss the solution space in terms of the gauge potential A µ .Upon imposing the Lorenz gauge ∇ • A = 0 and assuming the expansion the relations (3.4) and (3.10) yield while the gauge condition reads (3.17) The equations determining the log terms are obtained similarly.The solutions, for any D ≥ 4, are = 0 and odd D > 4 : In Lorenz gauge the parameter in δ λ A µ = ∇ µ λ is constrained by ∇ µ ∇ µ λ = 0. Thus, one can perform on λ the same analysis illustrated for scalar fields in section 2.1 and find the corresponding solutions (2.7).In particular, one can use the available free function λ (D−1) in the residual gauge parameter to gauge-fix A (D−2) z to zero.As we shall see below, see e.g.Eq. (3.55), λ (D−1) does not appear in the boundary charges (in contrast with λ (0) ), so that the transformation needed to achieve this gauge fixing is a small gauge transformation.This shows that the nontrivial boundary data consists of source, i.e. the free vector A whose solutions are, for D ≥ 4, with the extra constraints for n ̸ = 0, D − 3. Note also that as a consequence of (3.22a) and (3.24).Thus, for even dimension while for odd dimension The radial orders A (0) a and the divergence-free part of A (D−3) a are unconstrained by the above equations of motion.

Holographic renormalization
The starting point is the regularized action Upon expanding the fields for small z = ϵ, we see that this regulated action diverges for D ≥ 5 and one can isolate the divergent terms to then identify the counterterms needed to obtain the renormalized action.Once the latter is available, from its on-shell variation it is possible to compute the renormalized asymptotic charges.Note instead that (3.29) is finite for D = 4 thanks to the falloffs (3.7).

Even dimensions
In order to renormalize (3.29), we shall follow a strategy analogous to the one detailed for the scalar field below Eq. (2.14).Starting from in order to define an appropriate subtracted action, it is enough to find a counterterm action S ct such that where Ba is finite as ϵ → 0, since A a is itself finite in that limit.To this end, upon inserting the first of (3.7) in (3.29) we have where the sum runs formally over all n from 1 to D − 5 but only the odd terms are nonvanishing.Note that, in the present renormalization scheme, we conveniently choose not to expand A a = O(z 0 ).Similarly to the discussion for the scalar case below Eq. (2.16), this is because we do not aim to obtain the explicit expansion of the regulated action in terms of asymptotic field components A µ , but rather only its expression in terms of bulk fields, which will suffice in order to obtain a finite presymplectic potential.Let us also stress that this choice cannot affect the bulk-covariant form of the counterterms, except possibly for scheme-dependent terms in the finite piece (and only for odd D, as for the scalar field).This choice is particularly convenient because it allows us to only use the equations of motion for the field strength; below we will compare this with a different procedure, checking that a different choice only introduces finite local functionals of the source field.Using the equations of motion (3.4) together with near the boundary, we rewrite the regulated action in the form ) where the first term determines the most singular counterterm.Let us introduce the notation S (n) ct for the nth counterterm and S ≤n for the sum of the counterterms up to n (in decreasing order of divergence), for later convenience.So and The idea is to use (3.5) in order to write a recursion equation for the counterterms.Indeed, inserting (3.5) and shifting the sum by 2, one finds where, exploiting again (3.33), one can isolate the second counterterm, thus arriving at (3.39)The general structure is therefore and with Upon integrating by parts one finds the general form of the counterterm up to a boundary-of-the-boundary term, where the coefficients c (here N is just a label for "normalized", not an additional variable) can be found by means of the recursion relation (3.42) or by the equivalent one

