Asymptotic Structure of Electromagnetism in Higher Spacetime Dimensions

PHYSICAL REVIEW D 99, 125006 (2019)

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Asymptotic structure of electromagnetism in higher spacetime dimensions

Marc Henneaux1,* and C´edric Troessaert2 1Universit´e Libre de Bruxelles and International Solvay Institutes, ULB-Campus Plaine CP231, B-1050 Brussels, Belgium 2Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Am Mühlenberg 1, DE-14476 Potsdam, Germany

(Received 2 April 2019; published 10 June 2019)

We investigate the asymptotic structure of electromagnetism in Minkowski space in even and odd spacetime dimensions ≥ 4. We focus on d>4 since the case d ¼ 4 has been studied previously at length. We first consider spatial infinity where we provide explicit boundary conditions that admit the known physical solutions and make the formalism well defined (finite symplectic structure and charges). Contrary to the situation found in d ¼ 4 dimensions, there is no need to impose parity conditions under the antipodal map on the leading order of the fields when d>4. There is, however, the same need to modify the standard bulk symplectic form by a boundary term at infinity involving a surface degree of freedom. This step makes the Lorentz boosts act canonically. Because of the absence of parity conditions, the theory is found to be invariant under two independent algebras of angle-dependent uð1Þ transformations (d>4). We then integrate the equations of motion in order to find the behavior of the fields near null infinity. We exhibit the radiative and Coulomb branches, characterized by different decays and parities. The analysis yields generalized matching conditions between the past of I þ and the future of I −.

DOI: 10.1103/PhysRevD.99.125006

I. INTRODUCTION of the curvature that may be too stringent and thus may fail to be satisfied by any solution that would allow gravita- Most of the studies of the asymptotic properties of tional waves.” This point has also been forcefully stressed gravity in the asymptotically flat context have been in the recent work [9]. performed at null infinity [1–4] (for recent useful reviews, One remarkable by-product of the studies at null infinity see [5–7]). This is quite natural, and would seem to be even was the discovery that the asymptotic symmetry group of mandatory, in order to decipher the intricate properties of gravity in the asymptotically flat context was the infinite- gravitational radiation. One conceptual difficulty with analyses at null infinity, dimensional Bondi-Metzner-Sachs (BMS) group. Initially however, is that the existence of a null infinity with the received with some skepticism because the physical sig- smoothness properties usually assumed in the asymptotic nificance of this infinite-dimensional enlargement of the treatments is a difficult dynamical question: given reason- Poincaré group was not clear, the emergence of the BMS able initial data on a Cauchy hypersurface, will their group was understood recently to be related to profound Cauchy development give rise to a null infinity with the infrared properties of gravity having to do with soft – requested properties? Strong doubts that this would be the graviton theorems and memory effects [10 16] (see [17] – case have been expressed in [8], which we quote verbatim: for an exposition of this recent work and [18 20] for earlier “...it remains questionable whether there exists any non- investigations). The conclusion of the huge amount of trivial solution of the field equations that satisfies the activity that flourished since then is that the BMS group is a Penrose requirements [of asymptotic simplicity]. Indeed, gift rather than an embarrassment. An even further enlarge- “ ” his regularity assumptions translate into fall-off conditions ment of the Poincaré group including super-rotations have even been argued to be useful [21–23]. The BMS transformations are diffeomorphisms leaving *On leave from Collège de France, 11 place Marcelin the boundary conditions at null infinity invariant. They Berthelot, 75005 Paris, France. are exact symmetries of the theory. That is, they leave the action exactly invariant up to a surface term, without Published by the American Physical Society under the terms of having to make approximations. As exact symmetries the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to of the theory, they should be visible in any description, the author(s) and the published article’s title, journal citation, and, in particular, in slicings of spacetime adapted to and DOI. Funded by SCOAP3. spatial infinity. In such slicings, they would appear as

2470-0010=2019=99(12)=125006(21) 125006-1 Published by the American Physical Society MARC HENNEAUX and CEDRIC´ TROESSAERT PHYS. REV. D 99, 125006 (2019) diffeomorphisms leaving the boundary conditions at this problem [30], we put forward alternative boundary spatial infinity invariant. For this to be the case, however, conditions that were also BMS invariant, but which yielded the boundary conditions at spatial infinity should be a singular behavior (∼ ln r) for some components of the equivalent, or at least compatible in a sense that we shall Weyl tensor as one went to null infinity. In spite of this make more precise below, with the boundary conditions singular behavior at null infinity, nothing spectacular at null infinity. This brings us back to the dynamical occurred at spatial infinity and the BMS symmetry was question on null infinity mentioned above. untouched.] Earlier investigations of the asymptotic symmetries at Similar features arise in the discussion of the asymptotic spatial infinity showed no sign of the BMS group. One behavior of the electromagnetic field, where the null either found the Poincar´e group with no enlargement infinity analysis [32–34] seemed to be at variance with [24,25]; the smaller homogeneous Lorentz group [26]; the spatial infinity analysis [35]. The tension was solved in or an even larger extension, the Spi group [27,28], but in no [36], again by providing appropriate boundary conditions case the BMS group uncovered at null infinity. One logical at spatial infinity. The null infinity matching conditions of possibility for this discrepancy would be that the set of electromagnetism were also shown there to be a conse- boundary conditions at spatial infinity is incompatible with quence of the boundary conditions at spatial infinity. the set of boundary conditions at null infinity, so that there Extension of the asymptotic analysis to higher dimen- exists transformations preserving one set but not the other. sions raises interesting issues, which have led to a some- (Invariance of the action cannot be the issue—provided what unclear situation at null infinity where some studies the action is well defined—since we are dealing with yield infinite-dimensional asymptotic symmetries as in four diffeomorphisms.) spacetime dimensions, while some others do not [37–46]. If true, this situation would be very disappointing and The question is further complicated in odd spacetime physically unsatisfactory. Motivated by the desire to under- dimensions because half-integer fractional powers of r−1 stand better these earlier puzzling results, we have mix with integer powers, leading to problems with the reexamined the asymptotic structure of gravity at spatial conformal definition of null infinity [47–49], and the infinity [29,30]. We have provided in [29] boundary frequent necessity to split the analysis according to whether conditions at spatial infinity that eliminate the previous the spacetime dimension is odd or even since only in the tensions between spatial infinity and null infinity analyses, latter case does one avoid nonanalytic functions at null in the sense that (i) these boundary conditions are invariant infinity. under the BMS group, which acts nontrivially on the fields This provides strong motivations for investigating the and has generically nonvanishing conserved charges, and asymptotic structure of the electromagnetic and gravita- (ii) integration of the symmetry generators from spatial to tional fields at spatial infinity in higher dimensions, where null infinity enables one to show that it is the same BMS the falloff of the fields is more uniform (no fractional group that acts both at spatial infinity and at null infinity, powers of r). This is done here for electromagnetism. We expressed in different parametrizations that can be explic- show that the methods developed in our previous work [36] itly related [31]. generalize straightforwardly to higher dimensions, with no Furthermore, the matching conditions imposed at null new conceptual difficulty. The discussion proceeds along infinity on the leading order of the gravitational field [17] similar lines independently of the spacetime dimension. are automatic consequences of the asymptotic behavior of One finds in particular the same need to modify the the Cauchy data at spatial infinity. Although not equivalent standard bulk symplectic structure by a surface term, as (they are stronger), the boundary conditions at null infinity shown necessary also by different methods in d ¼ 4 are compatible with those at spatial infinity, in the sense spacetime dimensions [50,51]. that they can be shown to obey the conditions that are One remarkable feature, however, is that a second angle- implied at null infinity by the behavior at spatial infinity. It dependent uð1Þ asymptotic symmetry emerges. This sec- is of interest to point out in this respect that while the ond uð1Þ is eliminated in 4 spacetime dimensions because leading order of the Cauchy development of the gravita- of parity conditions that must be imposed to get rid of tional field coincides with the generally assumed leading divergences in the symplectic structure and of divergences order at null infinity, the subsequent terms in the expansion in some components of the fields as one goes to null infinity ln r differ in general, since subleading terms of the type rk [36], but these parity conditions turn out to be unnecessary (k ≥ 1) will develop from generic initial data. One conse- in higher dimensions (although it would be consistent to quence of our analysis is that these nonanalytic terms do impose them). not spoil the BMS symmetry—even if they spoil the The difference in the behavior of the fields according to usually assumed “peeling” behavior of the gravitational whether the dimension is even or odd appears when one field at null infinity [9]. This gives further robustness to the considers null infinity. We exhibit the behavior of the BMS symmetry. It also disentangles the BMS group from electromagnetic field near null infinity by integrating the gravitational radiation. [Incidentally, in our first work on equations of motion “from spatial infinity to null infinity.”

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This is done by going first to hyperbolic coordinates transformations that preserve the boundary conditions (and [27,52–54]. Hyperbolic coordinates are pathological in yield finite surface terms in the variation of the action so as the limit, however, and we thus go then to coordinates to have well-defined canonical generators, here at spatial introduced by Friedrich, which are better suited to that infinity). An asymptotic symmetry is nontrivial if its purpose [55–57]. We show that initial data fulfilling our generator is not identically zero, i.e., if there are allowed asymptotic conditions at spatial infinity, without parity configurations (configurations obeying the boundary con- conditions, lead to a nondivergent behavior at null infinity ditions) that make it not vanish. Such an asymptotic (d>4). The presence of terms with different parities leads symmetry is then called “improper gauge symmetry” to an interesting generalization of the matching conditions following the terminology introduced in the lucid paper between fields at the past of Iþ and the future of I −, which [58]. “Proper gauge transformations” have identically we give. vanishing generators and form an ideal. The true physical Our paper is organized as follows. Section II provides the asymptotic symmetry algebra is the quotient of all the boundary conditions for the standard canonical variables asymptotic symmetries by the proper ones. Note that no for (free) electromagnetism, i.e., the spatial components of gauge condition is involved in that definition, which is the vector potential and their conjugate momenta, which are therefore intrinsic since it does not view asymptotic the components of the electric field. We focus on the case of symmetries as residual gauge transformations preserving d ¼ 5 spacetime dimensions. The symplectic form is finite some gauge conditions. (Of course, the boundary condi- without parity conditions. We allow a gradient term ∂iΦ, tions might involve implicitly some gauge fixing as it is where Φ is of order Oðr0Þ in the asymptotic behavior of the usually difficult to formulate them in terms of gauge vector potential. This gives a gauge-invariant formulation invariants only. It is important to check gauge independ- of the boundary conditions and is crucial for exhibiting the ence. The boundary conditions given below for electro- full set of asymptotic symmetries. In Sec. III, we determine magnetism leave the freedom of making an arbitrary gauge the proper and improper [58] asymptotic symmetries for transformation with a finite generator.) these boundary conditions. We show in Sec. IV that because of the presence of a gradient term in the boundary II. ACTION AND BOUNDARY CONDITIONS: conditions, the Lorentz boosts are not canonical trans- PRELIMINARY CONSIDERATIONS formations. The problem can be cured by introducing a We start with the standard action of source-free electro- surface degree of freedom (d.o.f.) (which can ultimately be magnetism in d spacetime dimensions, which takes the identified with A0 at the boundary). This is just as in 4 canonical form spacetime dimensions [36]. We then give the complete 5 i formulation of the d ¼ theory in Sec. V where we write in SH½Ai;π ;A0 particular explicitly all the Poincar´e generators. In Sec. VI, Z Z 1 d−1 i we show how the second angle-dependent uð Þ symmetry ¼ dt d xπ ∂tAi emerges. Section VII generalizes the analysis to arbitrary Z Z 1 1 spacetime dimension ≥ 5. The detailed behavior of the d−1 i ij − dt d x π π þ F F þ A G þ F∞; ð2:1Þ fields as one goes to null infinity is derived in Sec. VIII, 2 i 4 ij t where we compare and contrast the situations in d ¼ 4 spacetime dimensions (where parity conditions are neces- where F∞ is a surface term at spatial infinity (r → ∞), sary to remove leading logarithmic divergences) and d>4 which depends on the boundary conditions and which will spacetime dimensions (where this is not necessary). The be discussed below. The dynamical variables to be varied in concluding Sec. IX gives further light on the emergence of the action are the spatial components Ai of the vector i the second angle-dependent uð1Þ. Three appendixes com- potential, their conjugate momenta π (equal to the electric ≡ plete the discussion. field), and the temporal component A0 At of the vector As it is common practice in such asymptotic investiga- potential which plays the role of the Lagrange multiplier for tions, we shall assume “uniform smoothness” [3] whenever the constraint −k −k−1 −k −k needed, i.e., ∂roðr Þ¼oðr Þ; ∂Aoðr Þ¼oðr Þ. − 1 − G −∂ πi ≈ 0 Similarly, the distinction between Oðr ðkþ ÞÞ and oðr kÞ ¼ i ð2:2Þ will usually not be important to the orders relevant to the (Gauss’s law). We use the symbol ≈ to denote equality on analysis. ’ We close this Introduction by recalling what is meant the constraint s surface. here by the concept of asymptotic symmetry. This concept is defined only in space with boundaries (which can be A. Asymptotic behavior of the fields: First conditions infinity), once boundary conditions are prescribed to We now specialize to 4 þ 1 spacetime dimensions for complete the definition of the theory [59], as particularly definiteness. In 4 þ 1 spacetime dimensions, the electric and 1 emphasized in [60]. Asymptotic symmetries are gauge magnetic fields decay at spatial infinity as r3. This implies that