.45)
Let us note that the expression (3.43) for the counterterms can be rewritten in the following manifestly diffeomorphism invariant way, where all explicit powers of ϵ have been reabsorbed into the induced metric γ ab = η ab /ϵ 2 on the surface z = ϵ and into □ γ = γ ab ∂ a ∂ b .There are two possibilities to do away with the boundary-of-the-boundary term.The first one is to assume that we are working with field configurations that fall off in the early past and in the far future at the boundary: The second one is to add a corner term to the action that cancels it, whose general form is Of course this will not interfere with the variational principle, since it only involves derivatives tangential to the boundary.Altogether, the on-shell variation of the subtracted action, S ct , is given by In this scheme, where we haven't expanded the Maxwell field and have inverted the expansions, we can provide a covariant expression for the presymplectic potential and symplectic structure in terms of the bulk fields: where (in both gauges).In particular, looking at the variation along a gauge parameter λ, we can rewrite the latter as where we used ∂ a F (D−4) za = 0 from (3.4).Correspondingly, the charge flux across the boundary is given by and the surface charges read Let us note that the time derivative ∂ 0 Q (λ) manifestly vanishes only for the standard electric charge (λ = 1).Let us now discuss in more detail the counterterm structure in a couple of examples.
In D = 6 there is only one counterterm which is given by so that, on shell Of the two options to do away with the boundary-of-the-boundary we simply use (3.47) to drop the first term in the second line so that the subtracted action is note that this expression is exactly gauge invariant.Its on-shell variation gives and expanding for small ϵ, we find where the divergent term vanishes due to (3.4).We conclude that We can thus identify the conserved current (3.51) and the surface charge (3.55), for D = 6.
In D = 8 a leading and a subleading term need to be taken into account.Indeed, there are two divergent terms Following the steps from (3.33) to (3.38) we obtain and integrating by parts where the second line identifies the counterterms.The subtracted action is therefore In the on-shell variation

Odd dimensions
From (3.29) and (3.12) we see that in odd D ≥ 5 there will be a single counterterm associated to a logarithmic divergence plus a collection of counterterms associated to the power-like divergences: with Using the equation of motion (3.11b), one finds while the structure of S st is the same as that of S reg in even D, i.e. its counterterms are given by (3.40) with coefficients (3.42).The recursion relation starts with where of course the n = D−5 terms is absent.Nevertheless, we do have a finite counterterm S (0) ct associated to c (D−5) .
Equivalently, for D > 5 (see (3.80) below for D = 5), in order to regularize the on-shell action, we can add a counterterm action that differs from the even-dimensional one only by the presence of a term in log ϵ, (3.72) and takes this suitable expression thanks to the further addition of a corner term, (3.74) Let us also rewrite (3.72) by introducing the induced metric γ ab on the cutoff surface, in a manner similar to (3.46), ) where the only leftover explicit dependence on ϵ enters via log ϵ.The on-shell variation of the subtracted action yields δS (3.76)In the limit ϵ → 0, the last equation gives us the following variation of the renormalized on-shell action In this case, F (in both gauges).Along a gauge parameter λ, the latter reduces on-shell to a corner term so that the conserved current and the charge are given by the same expressions as in the even dimensional case, (3.51) and (3.55), respectively.The case of D = 5 provides the simplest example to this effect, with the only the divergent term being the logarithmic one Assuming that (3.47) holds, we can integrate by parts this term and write the subtracted action as follows Taking the variation on shell and using the condition (3.47) to drop the boundary of the boundary contribution, Expanding the quantity within round parentheses, and the coefficient of the singularity vanishes on the equations of motion thanks to (3.10) evaluated for n = 1.In conclusion, and therefore and the equations of motion indeed ensure that this is a conserved current, since ∂ a F (n) za ≈ 0. Correspondingly, the charge flux that we get is where we have used the equations of motion ∂ a F (n) za ≈ 0. The first case in which both a power divergence and a logarithm need to be taken into account simultaneously is the case D = 7. Starting again from (3.29) for D = 7, and substituting the falloffs (3.12) we have The equation of motion (3.10), setting n = 1 and n = 3, gives while the recursion relation (3.11b) yields Substituting into (3.87) and using after integrating by parts in the usual way.Note that the second term in the round parentheses gives a finite contribution as ϵ → 0. Since in the minimal subtraction scheme we simply identify all divergent terms of the regularized action, it could be natural to drop it.However, we may also keep it because it actually cancels a finite term arising from the first term when re-expanded in ϵ.We are thus led to define the renormalized action as (3.91) Taking the variation on shell and using the condition (3.47) to drop the boundary of the boundary contribution, Expanding the quantity within square brackets, we have All round parenthesis in this expression vanish by the equations of motion (3.88), (3.89), including the "spurious" finite term coming from the variation of the counterterm.In conclusion, and therefore and the equations of motion indeed ensure that this is a conserved current, since ∂ a F (n) za ≈ 0. Correspondingly, the charge flux that we get is where we have used the equations of motion ∂ a F (n) za ≈ 0.

Symplectic renormalization
In order to obtain finite charges an alternative procedure is to directly renormalize the symplectic structure of the theory, as we saw in section 2.1 for the scalar case.To this end, from the Maxwell Lagrangian one derives the presymplectic potential (see appendix B) The main idea of the symplectic renormalization is to exploit the ambiguities of the presymplectic potential (B.7) in order to cancel radial divergences.In general, explicitly determining the form of the counterterm ambiguities in (B.7) can be impractical.However, a systematic way of obtaining such terms has been proposed in [52,57].In this approach, contrary to the holographic renormalization, we need not only the solution space of the field strength but also that of the gauge field and therefore this procedure depends in principle on the gauge.Again, since the boundary has the orientation n µ = δ z µ in the Poincaré patch, the charges are computed using Θ z .If we factor out the off-shell radial dependence of Θ µ and similarly for the Lagrangian we obtain the asymptotic renormalization equation Upon further expanding radially Θ µ and L , Therefore, one can see that the orders n < D − 4 of Θ z (n) , which are the ones that come with divergent prefactors in Θ z , are fixed on-shell to be total derivatives plus total variations that can be removed order by order, while Θ z (D−4) , which gives the finite order of Θ z , is undetermined by this equation.In order to see how the procedure concretely works one has to distinguish even-and odd-dimensional cases.

Even dimensions
In the generic even dimensional diverging case, i.e. for D ≥ 6, the asymptotic radial expansion of the presymplectic potential radial component is where the divergent orders are given in terms of the asymptotic solution space of the massless spin-1 field and its field strength by while the finite order is According to (3.103), the divergent orders can be expressed as ambiguities built on and δ L (2n) , where and odd n : even n : The counterterm is therefore and the renormalized presymplectic potential is This procedure involves a choice of the gauge fixation of the A−field in the asymptotic solution spaces.However, in the way we have written it above, the systematics is gauge independent, as is the final result for the finite renormalized part of the potential.

Odd dimensions
For D ≥ 5, the divergent terms of the presymplectic potential read where, in terms of the asymptotic solution spaces, the non-logarithmic terms are given by (3.105) and the logarithmic one by such that the renormalized presymplectic potential takes the form Although this finite order is not subject to the renormalization equation, we can still add finite ambiguities to it.In particular, if we add the following corner term ) we obtain Note that, despite involving a gauge choice for the spin-one field, the final result and the entire procedure above are gauge-independent.The only concrete effect of the Lorenz gauge that we employed above is in the factor 2 D−3 of (3.115) that in the radial gauge, for instance, would be equal to 1.We can further enhance the last expression with the addition of the following terms to end up with the expected result 4)   za . (3.118) Let us consider explicitly the case of D = 7, so as to also highlight the role played by different gauge choices, for the two instances of the Lorenz gauge and the radial gauge.
Lorenz gauge In this case, the asymptotic radial expansion of Θ z is where the diverging orders are and where in particular the solution space of A µ is such that with a a free function of x a .The divergences are cancelled thanks to the following counterterm since the asymptotic renormalization equation (3.103) yields where and Therefore, the renormalized presymplectic potential (3.114) is If we add the following corner and boundary terms ab δF ab (0) , (3.127a) the renormalized presymplectic potential reads and matches the result we obtained by renormalizing the variational principle.Note that the ambiguity in δ-exact must be justified by a boundary term to add to the bulk action.
Radial gauge In the radial gauge A z = 0, the solution space of the A−field takes the form where A (0) a is an arbitrary function of the boundary coordinates.The renormalization is identical to what we performed for the Lorenz gauge.The radial component of the presymplectic potential then radially expands as in (3.119),where the on-shell values of the diverging orders are given in (3.120).Using (3.103), we have shown that the latter can be cancelled if we add the counterterm (3.122)-(3.125)to the presymplectic potential, except that in the present case This leads us to the on-shell renormalized symplectic structure which differs from (3.126) obtained in the Lorenz gauge by a pure corner term We can then conclude that we obtain the same result in both gauges.Note that we can still fix the ambiguities of the presymplectic potential for the finite order.Hence, if we add the same finite boundary and corner terms to the renormalized potential as in the Lorenz gauge, it can finally be rewritten as (3.128).