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1 the electromagnetic potential behaves as r2 up to a gauge One gets for the asymptotic falloff in polar coordinates, transformation, i.e., up to a gradient ∂iΦ. Under a gauge recalling that the momenta carry a unit density weight, transformation A → A þ ∂ ϵ, Φ is shifted by ϵ.This i i i 1 1 transformation will have a well-defined generator if ϵ is ¯ ð1Þ −3 A ¼ A þ Ar þ oðr Þ; ð2:7Þ of order unity at infinity (see below). It is therefore natural r r2 r r3 to request that Φ be also of order unity at infinity (which 1 1 ∂ Φ ∼ r r ð1Þr −1 implies i r). We thus impose the following decay at π ¼ π¯ þ π þ oðr Þ; ð2:8Þ spatial infinity: r 1 1 ¯ ¯ ð1Þ −2 1 1 1 A ¼ ∂ Φ þ A þ A þ oðr Þ; ð2:9Þ A ¼ ∂ Φ¯ þ A¯ þ Að Þ þ oðr−3Þ; A A r A r2 A i i r2 i r3 i 1 1 1 1 πi ¼ π¯ i þ πð1Þi þ oðr−4Þ; ð2:3Þ πA ¼ π¯ A þ πð1ÞA þ oðr−2Þ; ð2:10Þ r3 r4 r r2 where Φ¯ and the coefficients of the various powers of r−1 where the coefficients of the various powers of 1=r are are arbitrary functions on the 3-sphere, i.e., of the angles xA functions of the angles xA. Furthermore, used to parametrize it. We have kept only the leading term Φð1Þ 1 A Φ Φ¯ ∂Aπ¯ ¼ 0: ð2:11Þ in ¼ þ r þ Oðr2Þ, since the subsequent terms can be absorbed by redefinitions. It is only in 3 þ 1 dimensions ∂ Φ 1 ∂ Φ¯ ¯ Note that r is of order Oð 2Þ and has been absorbed that i and the first term containing Ai are of the same ¯ r ¯ in Ar. order. Here, ∂iΦ decays more slowly by one power of r. The boundary conditions (2.3) make the kinetic term in the action well-defined provided we impose that Gauss’s C. Relativistic invariance of the boundary conditions law hold at infinity one order faster than expected, i.e., The boundary conditions (2.3) and (2.4) are easily verified to be Lorentz invariant. A general deformation 1 ∂ πi ¼ o ð2:4Þ of a spacelike hyperplane can be decomposed into normal i r4 and tangential components, denoted by ξ and ξi, respectively. A Poincar´e transformation corresponds to the 1 [the orderR implied by (2.3) is oðr3Þ]. This ensures that the deformation 4 i term d xπ ∂t∂iΦ has no logarithmic singularity and i ⊥ clearly does not eliminate any physical solution. The ξ ¼ bix þ a ð2:12Þ condition (2.4) is an integral part of our boundary ξi i j i conditions. ¼ b jx þ a ; ð2:13Þ The asymptotic conditions (2.4) are sufficient by them- − ⊥ i selves to make the symplectic form finite. There is no need where bi, bij ¼ bji, a and a are arbitrary constants. The to impose parity conditions on the fields, contrary to what constants bi parametrize the Lorentz boosts, whereas the − was found in 4 spacetime dimensions [36], where an antisymmetric constants bij ¼ bji parametrize the spatial ⊥ i appropriate generalization of the parity conditions of rotations. The constants a and a are standard translations. [25,35] was necessary. Under such a deformation, the fields transform as

δA ¼ ξπ þ ξjF þ ∂ ζ ð2:14Þ B. Polar coordinates i i ji i i mi m i i m i m For later purposes, we rewrite the boundary conditions in δπ ¼ ∂mðF ξÞþ∂mðξ π Þ − ð∂mξ Þπ − ξ ∂mπ : spherical coordinates, in which the Minkowski metric reads ð2:15Þ

2 2 2 A B ds ¼ −dt þ dr þ gABdx dx ; ð2:5Þ The transformation of the fields is really defined up to a gauge transformation. This is the reason why we have with included the term ∂iζ in the transformation of Ai. A definite choice of accompanying gauge transformation will 2 gAB ¼ r γ¯AB; ð2:6Þ be made below to get simple expressions for the algebra. It is clear that the falloff in (2.3) and (2.4) is γ ζ where AB is the round metric in the unit sphere. In the preserved under these1 transformations provided behaves “ ” ζ ζ¯ A ζ ðxAÞ 1 sequel, angular indices on a bar quantity will be raised or as ¼ ðx Þþ r þ oðr2Þ. A AB lowered with γ¯AB, e.g., v¯ ¼ γ¯ v¯ B. Bar quantities live on The above transformation rules imply that the leading the unit sphere. order of the fields transforms only under boosts and

125006-4 ASYMPTOTIC STRUCTURE OF ELECTROMAGNETISM IN … PHYS. REV. D 99, 125006 (2019) rotations (their variations under translations are of lower preserved under the Poincaré group if we also impose order). One gets explicitly the following changes of the π¯ A ¼ 0. Thus, we complete (2.3) and (2.4) by leading orders of the fields, which we write in polar coordinates: ¯ ¯ A AA ¼ ∂AΘ; π¯ ¼ 0: ð2:24Þ b ¯ ffiffiffi r A ¯ ¯ 1 δb;YAr ¼ p π¯ þ Y ð∂AAr þ AAÞ − ζ ; ð2:16Þ ¯ ¯ γ¯ The requirement AA ¼ ∂AΘ, which is new with respect to d ¼ 4 where it is not needed, is equivalent to the ¯ ¯ b condition F ¼ 0. Imposing F ¼ 0 does not seem δ ¯ ffiffiffi γ¯ π¯ B B ∂ ¯ − ∂ ¯ ∂ ζ1 AB AB b;YAA ¼ pγ¯ AB þ Y ð BAA AABÞþ A ; ð2:17Þ unreasonable in 5 (and higher) spacetime dimensions. If there are only electric sources, this condition is certainly ¯ ¯ fulfilled. So the question is whether it eliminates interesting δb;YΦ ¼ ζ; ð2:18Þ magnetic configurations. In 4 þ 1 dimensions, magnetic pffiffiffi δ π¯ r ¼ γ¯D¯ Aðbð∂ A¯ þ A¯ ÞÞ þ ∂ ðYAπ¯ rÞ; ð2:19Þ sources are extended objects, namely, strings. They can be b;Y A r A A of two different types: (i) infinite or (ii) closed (strings with pffiffiffi δ π¯ A ¼ γ¯D¯ ðbγ¯BCγ¯ADð∂ A¯ − ∂ A¯ ÞÞ þ ∂ ðYBπ¯ AÞ a boundary do not yield a conservation law). Our formalism b;Y B C D D C B only applies to the second case, since we are not covering A B A B − ∂BY π¯ − Y ∂Bπ¯ : ð2:20Þ infinitely extended strings going all the way to infinity, for which there are extra d.o.f. at infinity—the end points of the Here, we have set string—that must be taken into account (on which the Poincaré group acts). But if the strings are closed, their field 1 ξ ¼ rb þ T; ξr ¼ W; ξA ¼ YA þ D¯ AW; ð2:21Þ is more like the field of a dipole because the total charge r that they carry is zero [61]. The magnetic field decays thus ¯ ¯ ¯ ¯ faster at infinity (see Appendix A) and it is legitimate to DADBW þ γ¯ABW ¼ 0; DADBb þ γ¯ABb ¼ 0; ð2:22Þ ¯ assume FAB ¼ 0.

LYγ¯AB ¼ 0; ∂AT ¼ 0: ð2:23Þ E. Summary: Complete set of boundary A ! The quantities b, Y , T, and W are functions on the sphere. conditions at spatial infinity on Ai, π A The first two, b and Y , describe the homogeneous Lorentz To summarize, the complete set of boundary conditions transformations, while T and W, which do not appear in the on the canonical variables A , πi is given by (2.7)–(2.10) transformation laws (2.16)–(2.20) of the leading orders, i ¯ supplemented by (2.24) [which automatically implies describe the translations. In these equations, DA is the ¯ A AB ¯ (2.11)]. We rewrite them here: covariant derivative associated with γ¯AB and D ¼ γ¯ DB. Contrary to what happens in 4 spacetime dimensions, the 1 1 ¯ ¯ ð1Þ −3 leading order A is not gauge invariant. Furthermore, the Ar ¼ 2 Ar þ 3 Ar þ oðr Þ; ð2:25Þ r r r boosts mix radial and angular components, which do not have independent dynamics anymore. Finally, we note that 1 πr ¼ π¯ r þ πð1Þr þ oðr−1Þ; ð2:26Þ the spacetime translations have no action on the leading r orders of the fields. The symmetry group at this order is 1 1 ∂ Φ¯ ∂ Θ¯ ð1Þ −2 thus the homogeneous Lorentz group. To get the full AA ¼ A þ A þ 2 AA þ oðr Þ; ð2:27Þ Poincar´e group, one needs to consider the subsequent r r terms in the asymptotic expansion of the fields. In this 1 πA πð1ÞA −2 context, it is actually well known that in the case of gravity, ¼ 2 þ oðr Þ; ð2:28Þ the next-to-leading terms contribute crucially to the angular r momentum and the boost charges and so cannot be ignored. in order to have them conveniently grouped together. Given the vector potential, the functions Φ¯ ðxAÞ and D. Further strengthening of the boundary conditions Θ¯ ðxAÞ are determined up to a constant. We shall come back It turns out that while the boosts preserve the boundary to this point later. conditions, they fail to be canonical transformations (see Sec. IV below). In order to recover a canonical action for III. PROPER AND IMPROPER GAUGE the boosts, one adds new surface d.o.f. This can be achieved TRANSFORMATIONS along the lines of [36] if one strengthens further the boundary conditions. The boundary conditions (2.3) are invariant under gauge ¯ We impose that the leading term AA of the angular part of transformations generated by the first-class constraint the vector potential be a pure gradient. This condition is generator G:

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i δϵAi ¼ ∂iϵ; δϵπ ¼ 0; ð3:1Þ IV. BOOSTS AND SYMPLECTIC STRUCTURE We now turn to the question as to whether the Poincaré provided the gauge parameter ϵ has the asymptotic behav- transformations, which preserve the boundary conditions as ior we have just seen, are canonical transformations. That is, 1 we analyze whether they are true symmetries (invariance of ϵ ¼ ϵ¯ðxAÞþ ϵð1ÞðxAÞþoðr−1Þ: ð3:2Þ the symplectic form is a consequence of the invariance of r the action up to a total time derivative). We focus on boosts, which are the only transformations As already indicated above, the leading term A¯ in the r presenting difficulties. For boosts, the above transforma- expansion of the radial component of the potential is not tions reduce to gauge invariant, contrary to what happens in 3 þ 1 space- ¯ ¯ ð1Þ time dimensions. It transforms as Ar → Ar − ϵ . The −2 δAi ¼ ξπi þ ∂iζ ð4:1Þ leading [Oðr Þ] term of the field strength FAr is of course ¯ ∂ ¯ ¯ invariant. It is given by FAr ¼ AAr þ AA. One has δπi ∂ Fmiξ 4:2 ¯ ¯ ð1Þ ¯ ¯ ¼ mð ÞðÞ AA → AA þ ∂Aϵ so that FAr → FAr. The generator of (3.1) reads explicitly where ξ ¼ br. As we now show, these fail to be canonical Z I I transformations with the symplectic 2-form derived from ϵ 4 ϵG 3 ϵ¯π¯ i ≈ 3 ϵ¯π¯ i (2.1), G½ ¼ d x þ d Si d Si ð3:3Þ Z Ω 4 πi I ¼ d xdV dVAi; ð4:3Þ ≈ 3 ϵ¯π¯ r d x ð3:4Þ where the product is the exterior product ∧ of forms which we are not writing explicitly and where we use the symbol (where the integrations in the last terms are over the dV for the exterior derivative in phase space in order not to 3-sphere at infinity, i.e., over the angular variables xA). introduce confusion with the spacetime exterior derivative. Another crucial difference with respect to 3 þ 1 space- The transformation defined by the vector field X is Ω 0 time dimensions is that since the leading order of the radial canonical if dVðiX Þ¼ . Evaluating this expression for component πr does not fulfill particular parity conditions, the boosts, one finds, by a computation that parallels the 3 þ 1 case, both even and odd parts of the gauge parameters contribute Z to the charge G½ϵ. Nonvanishing values of ϵ¯ define ffiffiffi Ω 4 ∂ p ξ mi “improper gauge transformations” [58], no matter what dVðib Þ¼ d x mð g dVF ÞdVAi; ð4:4Þ the parity of ϵ¯ ≠ 0 is. An improper gauge transformation πi π 0 shifts the function Φ¯ by ϵ¯. where we have used dV dV i ¼ . Integrating by parts and using d F d Fij ¼ 0, we get that d ði ΩÞ reduces to a The value of the generator G½ϵ for ϵ¯ ¼ ϵ¯0 (constant) is V ij V V b surface term, the total electric charge Q. In the absence of charged matter I A field, Q ¼ G½ϵ0¼0, so that constant shifts of ϵ¯ðx Þ are pffiffiffi d i Ω d3x gξd Frid A ; : unobservable (no charged states). To have Q ≠ 0, one Vð b Þ¼ V V i ð4 5Þ needs to couple charged sources to the electromagnetic k 0 an expression that can be transformed to field, in which case Gauss’s law becomes −∂kπ þ j ≈ 0. I pffiffiffi The asymptotic analysis is unchanged provided the sources d ði ΩÞ¼− d3x γ¯bd ðD¯ AA¯ þ A¯ AÞd ∂ Φ¯ ; ð4:6Þ are localized or decrease sufficiently rapidly at infinity. We V b V r V A 0 should stress, however, that even if Q ¼ , the generator using the asymptotic form of the fields. ϵ ϵ¯ A G½ for nonconstant functions ðx Þ on the sphere will This expression would vanish if we had not allowed a generically be different from zero. gradient term ∂iΦ in Ai. But with such a term, the variation To complete the description of the asymptotic behavior, of the symplectic form is generically nonzero and the we need to specify the falloff of the Lagrange multiplier At. boosts are accordingly noncanonical transformations. Since A parametrizes the gauge transformation performed ¯ t In 3 þ 1 dimensions, the coefficient of the dV∂AΦ-term in the course of the evolution, we take for At the same reduces to d D¯ AA¯ because the contribution from the ϵ V r falloff as for the gauge parameter , angular part is subleading. This enables one to integrate by parts and replace the surface term by 1 ð0Þ A ¯ A −1 I At ¼ At ðx Þþ Atðx Þþoðr Þ: ð3:5Þ ffiffiffi r 2 p ¯ ¯ A ¯ dVðibΩÞ¼ d x γ¯dV ArD ðbdVAAÞ 0 If Að ÞðxAÞ ≠ 0, the time evolution involves a nontrivial t 3 1 improper gauge transformation. ðD ¼ þ dimensionsÞ: ð4:7Þ

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¯ By introducing a single surface d.o.f. Ψ that transforms where Ψ and πΨ are new fields which behave at infinity as appropriatelyH underpffiffiffi boosts and adding the surface contribution − d2x γ¯d A¯ d Ψ¯ to the symplectic form, one can Ψ¯ Ψð1Þ V r V Ψ ¼ þ þ oðr−2Þ; ð5:2Þ make the Lorentz boosts canonical [36]. r r2 A similar route can be followed here provided ¯ ¯ 1 1 AA ¼ ∂AΘ. The variation of the bulk symplectic form ð Þ −4 πΨ ¼ π þ oðr Þð5:3Þ can then be transformed into r4 Ψ I pffiffiffi (in Cartesian coordinates) and where λ is a Lagrange Ω 3 γ¯ ¯ Θ¯ ¯ A ∂ Φ¯ dV ðib Þ¼ d x dVðAr þ ÞD ðbdV A Þð4:8Þ multiplier for the constraint ¯ and by introducing a surface d.o.f. Ψ at infinity, trans- πΨ ≈ 0: ð5:4Þ forming under boosts as Since πΨ is constrained to vanish on shell, its precise decay ¯ ¯ A ¯ at infinity can in fact be strengthened without eliminating δbΨ ¼ D ðb∂AΦÞ; ð4:9Þ classical solutions. ¯ ¯ and adding the surface term We have written explicitly the zero mode Θ0 of Θ among I the arguments of the action functional to emphasize that SH pffiffiffi depends not just on A¯ ¼ ∂ Θ¯ , but on the full Θ¯ . Shifts of − d3x γ¯d A¯ Θ¯ d Ψ¯ ; 4:10 A A Vð r þ Þ V ð Þ Θ by constants will be seen to be symmetries. The equations of motion that follow from the action are one finds that the boosts are canonical (see the detailed the original equations of motion for the original fields Ai, computation in Sec. VB below). πi, which imply The discussion proceeds in fact as in the 3 þ 1 case, with ¯ the 3 þ 1 gauge-invariant A replaced by the 4 þ 1 gauge- ∂ ðA¯ þ Θ¯ Þ¼0; ð5:5Þ ¯ ¯ r t r invariant Ar þ Θ. The development parallels exactly the discussion in 4 dimensions and is given in the next section. an equation which also follows by varying Ψ. One also gets There is, however, one important difference with respect ¯ to the 3 þ 1-dimensional case: it is that both even and odd ∂tΨ ¼ 0 ð5:6Þ parts of the the field Ψ¯ carry physical d.o.f. This is because ¯ ¯ ¯ the coefficient Ar þ Θ of dVΨ in (4.10) has no particular by varying with respect to the vector potential. This parity property (while it is odd in 3 þ 1 dimensions). equation is compatible with the equation obtained by 1 varying with respect to πΨ, provided λ ∼ 2, which we 1 r λ V. COMPLETE FORMULATION shall assume. One can in fact allow for a r-term in if one introduces at the same time a nonvanishing surface We shall thus give here only the salient features. At this Hamiltonian that reflects that the motion would then Ψ¯ stage, the field is a field living on the 3-sphere at infinity, involve an improper gauge transformation for Ψ¯ (see 3 1 which can depend on time. As in þ dimensions, one can Sec. VI), but we shall not do so here. Finally, varying extend it inside the bulk to a “normal” field with conjugate π with respect to the Lagrange multipliers implies the momentum Ψ constrained to vanish. It is to this formu- constraints. lation that we shall immediately proceed.

A. Action B. Poincaré charges The complete action is With the boundary modification of the symplectic charges, all Poincaré transformations are canonical trans- i ¯ SH½Ai; π ; Ψ; πΨ; Θ0; At; λ formations with a well-defined generator. This generator Z Z can be written in terms of local diffeomorphism generators 4 i ¼ dt d xπ ∂tAi þ πΨ ∂tΨ in the following way: I Z ffiffiffi 3 p 4 EM i EM EM − γ¯ ¯ Θ¯ ∂ Ψ¯ i ξH ξ H B d x ðAr þ Þ t Pξ;ξ ¼ d xð þ i Þþ ðξ;ξiÞ; ð5:7Þ Z pffiffiffi 1 g pffiffiffi − 4 ffiffiffi πiπ ij 1 g d x p i þ F Fij HEM −Ψ∂ πi − ∇iπ π πi ij 2 g 4 ¼ i Ai Ψ þ pffiffiffi i þ FijF ; ð5:8Þ Z 2 g 4 − 4 λπ G d xð Ψ þ At Þ ; ð5:1Þ HEM πj − ∂ πj π ∂ Ψ i ¼ Fij Ai j þ Ψ i ; ð5:9Þ

125006-7 MARC HENNEAUX and CEDRIC´ TROESSAERT PHYS. REV. D 99, 125006 (2019) Z I I ffiffiffi Ωbulk − 4 ξHEM − 3 π¯ r Ψ¯ EM 3 ¯ r p ¯ ¯ A ¯ ¯ ib ¼ dV d x d xbðdV Þ B i ¼ d xfbðΨπ¯ þ γ¯∂ ΦD ðA þ ΘÞÞ ξ;ξ A r I pffiffiffi pffiffiffi A ∂ Φ¯ π¯ r γ¯Ψ¯ ∂ ¯ Θ¯ 3 ¯ A ¯ ¯ ¯ þY ð A þ AðAr þ ÞÞg: ð5:10Þ − d x γ¯bD ðAr þ ΘÞdV∂AΦ ð5:19Þ

As is well known, transformations of the fields under a and symmetry are defined up to a gauge transformation in any I gauge theory. The choice implicitly made in (5.7) leads to a Ωboundary − 3 π¯ r Ψ¯ simple algebra. For the kinematical transformations (spatial ib ¼ d xb dV I translations and rotations), it is such that the action of these ffiffiffi 3 p ¯ A ¯ ¯ ¯ spatial symmetries on the fields is the ordinary Lie − d x γ¯bD ðdVðAr þ ΘÞÞ∂AΦ; ð5:20Þ derivative, i.e., so that δξk Ai ¼ Lξk Ai; δξk Ψ ¼ Lξk Ψ; ð5:11Þ Z I k Ω − 4 ξHEM 3 π¯ rΨ¯ where Ψ is a spatial scalar so that LξΨ ¼ ξ ∂kΨ and where ib ¼ dV d x þ d xb the spatial vector ξk is given by (2.21). Note that the I ¯ ffiffiffi Poincaré charges are invariant under shifts of Θ by 3 p þ d x γ¯bD¯ AðA¯ þ Θ¯ Þ∂ Φ¯ : ð5:21Þ constants. r A To illustrate the derivation, consider the boosts (ξ ¼ br, ξi 0 ¼ ). The variations of all the canonical fields are In agreement with (5.7), ibΩ is the exterior derivative of given by minus the canonical generator of the transformation. Similarly, one finds for spatial rotations (ξ ¼ 0, ξr ¼ 0, A A ξπi ξ ¼ Y ), δ A ¼ pffiffiffi þ ∂ ðξΨÞ; ð5:12Þ b i g i j j δRAi ¼ ξ ∂jAi þ ∂iξ Aj; ð5:22Þ ffiffiffi i mip i δbπ ¼ ∂mðF gξÞþξ∇ πΨ ð5:13Þ i m i i m δRπ ¼ ∂mðξ π Þ − ∂mξ π ð5:23Þ i i δbΨ ¼ ∇ ðξAiÞ; δbπΨ ¼ ξ∂iπ ; ð5:14Þ i i δRΨ ¼ ξ ∂iΨ; δRπΨ ¼ ∂iðξ πΨÞ; ð5:24Þ from which one gets from which one gets bπ¯ r δ A¯ Θ¯ ffiffiffi ; δ Ψ¯ D¯ A b∂ Φ¯ ; : δ ¯ Θ¯ A∂ ¯ Θ¯ δ Ψ¯ A∂ Ψ¯ bð r þ Þ¼pγ¯ b ¼ ð A Þ ð5 15Þ RðAr þ Þ¼Y AðAr þ Þ; R ¼ Y A ð5:25Þ ¯ (with again a definite choice of the constant in δRΘ). It upon appropriate choice of the constant that characterizes follows that the ambiguity in Θ¯ . Under these transformations, the symplectic 2-form Z I Ωbulk − 4 ξiHEM 3 ξAπ¯ r∂ Φ¯ iR ¼ dV d x i þ d x A ð5:26Þ Ω ¼ Ωbulk þ Ωboundary; ð5:16Þ and that with I ffiffiffi Z 3 p ¯ i Ωboundary ¼ −d d x γ¯YA∂ ðA¯ þ Θ¯ ÞΨ ; ð5:27Þ bulk 4 i R V A r Ω ¼ d xðdVπ dVAi þ dVπΨdVΨÞð5:17Þ

yielding the above generator for spatial rotations. Note that and here, the bulk and boundary contributions to the symplectic I form are separately invariant. ffiffiffi 3 p ¯ Ωboundary ¼ − d x γ¯d ðA¯ þ Θ¯ Þd Ψ; ð5:18Þ The computation of the time and spatial translations is V r V simpler and leads to generators that have only a bulk piece. This is because the relevant leading orders are invariant is invariant, δbΩ ¼ dVibΩ ¼ 0, since one finds [it should be observed that for spatial translations