Spin-one fields in Bondi coordinates
We want to determine the renormalized asymptotic spin-one charges in Bondi coordinates A.3.One motivation is that in this framework computing the flat limit of the charge becomes essentially trivial.However, in Bondi coordinates the analysis in arbitrary dimensions turns out to be more involved than in the Poincaré patch and we will detail only a few specific cases.

Solution space
The equations of motion for the field strength read In analogy with (4.2), assuming we obtain

Holographic renormalization
We introduce the regulated on-shell action Since the orientation of the boundary is given by n µ = δ r µ , the above action reduces to Let us show how to obtain its renormalized on-shell variation in two explicit instances.

D = 6
By solving the equations (4.4)-(4.7)with the additional that all positive powers of r vanish, one obtains the asymptotic solution space for D = 6 where ij and F iu are arbitrary functions of (u, x i ), while for instance and where the free data are ij and F (0) iu , and We shall omit further relations among the coefficients that are not useful to our purposes.Using the inverse expansion and adding the corner action one concludes that the counterterm action is This leads us to the following variation of the renormalized on-shell action

.29)
If one evaluates the latter along a gauge parameter λ and uses (4.25), it yields Again the square bracket drops out, and we obtain the charge 3)  ur . (4.31)

Symplectic renormalization
As in (2.66) and (4.34) we factor out the radial off-shell dependence of the latter and of the Lagrangian, where and so as to get the asymptotic renormalization equation in the form Similarly, we break down the unit vector XI (x i ) parametrized by the D − 2 angles x i , with i = 1, . . ., D − 2, by isolating the angle with respect to the (D − 1)th direction, so that Then we can express the Poincaré coordinates in terms of the Bondi coordinates as follows x I ′ = ℓ XI ′ (x i ′ ) sin θ This relation highlights rather clearly that, depending on the retarded time at which one approaches the boundary, only a portion of the sphere at infinity is covered by the Poincaré coordinates, except when u = −πℓ/2 (i.e.X 0 = 0 for large r) in which case the whole sphere is covered except for the North pole (see Fig. 2b).Using the change of coordinates given above we can calculate the component transformation for the field strength for large r.In particular, we find F ru ∼ ℓ F z0 1 + cos θ sin u ℓ + F zI ′ XI ′ sin θ cos u ℓ r 2 (sin u ℓ + cos θ) 3 . (5.7) We also note that, for field configurations such that, for small z in Poincaré coordinates ( 5.11) This shows that the charges constructed in the two setups are indeed identical, as expected.
Let us now briefly comment on the role of the independent free functions appearing in the solutions of the equations of motion.The relation between the "VEV" or "Coulombic" branches in the two coordinate systems is discussed above.The information about the "source" or radiation branch, in Poincaré coordinates is contained in the leading, O(z 0 ) component of F ab , F ab ∼ F ab . (5.12) This translates into Bondi components via sin θ −F 0J ′ 1 + cos θ sin u ℓ + F J ′ I ′ XI ′ sin θ cos u ℓ (sin u ℓ + cos θ) 3   , so inducing O(r 0 ) terms in F ui and F ij , which thus encode the dependence on the radiation data.Note that Eq. (5.13) does not involve any additional factor of 1/r to leading order, in contrast with (5.7).