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¯ ¯ ¯ A ¯ ¯ δWAr ¼ −∂AΦD W and one can take δWΘ ¼þ∂AΦ asymptotic symmetries. These are transformations that shift ¯ A ¯ ¯ ¯ D W so that δWðAr þ ΘÞ¼0]. Ψ by an arbitrary (time-independent) function μ¯ of the angles. They can be extended in the bulk as C. Poincaré algebra δμΨ μ; δμ anything else 0; 6:2 In addition to the Poincaré symmetries, the theory is ¼ ð Þ¼ ð Þ invariant under with Θ¯ → Θ¯ þ c ð5:28Þ μ¯ (everything else fixed), where c is a constant. This trans- μ ¼ þ oðr−1Þ: ð6:3Þ formation is generated by r I ffiffiffi These are easily checked to leave the action invariant. [One 3 p ¯ J ¼ d x γ¯Ψ ð5:29Þ can include subleading terms in μ, parametrized by arbitrary functions of the angles and of time, but these are proper gauge transformations leaving the action invariant since one has λ δ λ ∂ μ I provided is transformed as μ ¼ t .] pffiffiffi These transformations are canonical transformations Ω Ωboundary − 3 γ¯ Ψ¯ − iJ ¼ iJ ¼ d x cdV ¼ dVðcJÞ: ð5:30Þ with canonical generator given by Z ¯ This quantity vanishes in 4 spacetime dimensions where Ψ 4 G½μ¼ d xμπΨ þ B; is odd due to the parity conditions (and no “naked” Θ¯ appears anyway). μ ∼ 1 One can easily compute the algebra of the various where the surface term is necessary when r, even generators. One finds though the corresponding bulk constraint generator is algebraic. This is just as in 3 þ 1 dimensions. P i ;P i Pˆ î ; 5:31 One finds explicitly f ξ1;ξ1 ξ2;ξ2 g¼ ξ;ξ ð Þ Z ˆ i i ξ ¼ ξ1∂ ξ2 − ξ2∂ ξ1; ð5:32Þ bulk 4 i i iμΩ ¼ −dV d xμπΨ ; ð6:4Þ

î j i j i ij ξ ¼ ξ ∂ ξ − ξ ∂ ξ þ g ðξ1∂ ξ2 − ξ1∂ ξ2Þ; ð5:33Þ I 1 j 2 2 j 1 j j ffiffiffi boundary 3 p ¯ ¯ iμΩ ¼ −d − d x γ¯ðA þ ΘÞμ¯ ; ð6:5Þ which is the Poincaré algebra. V r Furthermore, yielding as generator fPξ ξi ;Jg¼0: ð5:34Þ Z I ; ffiffiffi 4 3 p ¯ ¯ G½μ¼ d xμπΨ − d x γ¯ðAr þ ΘÞμ¯; ð6:6Þ VI. ASYMPTOTIC SYMMETRIES A. Two sets of angle-dependent uð1Þ symmetries an expression that reduces on shell to I We have already identified above an infinite set of pffiffiffi μ − 3 γ¯ ¯ Θ¯ μ¯ asymptotic symmetries, which are gauge transformations G½ ¼ d x ðAr þ Þ : ð6:7Þ parametrized by a gauge parameter that tends to a function ϵ¯ on the 3-sphere at infinity. Contrary to the situation encoun- The generator G½μ, which generically does not vanish, is 3 1 tered in þ dimensions, all of these transformations are invariant under proper and improper gauge transformations. improper gauge transformations, i.e., nontrivial symmetry ¯ ¯ ¯ One way to see this is to recall that ∂AðAr þ ΘÞ¼FrA,so transformations that generically change the physical state of ¯ that the gauge-invariant asymptotic field strength FrA the system, independently of their parity properties. This ¯ Θ¯ 1 determines Ar þ up to a constant, and we fix the variation brings a full angle-dependent uð Þ set of asymptotic sym- Θ¯ δ ¯ −δΘ¯ metries, with charge-generator equal (on shell) to of the constant in so that Ar ¼ . We thus come to the remarkable conclusion that the I theory is invariant under angle-dependent u 1 ⊕ u 1 3 ð Þ ð Þ G½ϵ¼ d xϵ¯π¯ r: ð6:1Þ asymptotic symmetries, with generators given on shell by I ¯ pffiffiffi The introduction of the surface field Ψ at infinity brings 3 r ¯ ¯ Gμ;ϵ ¼ d xðϵ¯π¯ − γ¯ðAr þ ΘÞμ¯Þ: ð6:8Þ in an independent second set of angle-dependent uð1Þ

125006-9 MARC HENNEAUX and CEDRIC´ TROESSAERT PHYS. REV. D 99, 125006 (2019) R 1 2 2 B. Algebra added to the motion generated by 2 ðE þ B Þ. 1 The algebra of the global symmetries with themselves [This Oð Þ-piece can easily be reinserted if needed, and the Poincar´e generators are easily worked out. One e.g., in discussing black hole thermodynamics.] Ψ finds (ii) With the identification At ¼ , the action reduces to

i 0 Z Z fGμ;ϵ;Pξ;ξ g¼Gμˆ;ϵˆ; fGμ1;ϵ1 ;Gμ2;ϵ2 g¼ ; ð6:9Þ i 0 4 i 0 SH½Ai;π ;At;π ;λ¼ dt d xπ ∂tAi þπ ∂tAt μˆ −∇i ξ∂ ϵ − ξi∂ μ ϵˆ −ξμ − ξi∂ ϵ ¼ ð i Þ i ; ¼ i : ð6:10Þ I pffiffiffi − d3x γ¯ðA¯ þΘ¯ Þ∂ A¯ It follows from these equations that the algebra of the r t t symmetries is a semidirect sum of the Poincaré algebra and Z pffiffiffi 4 1 g the Abelian algebra parametrized by μ¯ and ϵ¯. The action of − d x pffiffiffiπiπ þ FijF 2 g i 4 ij the Poincaré subalgebra characterizing this semidirect sum Z can be read off from (6.9): 4 0 − d xðλπ þAtGÞ ; ð6:16Þ δ μ¯ A∂ μ¯ ¯ ¯ Aϵ¯ ðY;b;T;WÞ ¼ Y A þ DAðbD Þ; ð6:11Þ

δ ϵ¯ A∂ ϵ¯ μ¯ π0 ≡ π ðY;b;T;WÞ ¼ Y A þ b : ð6:12Þ where Ψ. (iii) The asymptotic equation ∂ A¯ ¼ 0, which follows ¯ t t One also finds by direct computation from ∂tΨt ¼ 0, is asymptotically equivalent to the ∂t ∂i 0 ∂t 1 I Lorenz gauge At þ Ai ¼ , since At ¼ OðrÞ, ffiffiffi i 1 3 p while ∂ Ai ¼ Oð 2Þ. Gμ ϵ;J − d x γμ¯; 6:13 r f ; g¼ ð Þ The action (6.16) is in fact the action one obtains by direct application of the Dirac constrained Hamiltonian formal- i.e., a central charge appears in this Poisson bracket ism. The primary constraints (here π0 ¼ 0) are enforced relation. explicitly with their own Lagrange multipliers (here λ). The It should be observed that the even-parity component secondary constraints (here G ¼ 0) follow from the pres- (under the antipodal map) of ϵ¯ and the odd-parity compo- ervation in time of the primary constraints. The parameters nent of μ¯ transform into each other under Poincaré trans- of the gauge transformations generated by π0 and G are formations. These are the only nontrivial components related in this “unextended formulation” by ϵ_ ¼ μ (see present in 4 spacetime dimensions. Similarly, the odd- more in [62] and also [63]). parity component of ϵ¯ and the even-parity component of μ¯ For that reason, the introduction of Ψ is not really the transform also into each other. These define proper gauge introduction of a new physical d.o.f. It corresponds rather to transformations in 4 spacetime dimensions, but improper 1 the explicit recognition that the OðrÞ-part of A0 is not pure ones in d>4 spacetime dimensions when no parity gauge, which must therefore be kept. It would be incorrect condition need be imposed on the asymptotic fields. to set it equal to zero since this would require the use of an ¯ improper gauge transformation. That shifts of At are not C. Time evolution proper gauge transformations is reflected throughR the d4xμπ0 − Up to now, the field A is completely arbitrary. A definite factH p thatffiffiffi the corresponding generator is t d3x γ¯ A¯ Θ¯ μ¯ choice of A amounts to a choice of which gauge trans- ð r þ Þ and does not generically vanish, even t on shell. The advantage of including the Ψ field is that this formation accompanies theR time evolution generated by the 1 2 2 important property is clearly put into the foreground. But standard Hamiltonian 2 ðE þ B Þ. It is convenient to choose once understood, one can equivalently work with the more familiar action (6.16). This action reproduces the standard A ¼ Ψ ð6:14Þ Maxwell action (up to a surface term) with Aμ as the sole t dynamical variable if one performs the Legendre trans- and we shall adopt this condition in the sequel. This has a formation in reverse, back to the Lagrange formalism. number of consequences: (i) The behavior of At at spatial infinity takes the form VII. MORE THAN 5 SPACETIME DIMENSIONS 1 The extension of the formalism to other dimensions A ¼ A¯ ðxAÞþoðr−1Þ; ð6:15Þ “ ” t r t higher than 4 is direct and remarkably uneventful. There is in particular no difference between odd and ð0Þ A i.e., the Oð1Þ-piece At ðx Þ is set to zero, which even spacetime dimensions. We give below the relevant means that no improper gauge transformation is formulas.