Flat limit
In order to take the flat-space limit ℓ → ∞, we need to fix a choice of coordinate system, and the factor of 1 ℓ on the left-hand side of (5.11) suggests that the flat limit be better defined in Bondi coordinates than in Poincaré ones.
Actually, from (4.30) for D = 5 and (4.22) for D = 6, we see that even in these more convenient coordinates the symplectic structure itself still contains potentially dangerous terms that scale as ℓ 2 .However, such offending terms actually are total derivatives on the sphere and vanish identically.We can then safely take the limit ℓ → ∞ and find the standard expression ur .
A vector V ∈ T Γ is called symplectomorphism or Hamiltonian vector field if L V ω = 0. Since L V ω = δI V ω, given a Hamiltonian vector field one can define a current J V and the associated global functional H V as If this vector corresponds to a spacetime symmetry ξ, we denote it V = V ξ and we have that L V ξ = L ξ .This implies that we can write the current as follows,4 modulo δ-exact terms.This current is called the local weakly-vanishing Noether current.If, in addition, the vector V ξ is a gauge symmetry, we have also that L V S = L ξ S = 0, where Equating the two we get the fundamental theorem of CPS, stating that for gauge symmetries the Noether charge is a corner term: Moreover, one can add to Q ξ the divergence of a (D − 3)-form without modifying H ξ , due to Stokes' theorem.

(
D−3+n) ab for positive even n in terms of F (D−3) ab .19) The arbitrary functions of the boundary coordinates are A (0) a , A (D−2) z and the transverse part of A (D−3) a (while ∂ • A (D−3) = □A (D−4) z D−3 ), and the corresponding radial expansions are given by even D ≥ 4 : i.e. the divergence-free part of A (D−3) a .As an alternative, in the radial gauge A z = 0, (3.4) and (3.10) become 49) which, after taking the limit ϵ → 0, gives us the following variation of the renormalized on-shell action δS ren = lim ϵ→0 δS

N 2 Γ D−5 2 .
determined by the recursive relation (3.44), while the coefficient for the logarithmic term is c L = 2 6−D Γ D−3 .112) Using the radial equation (3.103) and the expressions (3.107)-(3.108),we can cancel the above divergences with the following counterterm

.35) 5 observations 5 . 1
Further Dictionary between Bondi and PoincaréWe would like to compare our results obtained in the two different coordinate systems presented in the previous sections.To this end, let us write down the associated change of coordinates.We introduce a splitting of the index I, which labels the D − 1 spatial directions 1, . . ., D − 1 by breaking it down as I = (I ′ , D − 1) with I ′ = 1, . . ., D − 2.

14 )
Whereas we have explicitly performed the derivation in Bondi coordinates only for D = 5, 6, but our generic-D results of section 3, in particular the charge (3.55), combined with(5.11)allow us to conclude that the expression (5.14) applies in any dimensions, for both AdS and flat spacetime.computes their effects when applied in different orders along V ∈ T Γ.A theory is specified by an action L = L d D x is the Lagrangian form.An arbitrary field variation yields δL = (eom)δΨ + dΘ , (B.4) where (eom) denotes the equations of motion, Ψ ∈ Γ and Θ = Θ µ (d n−1 x) µ is the local presymplectic potential form.One defines the local presymplectic two-form ω = δΘ (B.5) in terms of which one computes the presymplectic two-form Ω = Σ ω , (B.6) as the integral over an (arbitrary) Cauchy surface Σ ⊂ M. The presymplectic potential admits two types of modifications -often referred to as ambiguities in the literature although in general they lead to inequivalent presymplectic potentials -which do not alter the equations of motion: Θ → Θ + δB − dC .(B.7)