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A. Action, boundary conditions, Poincaré generators The Poincaré generators are The action generalizing (6.16) is Z d−1 EM i EM EM i ξH ξ H B Z Z Pξ;ξ ¼ d xð þ i Þþ ðξ;ξiÞ; ð7:12Þ i 0 ¯ d−1 i 0 S ½A ;π ;A ;π ;Θ0;λ¼ dt d xπ ∂ A þπ ∂ A H i t t i t t pffiffiffi I 1 g ffiffiffi HEM ¼−A ∂ πi −A ∇iπ0 þ ffiffiffiπ πi þ F Fij; ð7:13Þ d−2 p ¯ ¯ ¯ t i i 2p i 4 ij − d x γ¯ðAr þðd−4ÞΘÞ∂tAt g Z pffiffiffi 1 g HEM πj − ∂ πj π0∂ d−1 i ij i ¼ Fij Ai j þ iAt; ð7:14Þ − d x pffiffiffiπ πi þ F Fij 2 g 4 I Z ffiffiffi EM d−2 ¯ r p ¯ ¯ A d−1 0 B d x b A π¯ γ¯∂ ΦD A d−4 Θ − d xðλπ þA GÞ : ð7:1Þ ξ;ξi ¼ ð ð t þ A ð r þð Þ ÞÞÞ t I pffiffiffi þ dd−2xðYAð∂ Φ¯ π¯ r þ γ¯A¯ ∂ ðA¯ þðd−4ÞΘ¯ ÞÞÞ: The asymptotic conditions are (at a given time) A t A r 1 1 ð7:15Þ ∂ Φ ¯ ð1Þ 2−d Ar ¼ r þ d−3 Ar þ d−2 Ar þ oðr Þ; ð7:2Þ r r There is also the symmetry Θ¯ → Θ¯ þ c generated by 1 I πr π¯ r πð1Þr −1 ffiffiffi ¼ þ þ oðr Þ; ð7:3Þ d−2 p ¯ r J ¼ d x γ¯At: ð7:16Þ

1 1 1 A ¼ ∂ Φ þ ∂ Θ¯ þ Að Þ þ oðr3−dÞ; ð7:4Þ A A rd−4 A rd−3 A 1 B. Asymptotic symmetries πA ¼ πð1ÞA þ oðr−2Þ; ð7:5Þ r2 The theory has gauge symmetries generated by Z 1 1 ð2Þ 1 ðd−4Þ 4− d−1 0 ¯ A d Gϵ μ ¼ d xðϵG þ μπ Þ At ¼ Atðx Þþ 2 At þ d−4 At þ oðr Þð7:6Þ ; r r r I pffiffiffi 1 þ dd−2xðϵ¯π¯ r − γ¯μ¯ðA¯ þðd − 4ÞΘ¯ ÞÞ ð7:17Þ π0 ¼ π0 þ oðr−1Þð7:7Þ r r ð1Þ I pffiffiffi 1 1 ≈ d−2 ϵ¯π¯ r − γ¯μ¯ ¯ − 4 Θ¯ Φ ¼ Φ¯ þ Φð1Þ þþ Φðd−5Þ ð7:8Þ d xð ðAr þðd Þ ÞÞ; ð7:18Þ r rd−5 and involve no parity conditions (d>4). The Lagrange with multiplier λ fulfills 1 1 ϵ ¼ ϵ¯ðxAÞþ ϵð1Þ þ O ð7:19Þ 1 r r2 λ ¼ O : ð7:9Þ r2 and We stress that we have given the asymptotic behavior of the fields, in particular of the density π0, in polar coordinates μ¯ðxAÞ 1 μ ¼ þ O : ð7:20Þ (in Cartesian coordinates, π0 ∼ r1−d). r r2 The equations of motion imply ¯ The action is invariant provided the following (noninde- ∂tAt ¼ 0; ð7:10Þ pendent) conditions hold: (i) ∂tϵ¯ ¼ 0, (ii) ∂tμ¯ ¼ 0, which is equivalent to and (iii) ∂tϵ ¼ μ. 1 The unwritten Oð 2Þ-terms in (7.19) are arbitrary func- 1 r ∂μ tions of time and define proper gauge transformations. The Aμ ¼ O 2 ð7:11Þ 1 r corresponding Oðr2Þ-terms in (7.20) are determined by the condition μ ¼ ϵ_ (in the unextended canonical formalism 1 [instead of the expected OðrÞ that would follow from a considered here). One could use these proper gauge trans- generic time dependence]. The Lorenz gauge is therefore formations to eliminate all terms in Φ but the leading one, fulfilled asymptotically. but this is not necessary.

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ϵ¯ A 1 By contrast, the leading terms parametrized by ðx Þ and η ffiffiffiffiffiffiffiffiffiffiffiffis η ffiffiffiffiffiffiffiffiffiffiffiffi μ¯ A t ¼ p 2 ;r¼ p 2 : ð8:4Þ ðx Þ, which are arbitrary time-independent functions of 1 − s 1 − s the angles restricted by no parity condition (d>4) define improper gauge transformations. The asymptotic sym- The use of hyperbolic coordinates in this context is known metries are thus the direct sum of two independent sets to be very useful [27,52–57]. of angle-dependent uð1Þ transformations. Note that ϵð1Þ is The hypersurface s ¼ 0 in hyperbolic coordinates, on ð1Þ A determined by the condition ∂tϵ ¼ μ to be ϵ ¼ tμ¯ðx Þ which η ¼ r, coincides with the hypersurface t ¼ 0 and is and that the generator (7.17)–(7.18) is just the correspond- therefore a Cauchy hypersurface on which we have already ing Noether charge. studied the behavior of the fields at infinity. This leads to The algebra of the generators of asymptotic symmetries the following falloff for the electromagnetic potential: and Poincaré transformations is the same as in Sec. VI B. ∂2ϵ 1 ¯ −dþ3 −dþ2 We close this section by observing that t ¼ Oð 2Þ is Aη ¼ ∂ηΦ þ Aηη þ Oðη Þ; ð8:5Þ □ϵ 1 □ϵr 0 equivalent to ¼ Oðr2Þ. Now, the condition ¼ defines the residual gauge freedom in the Lorenz gauge. ¯ −dþ4 −dþ3 Aa ¼ ∂aΦ þ Aaη þ Oðη Þ; ð8:6Þ The asymptotic symmetries are therefore residual gauge transformations of the asymptotic Lorenz gauge. What with field strengths taking the asymptotic form makes them nontrivial symmetry transformations, however, is not that they are residual gauge transformations for some −dþ3 ¯ ¯ Eb ¼ Fbη ¼ η ð∂bAη þðd − 4ÞAbÞ gauge conditions, but that their canonical generator has a −dþ2 nonvanishing value. Note in that respect that one can þ Oðη Þ; ð8:7Þ impose the Lorentz gauge to all orders. In that case, the η−dþ4 ¯ η−dþ3 residual gauge transformations fulfill □ϵ ¼ 0 to all orders. Fab ¼ Fab þ Oð Þ: ð8:8Þ The generalP solution to that equation that starts like ϵ ¼ ϵ¯ tμ¯ ϵðnÞ ϵðnÞ ∼ n ϵðnÞ Here, þ r þ n≥2 rn will have generically t . Each will involve two new integration “constants” C ðxAÞ N 1 (functions of the angles), but these integration constants Φ ¼ Φ¯ þþ Φðd−5Þ: ð8:9Þ correspond to pure gauge transformations, even though ηd−5 they are associated with residual gauge transformations. Note that the temporal component As of the vector potential in hyperbolic coordinates is boosted by one power of η with VIII. CONNECTION WITH NULL INFINITY ¯ respect to At. As above, we will also assume Fab ¼ 0 (i.e., The formulation on Cauchy hyperplanes in Minkowski FsA ¼ 0 and FAB ¼ 0), which means that there exists a spacetime Rd−1;1 ≃ R × Rd−1 is complete and fully spec- ¯ boundary field Θ such that Aa ¼ ∂aΘ. ifies the system. It is self-contained and sufficient to answer The equations of motion are given by all dynamical questions, including asymptotic ones for t → ∞ or any other limit. a d−5 b d−3 D Ea ¼ 0; η D Fba − ∂ηðη EaÞ¼0; ð8:10Þ A. Hyperbolic coordinates with Eb ≡ Fbη. Here, Da is the covariant derivative It is of interest in particular to derive from the present associated with the hyperboloid metric hab, while latin ab formulation the behavior of the fields as one goes to null indices are raised or lowered with h and hab, respectively. infinity. To that end, we first integrate the equations in Although the asymptotic analysis only needs the leading hyperbolic coordinates [27], where the metric is given by orders, it is instructive to assume that the asymptotic behavior of the fields can be expressed in terms of an 2 2 2 a b η−1 ds ¼ dη þ η habdx dx ; ð8:1Þ expansion in : X 1 γ¯ η−dþ3−k ðkÞ a b − 2 AB A B Eb ¼ Fbη ¼ E ; ð8:11Þ habdx dx ¼ 2 2 ds þ 2 dx dx : ð8:2Þ b ð1 − s Þ 1 − s k≥0 X − 4− ðkÞ ð0Þ Here, hab is the metric on the (d − 1) hyperboloid. The η dþ k 0 Fab ¼ Fab ;Fab ¼ : ð8:12Þ explicit change of variables is k≥0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t η ¼ −t2 þ r2;s¼ : ð8:3Þ This is because the equations of motion imply a decoupling r order by order. Indeed, one gets The hyperbolic patch covers the region r>jtj. The inverse Da ðkÞ 0 Db ðkÞ ðkÞ 0 transformation reads Ea ¼ ; Fba þ kEa ¼ : ð8:13Þ

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Similarly, the Bianchi identity becomes discussed in [64], where the necessary mathematical back- ground was recalled. For the sake of completeness of this − 4 ðkÞ ∂ ðkÞ − ∂ ðkÞ paper, the needed results are reproduced in Appendix B2. ðd þ kÞFab ¼ aEb bEa ; ð8:14Þ As shown in Appendix B2, the general solution to ∂ ðkÞ 0 Eq. (8.23) takes the form ½aFbc ¼ : ð8:15Þ ΞðkÞ ΞPðkÞ ˜ ðλÞ ΞQðkÞ ˜ ðλÞ This leads to the second order equation lm ¼ lm Pn ðsÞþ lm Qn ðsÞ: ð8:24Þ In the limit s → 1, the resulting function ΞðkÞ tends to a DbD EðkÞ − d − 2 EðkÞ k d − 4 k EðkÞ 0 : lm b a ð Þ a þ ð þ Þ a ¼ ð8 16Þ constant. When l ≥ k, the contribution from the P-branch is subleading. containing EðkÞ only. a The complete analysis given in Appendix C leads to We now focus on the component EðkÞ of the curvature. s similar expressions for the other components of the As in the 4D case, this is the most interesting one as it curvature. An interesting feature that emerges from the carries the information about the charges. The analysis of ðkÞ analysis is that the divergenceless condition on Ea implies the other components of the field strength can be found in that most of the zero modes of EðkÞ are zero: Appendix C. s ðkÞ Using the fact that E is divergenceless, the component 0 a ΞPðkÞ 0; ΞQð Þ 0; ΞQðkÞ 0; ∀ k>0: 8:25 a ¼ s of Eq. (8.16) reads 00 ¼ 00 ¼ 00 ¼ ð Þ

ð1 − s2Þ∂2EðkÞ þðd − 6Þs∂ EðkÞ s s s s B. Behavior at null infinity − 4 ðkÞ − A ðkÞ þðd ÞEs DAD Es In order to make the link with null infinity, it is necessary pffiffiffiffiffiffiffiffiffiffiffiffi 2 −1 ðkÞ ρ η 1 − 2 – − ð1 − s Þ kðd − 4 þ kÞEs ¼ 0: ð8:17Þ to rescale the radial coordinate ¼ s [55 57]. Indeed, the hyperbolic coordinates badly describe the limit Making then the following rescaling of the components, to null infinity: the pair ðs; ηÞ always tends to ð1; ∞Þ on all outgoing null geodesics t ¼ r þ b, no matter what b is. By ðkÞ 2 k ðkÞ contrast, the coordinate ρ tends to 2jbj and can distinguish Ξ ¼ð1 − s Þ2Es ; ð8:18Þ between the various null geodesics. The detailed procedure leads to on how to derive the evolution of the Cauchy data as one goes to null infinity is discussed at length in [55–57],to 1 − 2 ∂2ΞðkÞ − 6 2 ∂ ΞðkÞ which we refer the reader. Note the use of the radial ð s Þ s þðd þ kÞs s 1 coordinate ρ there, which goes to zero in the null infin- − k − 1 k d − 4 ΞðkÞ − D DAΞðkÞ 0: 8:19 ð Þð þ Þ A ¼ ð Þ ity limit. A relevant quantity is then the electric density: Using a basis of spherical harmonics for the (d − 2)-sphere ffiffiffi (see Appendix B1), we get pffiffiffiffiffiffi p 7 ρs d−3 2 −dþ2 A −gF ¼ γ¯ρ ð1 − s Þ Esðρ;s;x Þ X ffiffiffi X p 2 −d 2 − ΞðkÞ ΞðkÞ A γ¯ 1 − 2þ ρ kΞðkÞ ¼ lm ðsÞYlmðx Þ; ð8:20Þ ¼ ð s Þ : ð8:26Þ l;m k

A Let us look at the leading term k ¼ 0; separating the two D DAYlm ¼ −lðl þ d − 3ÞYlm; ð8:21Þ branches, we get 2 2 ðkÞ ðkÞ 1− ∂ Ξ −6 2 ∂ Ξ ffiffiffiX −3 ð s Þ s lm þðd þ kÞs s lm pffiffiffiffiffiffi p P 0 d ðd Þ ρs ð Þ 2 −2þ2 ˜ 2 −gF ¼ γ¯ Ξ ð1 − s Þ P Ylm − −1 −4 ΞðkÞ −3 ΞðkÞ 0 lm l ðk Þðkþd Þ lm þlðlþd Þ lm ¼ : ð8:22Þ lm ffiffiffi X −3 p d Q 0 ðd Þ 2 −2þ2 ð Þ ˜ 2 −1 d−3 þ γ¯ð1 − s Þ Ξ Q Ylm þ Oðρ Þ: With λ ¼ k þ 2 (¼ integer or half-integer) and n ¼ l − k, lm l we can rewrite this equation as lm ð8:27Þ ð1 − s2Þ∂2ΞðkÞ þð2λ − 3Þs∂ ΞðkÞ s lm s lm In the limit s → 1, the first term will go to a constant 1 2λ − 1 ΞðkÞ 0 þðn þ Þð þ n Þ lm ¼ : ð8:23Þ while the second one will diverge. Surprisingly, when we look at the subleading contributions in ρ−1, all contribu- ðλÞ ðλÞ This equation is exactly the same as the equation that arises tions from Qn along with the contributions from Pn with in the discussion of the asymptotics of the scalar field n ¼ l − k<0 will diverge in the same way when s → 1.

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The rest of the contribution from the P’s will be subleading In 4 spacetime dimensions, the k ¼ 0 contribution of the when s → 1: Q branch acquires a leading logarithmic divergence [29,36]. Assuming that this contribution is absent, i.e., X X −3 0 pffiffiffiffiffiffi ρ pffiffiffi 2 −d 2 − PðkÞ ðkþd Þ Qð Þ − s γ 1 − 2þ ρ k Ξ ˜ 2 Ξ ¼ 0, then the above expansion is still valid for d ¼ 4. gF ¼ ð s Þ lm Pl−k lm k lm In that case, the remaining Q contributions and the leading −2 d−3 P contribution appear at the same order r . There is no ΞQðkÞ ˜ ðkþ 2 Þ þ lm Ql−k Ylm: ð8:28Þ such singularity in d>4 dimensions and so no need to remove the k ¼ 0 contribution of the Q branch. Performing the change of variables leading to standard Although this is not necessary, one might be tempted to retarded null coordinates, remove the Q branches at all orders along with all the P branch contributions with k ≥ l, in order to get rid of the −d 2− − 1 1 r 2-terms. The only component in F is then at r d, which sffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi ur u ¼ η p ;r¼ η p ; ð8:29Þ encodes as we mentioned the Coulomb part associated with 1 − s2 1 − s2 the parity even branch at the leading order at spatial so that infinity: the P-branch for k ¼ 0. However, when d>4, this contribution is constant along scri and it will not build a u u2 generic function of u. From our analysis, we see that the −d s ¼ 1 þ ; ρ ¼ −2u − ; ð8:30Þ 2 r r relevant order is r , which must be kept. A curiosity in 4 spacetime dimensions is that this is also the order at which (u<0), we get the Coulomb contribution appears.

2 1 − s −1 2 3 ρ 1 − 2 ρ A C. Generalized matching conditions Fur ¼ η Es ¼ ð s Þ Esð ;s;x Þ: ð8:31Þ To come back to the case of 5 spacetime dimensions, our Using the expansion we obtained above, this leads to analysis implies the following asymptotics at null infinity for d ¼ 5: X X d P k λ 2−d−k 2 2 ð Þ ˜ ð Þ Fur ¼ ρ ð1 − s Þ Ξ Pn ðsÞYlm 5 lm A −2 A −3 Fur ¼ Aðu; x Þr þ Cðx Þr þ …; ð8:34Þ kX lmX 2− − 2 d QðkÞ ˜ ðλÞ ρ d k 1 − s 2 Ξ Q s Y : 8:32 1 þ ð Þ lm n ð Þ lm ð Þ ¯ −3 Aðu; xAÞ¼pffiffiffiffiffiffi AðxAÞþOðð−uÞ 2Þ: ð8:35Þ k lm −u Substitution of s and ρ then yields Both C and A¯ have opposite matching conditions since A¯ X ð0Þ ð0Þ d−3 comes from the Q branch, while C comes from the P 2−d Pð0Þ ð 2 Þ 1−d Fur ¼ r Ξ limPn Ylm þ Oðr Þ lm s→1 branch. lm The generalized matching conditions will thus involve X X d d λ Q k −2 2−2 −k ˜ ð Þ ð Þ both even and odd matchings (P is even under the sphere þ r ð−2uÞ ð−2uÞ lim Qn Ξ Ylm s→1 lm antipodal map combined with s → −s,butQ is odd). These k l≥0;m X matching conditions are associated with different powers of ðλÞ PðkÞ −d−1 ˜ Ξ 2 r. The same procedure works in higher dimensions. In 4 þ Pn lm Ylm þ O r : ð8:33Þ ¯ k>l≥0;m dimensions where one must set A ¼ 0, there is only the even component [29,36]. We can thus conclude that in the r−1 expansion, the various η−1 orders in of the Q branch combined with some D. Gauge transformations spherical harmonic component of the P branch will build −d a function of u at order r 2, while the rest of the P branch In hyperbolic coordinates where the s-time is boosted by 2− η will only contribute to a u-constant term at order r d.It one power of , the two types of improper gauge trans- η0 should be noted that whereas there are no fractional powers formations appear at the same order Oð Þ. Technically, this t t k of η in hyperbolic coordinates, fractional powers of r follows from the fact that s ¼ r, so that terms such as ðrÞ , −k k appear in null coordinates, in the case of odd spacetime which decrease as r on constant t-slices, behave as s , dimensions. i.e., as constants on constant s-slices. 2− ϵ The leading term in r d represents the Coulomb part, Gauge transformations are generated by parameters of −d while the leading term in r 2 represents the radiation part. the form Our expansion in null coordinates is in agreement with the ϵ η a ϵ¯ a η−1 DaD ϵ¯ 0 results of [65]. ð ;x Þ¼ ðx ÞþOð Þ; a ¼ ; ð8:36Þ

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¯ where, as we have seen, the Lorenz gauge holds at infinity. The combination Aη þ Θ satisfies the same equation as ϵ¯ The generator is given on-shell by and takes the general form I X ffiffiffiffiffiffi 2 2 p ¯ Θ ΘP ˜ ð Þ ΘQ ˜ ð Þ d−2 sb ¯ ¯ Aη þ ¼ ð l;mP −1ðsÞþ Q −1ðsÞÞYl;m; ð8:44Þ Qϵ ≈ ðd xÞ −hh ½Ebϵ¯ − ðAη þðd − 4ÞΘÞ∂bϵ¯; l l;m l l;m ð8:37Þ ΘP ΘQ Ξð0ÞP where lm and lm can be expressed in terms of lm Ξð0ÞQ where the first term, associated with the gauge parameter and lm . undifferentiated with respect to time and linked with the In the limit s → 1, the ultraspherical functions with flux of the electric field Es ∼ Fηs, generates the first type of n ≥ 0 behave as improper gauge transformations, while the second one, ˜ ð2Þ 2 3 ð2Þ0 ð2Þ1 2 associated with the s-derivative of the gauge parameter, Pn ¼ð1 − s Þ2½Pn þ Pn ð1 − sÞþOðð1 − sÞ Þ; generates the second type of improper gauge transforma- ð8:45Þ tions. The generator Qϵ is conserved. The equation for ϵ¯ can be rewritten as ˜ ð2Þ ð2Þ0 ð2Þ1 2 Qn ¼ Qn þ Qn ð1 − sÞþOðð1 − sÞ Þ: ð8:46Þ 2 2 A ð1 − s Þ∂s ϵ¯ þðd − 4Þs∂sϵ¯ − DAD ϵ¯ ¼ 0; ð8:38Þ Taking into account these asymptotic behaviors, we can which leads to expand the charge in the neighborhood of future null infinity: X ϵ¯ ϵ a X ¼ l;mðsÞYl;mðx Þ; ð8:39Þ 2 −1 ð2Þ Q ð2Þ Q − 1 − 2 ∂ ΘP ˜ Θ ˜ l;m ϵ ¼ ð s Þ ½ sð l;mPl−1ðsÞþ l;mQl−1ðsÞÞ l>0;m 1 − 2 ∂2ϵ − 4 ∂ ϵ ϵP ˜ ð2Þ ϵQ ˜ ð2Þ − Θ ↔ ϵ ð s Þ s l;m þðd Þs s l;m ð l;mPl−1ðsÞþ l;mQl−1ðsÞÞ l l d − 3 ϵ 0: 8:40 − ΘP ϵQ ΘQ ϵP þ ð þ Þ l;m ¼ ð Þ 0;0 0;0 þ 0;0 0;0 ð8:47Þ

1 X λ − 1 2 −1 Q ð2Þ0 Q ð2Þ1 This corresponds to Eq. (B6) with ¼ 2 ðd Þ and ¼ − ð1 − s Þ 2½∂ ðΘ Q þ Θ Q ð1 − sÞ − 1 0 s l;m l−1 l;m l−1 n ¼ l .Forl> , we can write the solution as l>0;m ð2Þ0 2 3 2 ΘP 1 − 2 1 − d−1 d−1 þ l;mP −1 ð s Þ þ Oðð sÞ ÞÞ 2 d−1 P ð2 2Þ Q ð2 2Þ l ϵ ¼ð1 − s Þ2 ϵ P −1 ðsÞþϵ Q −1 ðsÞ : ð8:41Þ l;m l;m l l;m l ϵQ ð2Þ0 ϵQ ð2Þ1 1 − ð l;mQl−1 þ l;mQl−1 ð sÞ ð2Þ0 2 3 2 0 ϵP 1 − 2 1 − − Θ ↔ ϵ The term for l ¼ is given by þ l;mPl−1 ð s Þ þ Oðð sÞ ÞÞ Z − ΘP ϵQ ΘQ ϵP s 0;0 0;0 þ 0;0 0;0 2 d−2 Q ϵ ϵP 1 − 2 ϵ X 0;0ðsÞ¼ 0;0 ð x Þ dx þ 0;0: ð8:42Þ Q Q ð2Þ0 ð2Þ0 1 3 ϵ ΘP − ϵP Θ 1 − 2 0 ¼ ½ ð l;m l;m l;m l;mÞQl−1 Pl−1 þ Oðð sÞ ÞÞ l>0;m As for the radial electric field in the previous section, we − ΘP ϵQ þ ΘQ ϵP : ð8:48Þ have two branches of solutions for ϵ¯ characterized by their 0;0 0;0 0;0 0;0 behavior under parity. The solution parametrized by ϵQ is l;m As expected, the various modes pair up and we do recover odd while the one parametrized by ϵP is even. The odd l;m the two uð1Þ’s. solution tends to a finite function on the sphere in the limit s → 1 , while the even solution tends to zero except for the IX. CONCLUSIONS zero mode l ¼ 0. At first sight, the behavior of these uð1Þ transformations One striking result of our analysis is the emergence of an in the limit s → 1 might be surprising. In particular, it asymptotic symmetry algebra that is the direct sum of two would seem that the P branch falls off too fast and will lead angle-dependent uð1Þ’s for d>4, instead of just one angle- to a zero charge close to null infinity. Let us focus on d ¼ 5 dependent uð1Þ. and look at the behavior of the charges in the limit s → 1. This somewhat unexpected result can be understood as From our previous analysis, we have follows. It is customary to say that in electromagnetism (and for that matter, also in gravity), “the gauge symmetry ¯ ¯ a a ¯ strikes twice.” That is, both temporal and longitudinal Es ¼ ∂sðAη þ ΘÞ; D Ea ¼ D DaðAη þ ΘÞ¼0: components of Aμ are removed by the single gauge 8:43 ð Þ invariance δϵAμ ¼ ∂μϵ. This property follows from the

125006-15 MARC HENNEAUX and CEDRIC´ TROESSAERT PHYS. REV. D 99, 125006 (2019) fact that the gauge symmetry involves both ϵ and ϵ_, which encountered here (absence of parity conditions and are independent at any given time, so that invariance of the expected corresponding enlargement of the asymptotic physical states (defined on a spacelike surface) under gauge symmetry—in that case doubling of the BMS supertrans- transformations yields two conditions, one related to ϵ and lations) should appear in the gravitational context. one related to ϵ_. Classically, this is reflected in the presence of two first-class constraint generators per space point, π0 ≈ ACKNOWLEDGMENTS 0 and G ≈ 0 [62]. We thank Prahar Mitra for conversations that contributed The “obviousR ” angle-dependent uð1Þ symmetry is asso- −1 to initiate this work. This work was partially supported ciated with dd xϵG ≈ 0, which involves spacelike deriv- by the European Research Council Advanced Grant atives and clearly needs to be supplemented by a “High-Spin-Grav” and by the Fonds De La Recherche nonvanishing surface term in order to be a well-defined Scientifique-Belgium (convention IISN 4.4503.15). canonical generator when the gauge parameter ϵ does not go to zero. This surface term can be computed along the lines of Note added.—Recently, we received the interesting preprint [25]. This is the first global angle-dependent uð1ÞRsymmetry. It is traditionally assumed that the generator dd−1xϵπ_ 0, [68] that deals with similar questions. This work confirms 4 being purely algebraic, does not need to be supplemented the results of [36,29] that the parity conditions in d ¼ by a surface term, no matter how its parameter ϵ_ behaves at spacetime dimensions not only make the symplectic form infinity. This would be true with the standard bulk finite but also eliminate the divergence of the fields as one symplectic form, but is incorrect in the present case due goes to null infinity. The preprint [68] also imposes conditions in d>4 that are stronger than the conditions to the surface contributionR to the symplectic structure. As we d−1 ϵπ_ 0 discussed here, which explains why only one angle- have seen, the generator d x must be supplemented 1 by a nonvanishing surface term when ϵ_ goes to a non- dependent uð Þ is found in that work. Similarly, the recent vanishing function of the angles at infinity (at order 1). This is preprint [69] considers matching conditions equivalent to r the standard parity conditions of d ¼ 4 and also finds a the origin of the second angle-dependent global uð1Þ. single u 1 . Each first-class constraint kills locally one canonical ð Þ gauge pair and also brings at infinity its own angle- dependent global uð1Þ. There is therefore symmetry APPENDIX A: MAGNETIC between longitudinal and temporal directions. SOURCES IN d > 4 Another way to understand the necessity of the second Let us assume that there are only magnetic sources. The 1 1 uð Þ comes from Lorentz invariance, once the first uð Þ equations for the electromagnetic field take the form generated by G½ϵ is uncovered. Indeed, we have seen that ϵ μν μν …ν ν …ν while G½ 0¼Q (zero mode ¼ electric charge) is in the ∂μF ¼ 0; ∂μð⋆FÞ 1 n ¼ κ 1 n ;n¼ d − 3; trivial representation of the Poincar´e group, generators G½ϵ with a nontrivial angle-dependent parameter ϵ do not form ðA1Þ by themselves a representation. They transform into the μ 1 μ where we have assumed usual Euclidean coordinates generator G½ of the second uð Þ, so that G½ is needed μ 1 −1 ϵ x ¼ t; x ; …;xd . We will assume that the magnetic by Lorentz invariance once G½ with a nontrivial angle- ν …ν κ 1 n dependent parameter ϵ appears. As we pointed out, one has source is nonzero in a finite region only. The natural − 3 in fact two representations pairing G½ϵ and G½μ of source will be a (d )-dimensional object: opposite parities, namely, ðG½ϵeven;G½μoddÞ and Yν σ ∶ R Σ → Rd; ðG½ϵodd;G½μevenÞ. From the point of view of Lorentz ð Þ × ðA2Þ invariance, the two angle-dependent uð1Þ’s are more Z ν …ν μ ν ν μ μ conveniently split in this way. κ 1 n ðx Þ¼q dY 1 ∧ … ∧ dY n δdðx − Y ðσÞÞ; ðA3Þ In four dimensions, parity conditions eliminate one uð1Þ, ϵodd μeven the one generated by ðG½ ;G½ Þ. At the same time, Σ − 4 ¯ 0 where is a closed spatial manifold of dimension (d ). there is no need for imposing the condition FAB ¼ . This A static configuration will take the following form: enables one to consider magnetic sources with monopole decay at infinity. It has been argued in [66] that a second 0 0 i Y ¼ σ ; ∂σ0 Y ¼ 0; ðA4Þ magnetic uð1Þ was underlying some soft theorems, so there would be also two u 1 ’s in that case. i1…i tij…j −2 ð Þ κ n ¼ 0; ∂ κ n ¼ 0; ðA5Þ It would be of interest to display explicitly the action of i I the second uð1Þ at null infinity and study its quantum … −1 implications. Similarly, the generalization to gravity should κti1 in−1 ðxjÞ¼q dYi1 ∧ … ∧ dYin−1 δd ðxj − YjðσÞÞ; Σ be carried out. This will be done in a forthcoming paper [67], where we shall discuss how features similar to those ðA6Þ

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ΔΨA − − 2 − 1 ΨA where the time dependence drops out as we can evaluate the l ¼ ½lðl þ d Þ l : ðB4Þ integral over σ0. We can compute the “total” charge: Z I −1 … dd xκti1 in−1 ¼ q dYi1 ∧ … ∧ dYin−1 ¼ 0: ðA7Þ The above results for the decomposition of transverse Σ vector fields lead to a set of spherical harmonics for exact In this case, the dynamical equations (A1) are easily solved 2-forms ΘlAB ¼ ∂AΨlB − ∂BΨlA. The action of the Laplacian is given by in terms of a dual potential Bν1…νn :

⋆ 1 ∂ 0 ΔΘ ¼ −½lðl þ d − 2Þ − d þ 3Θ : ðB5Þ Fμν1…νn ¼ðn þ Þ ½μBν1…νn;Bi1…in ¼ ; ðA8Þ lAB lAB

… −1 … Bti1 in−1 ¼ Δ κti1 in−1 : ðA9Þ 2. Ultraspherical polynomials and functions of the ’ Using the relevant Green s function, we get second kind Z … −1 1 … This subsection follows very closely [64]. We have Bti1 in−1 ðt; xÞ¼ dd x0 κti1 in−1 ðx0ÞðA10Þ jx − x0jd−3 chosen to give this information again here (rather than Z referring to the relevant equations in [64]) for the sake of 1 −1 … 2− completeness and for the convenience of the reader. ¼ dd x0κti1 in−1 ðx0ÞþOðr dÞðA11Þ rd−3 The central equation for our analysis is (8.23), i.e.,

2−d 2 2 ðλÞ ðλÞ ¼ Oðr Þ: ðA12Þ ð1 − s Þ∂s Yn þð2λ − 3Þs∂sYn 1 2λ − 1 ðλÞ 0 The corresponding field strength is given by þðn þ Þðn þ ÞYn ¼ ; ðB6Þ

… 1− … − 3 ⋆Fti1 in ¼ Oðr dÞ; ⋆Fi1 inþ1 ¼ 0; ðA13Þ λ d − ¼ k þ 2 ;n¼ l k: ðB7Þ 1−d Fti ¼ Oðr Þ;Fij ¼ 0: ðA14Þ From its definition, the parameter λ is an integer (odd spacetime dimensions) or a half-integer (even spacetime APPENDIX B: SPECIAL FUNCTIONS 1 1 dimensions). Furthermore, λ ≥ 2, the minimum value λ ¼ 2 1 1. Spherical harmonics being achieved for the leading order of the η expansion in 4 Functions on the (n − 1)-sphere can be decomposed in spacetime dimensions (d ¼ 4 and k ¼ 0). This minimum A spherical harmonics Ylðx Þ, where l is the degree of the value has a special status as we shall see (appearance of corresponding polynomial [70]. Spherical harmonics of logarithmic singularities). degree l are eigenfunctions of the Laplacian on the sphere Equation (B6) can be brought into standard form [71] by appropriate changes of variables. The procedure is different Δ − − 2 Yl ¼ lðn þ l ÞYl; ðB1Þ according to whether the parameter n, which is an integer, is and they have the following parity properties under the positive or zero, n ≥ 0, or is negative, n<0. Both cases A → − A occur in our analysis since n ¼ l − k, so that n<0 antipodal map x x : 1 whenever the order k in the η-expansion exceeds the order − A − l A 1 Ylð x Þ¼ð Þ Ylðx Þ: ðB2Þ l of the spherical harmonics. For the leading η-order k ¼ 0, only the case n ≥ 0 occurs. Vectors on the (n − 1)-sphere can also be decomposed in (i) n ≥ 0. spherical harmonics. We have two families: The rescaling (i) longitudinal vector fields are decomposed in longi- ΦA tudinal vector spherical harmonics l . They are ðλÞ 2 λ−1 ðλÞ 1 − 2ψ eigenfunctions of the Laplacian with eigenvalues Yn ðsÞ¼ð s Þ n ðsÞðB8Þ given by brings the equation to the form ΔΦA − − 2 − 2 ΦA Φ ∂ ; l ¼ ½lðl þ d Þ d þ l ; lA ¼ AYl 2 2 ð1 − s Þ∂sψ n − ð2λ þ 1Þs∂sψ n þ nðn þ 2λÞψ n ¼ 0: ðB3Þ ðB9Þ (ii) transverse vector fields are decomposed in trans- λ 1 1 3 ∈ N verse vector spherical harmonics ΨA. They are In our case ( ¼ 2 ; ; 2 ; and n ), this equation l 1 eigenfunctions of the Laplacian with eigenvalues has a polynomial solution. For λ ¼ 2, the equation given by reduces to the Legendre equation and we recover the

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λ − 1 ðλÞ familiar Legendre polynomials. For general > 2, The general solution for Yn ( ∀ n ≥ 0) is then this equation takes a form analyzed, e.g., in [71], and given from (B8) by the polynomial solution is known as an ultraspher- ðλÞ ðλÞ ˜ ðλÞ ˜ ðλÞ ical polynomial or Gegenbauer’s polynomial Pn . Yn ðsÞ¼APn ðsÞþBQn ðsÞ; ðB18Þ These polynomials satisfy ˜ ðλÞ 2 λ−1 ðλÞ Pn ðsÞ¼ð1 − s Þ 2Pn ðsÞ; ðB19Þ ðλÞ n ðλÞ Pn ð−sÞ¼ð−Þ Pn ðsÞ; ðλÞ 2 λ−1 ðλÞ 2λ − 1 ˜ 1 − 2 ðλÞ n þ Qn ðsÞ¼ð s Þ Qn ðsÞ; ðB20Þ Pn ð1Þ¼ ðB10Þ n ˜ ðλÞ n ˜ ðλÞ Pn ð−sÞ¼ð−Þ Pn ðsÞ; ðB21Þ and can be constructed using the following recur- ˜ ðλÞ nþ1 ˜ ðλÞ rence formula: Qn ð−sÞ¼ð−Þ Qn ðsÞ: ðB22Þ ðλÞ 2 λ − 1 ðλÞ ˜ nPn ðsÞ¼ ðn þ ÞsPn−1ðsÞ For all values of λ, the Q branch of Y will dominate 1 λ ˜ ð Þ in the limit s → 1.Ifλ ¼ 2, the Q branch will − ðn þ 2λ − 2ÞP −2ðsÞ;n>1; ðB11Þ n diverge logarithmically while the P˜ branch will be λ λ finite. For all other values of λ (integers and half- Pð ÞðsÞ¼1;Pð ÞðsÞ¼2λs: ðB12Þ 0 1 integers ≥ 1), the Q˜ branch will be finite and will tend to a nonzero constant at s ¼1, while the P˜ ðλÞ branch will go to zero. The function of the second kind Qn is the (ii) n<0. solution of the differential equation (B9), which is In that case, taking into account that k ≥ 1 (in ðλÞ linearly independent of Pn . The full set can be order to have n ¼ l − k<0), one finds that constructed using the same recurrence relation with n þ 2λ − 1 ¼ l þ k þ d − 4 > 0, so that n>1–2λ. 3 a different starting point: Furthermore λ ≥ 2 > 1,

ðλÞ 2 λ − 1 ðλÞ 3 nQn ðsÞ¼ ðnþ ÞsQn−1ðsÞ 0 1 − 2λ λ ≥ >n> ; 2 : − 2λ − 2 ðλÞ 1 ðnþ ÞQn−2ðsÞ;n>; ðB13Þ Z The change of parameters n ¼ −r − 1 s ðλÞ 2 −λ−1 − − 1 0 1 2 … λ 1 − ρ Q0 ðsÞ¼ ð1 − x Þ 2dx; ðB14Þ (r ¼ n ¼ ; ; ; ) and ¼ brings 0 the equation to the form

ðλÞ ðλÞ 2 −λþ1 λ λ Q1 ðsÞ¼2λsQ0 ðsÞ − ð1 − s Þ 2: ðB15Þ 2 2 ð Þ ð Þ ð1 − s Þ∂sYn − ð2ρ þ 1Þs∂sYn ðλÞ They take the general form þ rðr þ 2ρÞYn ¼ 0; ðB23Þ

ðλÞ ðλÞ ðλÞ − − 1 0 1 2 … Qn ðsÞ¼Pn ðsÞQ0 ðsÞ r ¼ n ¼ ; ; ; ; ðB24Þ ðλÞ 2 −λ 1 þ R ðsÞð1 − s Þ þ2; ðB16Þ − 3 n ρ 1 − λ 1 − d ¼ ¼ ðk þ 2 Þ: ðB25Þ ðλÞ where Rn are polynomials of degree n − 1 and λ λ This is again Eq. (B9) for ultraspherical polyno- satisfy Qð Þ −s − nþ1Qð Þ s . For the values of n ð Þ¼ð Þ n ð Þ mials, but with a range of ρ which is not the usual λ relevant for our analysis (half-integers and inte- 1 one since ρ ≤− . This more general equation has ðλÞ 2 gers), the functions of the second kind Qn ðsÞ also been studied in [71], where it was found that the 1 λ 1 diverge at s ¼ .For ¼ 2, the Legendre function pattern is similar: there is a polynomial branch and a of the second kind diverges logarithmically while “second”-class branch, which is actually also poly- the other values of λ lead to nomial when ρ (and thus λ) is a half-integer. We give the solutions directly in terms of the 1 2 λ−1 ðλÞ original parameters λ and n appearing in the ex- limð1−s Þ 2Qn ðsÞ¼ ;n¼ 0;1;… ðB17Þ →1 s 2λ−1 pansion of the fields. The polynomial branch prðsÞ of solutions has the standard parity p ð−sÞ¼ λ 1 r ð Þ 2 −λþ2 r [Qn ðsÞ ∼ ð1 − s Þ ]. ð−Þ prðsÞ, while the other branch (which might

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also be polynomial as we pointed out) fulfills The equations to be solved are − − rþ1 qrð sÞ¼ð Þ qrðsÞ. Therefore, in order to keep X uniformity in the parity properties when expressed in −dþ3−k ðkÞ E ¼ F η ¼ η E ; ðC1Þ ˜ ðλÞ b b b terms of n, we denote the polynomial branch by Qn k≥0 ˜ ðλÞ and the other branch by Pn . X η−dþ4−k ðkÞ ð0Þ 0 The two sets of solutions can be constructed with Fab ¼ Fab ;Fab ¼ ; ðC2Þ the following recurrence relation: k≥0

2λ − 1 ˜ ðλÞ 2 λ ˜ ðλÞ Da ðkÞ 0 ð þ n ÞPn−1ðsÞ¼ ðn þ ÞsPn ðsÞ Ea ¼ ; ðC3Þ − 1 ˜ ðλÞ ≤−2 ðn þ ÞPn 1ðsÞ;n ; ðB26Þ þ ðd − 4 þ kÞFðkÞ ¼ ∂ EðkÞ − ∂ EðkÞ; ðC4Þ Z ab a b b a s ðλÞ 2 λ−3 ˜ 1 − 2 P−1ðsÞ¼ ð x Þ dx; ðB27Þ Db ðkÞ ðkÞ 0 ∂ ðkÞ 0 0 Fba þ kEa ¼ ; ½aFbc ¼ : ðC5Þ

ðλÞ ðλÞ 1 2 λ−1 ˜ ˜ 1 − 2 Combining these equations, we can obtain second order P−2ðsÞ¼sP−1ðsÞþ2 λ − 1 ð s Þ ; ðB28Þ ðkÞ ðkÞ ð Þ equations for both Ea and Fab :

the polynomial branch being given by the starting DaD EðkÞ − ðd − 2ÞEðkÞ þ kðd − 4 þ kÞEðkÞ ¼ 0; ðC6Þ point a b b b ˜ ðλÞ 1 ˜ ðλÞ DaD ðkÞ − 2 − 3 ðkÞ − 4 ðkÞ 0 Q−1ðsÞ¼ ; Q−2ðsÞ¼s: ðB29Þ aFbc ðd ÞFbc þ kðd þ kÞFbc ¼ : ðC7Þ

In order to prove this recurrence, one needs the As we saw in the main text, the first equation combined following relation: ðkÞ with the fact that Ea is divergenceless on the hyperboloid ðkÞ implies an evolution equation for Es : 1 − 2 ∂ ˜ ðλÞ 1 ˜ ðλÞ − ˜ ðλÞ ð s Þ sPn ¼ðn þ ÞðsPn Pnþ1ÞðB30Þ 2 2 ðkÞ ðkÞ ðkÞ ð1 − s Þ∂s Es þðd − 6Þs∂sEs þðd − 4ÞEs and its equivalent in terms of Q˜ . The parity con- A ðkÞ 2 −1 ðkÞ ditions read −DAD Es − ð1 − s Þ kðd − 4 þ kÞEs ¼ 0: ðC8Þ ˜ ðλÞ − − n ˜ ðλÞ Pn ð sÞ¼ð Þ Pn ðsÞ; ðB31Þ In a similar way, the second equation combined with the fact that FðkÞ is closed on the hyperboloid gives us an ˜ ðλÞ nþ1 ˜ ðλÞ ab Qn ð−sÞ¼ð−Þ Qn ðsÞ: ðB32Þ ðkÞ evolution equation for FAB:

1 − 2 ∂2 ðkÞ − 6 ∂ ðkÞ 2 − 4 ðkÞ ðλÞ ð s Þ s FAB þðd Þs sFAB þ ðd ÞFAB The general solution for Yn keeps the form − C ðkÞ − 1 − 2 −1 − 4 ðkÞ 0 D DCFAB ð s Þ kðd þ kÞFAB ¼ : ðC9Þ ðλÞ ˜ ðλÞ ˜ ðλÞ Yn ðsÞ¼APn ðsÞþBQn ðsÞ; The general solution for EðkÞ is given in terms of 0 1 − 2λ s ð >n> Þ: ðB33Þ ultraspherical polynomials as follows: X An important difference from the regime n ≥ 0 is ðkÞ 2 −k PðkÞ ˜ ðλÞ QðkÞ ˜ ðλÞ E ¼ð1 − s Þ 2 ðΞ P þ Ξ Q ÞY ; ðC10Þ that both branches have now the same asymptotic s lm n lm n lm l;m behavior in the limit s → 1: they both tend to a nonzero finite value. In particular, as we already where λ k d−3 and n l − k. In order to obtain the λ ¼ þ 2 ¼ mentioned, both branches are polynomials if is a general solution to Eq. (C9), we will use the same strategy half-integer. ðkÞ and decompose FAB in spherical harmonics: X ðkÞ 2 −k ðkÞ 1 − 2 α Θ APPENDIX C: EXTRA COMPONENTS IN FAB ¼ð s Þ lm lmAB: ðC11Þ 0 HYPERBOLIC DESCRIPTION l> ;m Θ ðkÞ In the main text, we have solved the equations of motion The exact spherical harmonics lmAB is enough as FAB is an s for one component of the curvature Fμν, namely, E . In this exact 2-form on the sphere. Equation (C9) then takes the appendix, we will solve for the other components. form

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1 − 2 ∂2αðkÞ 2 − 6 ∂ αðkÞ − 3 αðkÞ is not a solution of the dynamical equation of the zero mode ð s Þ s lm þðd þ k Þs s lm þ lðl þ d Þ lm of ΞðkÞ. This implies − − 1 − 4 αðkÞ 0 ðk Þðk þ d Þ lm ¼ ; ðC12Þ PðkÞ Pð0Þ k QðkÞ which has the following solution: Ξ00 ¼ Ξ00 δ0; Ξ00 ¼ 0: ðC18Þ αðkÞ αPðkÞ ˜ ðλÞ αQðkÞ ˜ ðλÞ lm ¼ lm Pn ðsÞþ lm Qn ðsÞ; ðC13Þ Using the symbols T and L to respectively denote the transverse and longitudinal part of vectors defined on the with the same λ and n. ðkÞ ðkÞ sphere, we obtain the following expressions: Using the expression for Es and FAB, we can now build the remaining components of the curvature in such a way ðkÞT 2 −k E ¼ðd − 4 þ kÞð1 − s Þ 2 that the main equations are satisfied. This is done by A X expending (C1): αPðkÞ ˜ ðλÞ αQðkÞ ˜ ðλÞ Ψ ð lm Pn þ lm Qn Þ lmA; ðC19Þ l>0;m A ðkÞ 1 − 2 ∂ ðkÞ − 4 ðkÞ D EA ¼ð s Þ sEs þðd ÞsEs ; ðC14Þ X 1 k k k ðkÞL 2 −k ð Þ − ∂ ð Þ − 4 ð Þ E − 1 − s 2 dAEB BEA ¼ðd þ kÞFAB; ðC15Þ A ¼ ð Þ − 3 l>0;m lðl þ d Þ 1 ðkÞ ðkÞ ðkÞ 1 − s2 ∂ d − 4 k s F ¼ ð∂ E − ∂ Es Þ: ðC16Þ ðð Þ s þð þ Þ Þ sA d − 4 þ k s A A ΞPðkÞ ˜ ðλÞ ΞQðkÞ ˜ ðλÞ Φ ð lm Pn þ lm Qn Þ lmA; ðC20Þ The first two lines give respectively the longitudinal and the ðkÞ ðkÞ ðkÞ transverse part of EA in terms of Es and FAB, while the X ðkÞ ðkÞT 2 −k−1 2 1 − 2 1 − ∂ last one then gives the last components FsA in terms of FsA ¼ð s Þ ðð s Þ s þ ksÞ the rest. The only consistency condition coming from these l>0;m equations concerns the zero mode of (C14). Integrating this PðkÞ ðλÞ QðkÞ ðλÞ ðα P˜ þ α Q˜ ÞΨ ; ðC21Þ equation on the sphere, we see that the zero mode of the rhs lm n lm n lmA has to be zero: X 1 ðkÞL 2 −k−1 Z − 1 − 2 FsA ¼ kð s Þ − 3 d−2 2 ðkÞ ðkÞ l>0;m lðl þ d Þ 0 ¼ d Ωfð1 − s Þ∂sEs þðd − 4ÞsEs g ΞPðkÞ ˜ ðλÞ ΞQðkÞ ˜ ðλÞ Φ k ð lm Pn þ lm Qn Þ lmA: ðC22Þ 2 −2 2 ¼ð1 − s Þ ðð1 − s Þ∂s þðd − 4 þ kÞsÞ ΞPðkÞ ˜ ðλÞ ΞQðkÞ ˜ ðλÞ This provides the general solution to the complete system ð 00 P−k þ 00 Q−kÞ: ðC17Þ ðkÞ ðkÞ as the tensors Fab and Ea built in this way satisfy the ΞPðkÞ ˜ ðλÞ ΞQðkÞ ˜ ðλÞ last two equations given in (C5). The second equation is This implies that 00 P−k þ 00 Q−k has to be propor- 2 d−4þk trivially satisfied while the first one can be shown to reduce tional to ð1 − s Þ 2 . One can easily check that, except for to combinations of the two evolution equations (C8) ðλÞ 2 d−4 2 d−4þk k ¼ 0, where P0 ¼ð1 − s Þ 2 , the function ð1 − s Þ 2 and (C9).

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