On asymptotic symmetries in higher dimensions for any spin
NNORDITA 2020-103
On asymptotic symmetries in higher dimensionsfor any spin
Andrea Campoleoni, a, Dario Francia, b,c and Carlo Heissenberg d,e a Service de Physique de l’Univers, Champs et Gravitation, Université de Mons, 20 place du Parc,7000 Mons, Belgium b Centro Studi e Ricerche E. Fermi, Piazza del Viminale 1, 00184 Roma, Italy c Roma Tre University and INFN, Via della Vasca Navale 84, 00146 Roma, Italy d Nordita, Stockholm University and KTH Royal Institute of Technology, Roslagstullsbacken 23,10691 Stockholm, Sweden e Department of Physics and Astronomy, Uppsala University, 75108 Uppsala, Sweden
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We investigate asymptotic symmetries in flat backgrounds of dimension higherthan or equal to four. For spin two we provide the counterpart of the extended BMStransformations found by Campiglia and Laddha in four-dimensional Minkowski space. Wethen identify higher-spin supertranslations and generalised superrotations in any dimen-sion. These symmetries are in one-to-one correspondence with spin- s partially-masslessrepresentations on the celestial sphere, with supertranslations corresponding in particularto the representations with maximal depth. We discuss the definition of the correspondingasymptotic charges and we exploit the supertranslational ones in order to prove the linkwith Weinberg’s soft theorem in even dimensions. Research Associate of the Fund for Scientific Research – FNRS, Belgium. a r X i v : . [ h e p - t h ] J a n ontents A Notation and conventions 21B Geometry of the sphere and polarisations 21
B.1 Properties of the n -sphere 21B.2 Spectrum of ∆ C Symmetries of the Bondi-like gauge 26D Stationary and static solutions of Fronsdal’s equations 29
D.1 Stationary solutions 29D.2 Static solutions 30
In this work we construct higher-spin supertranslations and generalised superrotations atnull infinity, in flat spacetimes of any dimension D ≥ . We thus extend the resultsof [1], where higher-spin supertranslations and superrotations have been identified in fourdimensions, and of [2], where global higher-spin symmetries have been studied in any D > .Following the seminal works [3–5], the asymptotic symmetry group of four-dimensionalasymptotically flat gravity, and later of spin-one gauge theories [6–9], was long identified ascomprising those transformations of the gauge potentials that preserve the falloffs typicalof radiation, where the norm of the corresponding fields scales to leading order as r − inretarded Bondi coordinates. (See [10] for a review.) However, in striking contrast with thefour-dimensional case, imposing the same requirement in higher-dimensional gravity, whereradiation scales asymptotically as r − D , effectively selects only the (global) transformationsof the Poincaré group within the full group of diffeomorphisms, thus apparently preventingBMS D> to be identified as a physically sensible asymptotic group [11, 12]. The absenceof gravitational memory effects to radiative order beyond D = 4 [13], moreover, providedfurther support to the idea that D = 4 was to be regarded as possessing a special status– 1 –s for what concerns the asymptotic structure of asymptotically flat spacetimes. In thesame fashion, in theories of photons or gluons whose associated potentials decay at nullinfinity as fast as r − D only global U (1) or SU ( N ) transformations are kept asymptotically.Superrotations, in their turn, were originally identified as the infinite-dimensional familyof vector fields providing local solutions to the conformal Killing equation on the two-dimensional celestial sphere [14, 15]. In this sense, their very existence appeared to besomewhat specific of four-dimensional Minkowski space.A different view was advocated for flat spaces in [16–29]. The interpretation of Wein-berg’s soft theorems as Ward identities of asymptotic symmetries in D = 4 rather naturallycalled for a similar correspondence in higher dimensions, thus suggesting the existence ofrelevant symmetries beyond the global ones. This picture eventually found two differentincarnations. Supertranslations were first recovered in any D by weakening the falloffs ofthe fields so as to match those of the four-dimensional case [16, 17, 23–25]. Memory effects,in their turn, were better identified as due to the leading components of the stationarysolutions of the field equations, whose typical Coulombic scale of O ( r − D ) is subleadingwith respect to the radiation falloffs in any D > . While in agreement with the observedabsence of memory effects to leading-order, the identification of such higher-dimensional,subleading, memory effects also led to consider another class of residual gauge symmetriesakin to supertranslations [20, 26] (see also [18, 19]). For spin-one gauge theories the pres-ence of angle-dependent asymptotic symmetries in higher dimensions was also confirmedby an analysis at space-like infinity [30, 31].Similarly, the idea that additional asymptotic symmetries, other than supertranslations,could be held responsible for subleading soft graviton theorems [32, 33] led to identifya different extension of the BMS group in four dimensions as the semidirect product ofsupertranslations and Diff ( S ) [34, 35], differently from the original proposal of [36] thatwould link the subleading soft amplitudes to the Ward identities of the superrotations of[14, 15]. (See also [37] for yet an alternative derivation of subleading soft graviton theorems.)What is relevant to our purposes is that the four-dimensional construction of [34], contraryto that of [32, 36], is amenable to be pursued in any D [38, 39].In the following we apply similar considerations both to low ( s = 1 , ) and to higher-spin ( s > ) gauge theories in D ≥ . In [2, 40] we showed that, if the asymptotic behaviourtypical of radiation is chosen as the leading falloff in D > , the corresponding asymptoticgroup only comprises the solutions to the global Killing equations. By contrast, herewe begin by imposing in any dimension the same falloffs as those allowing (higher-spin)supertranslations in D = 4 [1], i.e. we consider fields whose norm scales asymptotically as r − for any D ≥ . In our Bondi-like gauge (2.1), this choice naturally leads to asymptoticsymmetries depending on an arbitrary function on the celestial sphere, which we identifyas higher-spin counterparts of BMS supertranslations. In addition, we show that, on shell, Similar conclusions have been drawn for higher-spin fields in Anti de Sitter spacetimes in [41, 42]. The authors of [43] identify operators performing spin-dependent supertranslations in any D in theanalysis of the near-horizon symmetries of a black-hole background, although in the (putative) absence ofhigher-spin fields. It is conceivable that the chosen class of spin-dependent boundary conditions effectivelysubsume the presence of higher-spin fields in the corresponding thermal bath. – 2 –ll overleading configurations above the falloffs typical of radiation must be pure gauge andthen, following [26], we propose a prescription to associate finite surface charges to higher-spin supertranslations. These results suggest to interpret the additional overleading termsas new global degrees of freedom. We complete our analysis of higher-spin supertranslationsby showing that Weinberg’s factorisation theorems for soft particles of any spin [44, 45] canbe recovered as Ward identities for these asymptotic symmetries, thus extending to anyeven space-time dimension the results of [1].We then compute the full set of residual symmetries of the Bondi-like gauge, withoutany prior assumption on the allowed decay rates of the fields. In this way we discoverother classes of infinite-dimensional symmetries that depend on arbitrary traceless tensorson the celestial sphere of rank , . . . , s − and, for s = 2 , reduce to the superrotations of[34, 39]. Their scaling with r gets more and more relevant, so that if one wishes to keepall of them the norm of the fields should actually blow up as fast as r s − . However, fieldconfigurations that are overleading with respect to radiation can be shown to be anywaypure gauge on shell and thus, for instance, they won’t affect the decay rate of the higher-spin Weyl tensors that can be kept to be those typical of radiation. Interestingly, eachfamily of asymptotic symmetries appears to be in one-to-one correspondence with partiallymassless representations on the celestial sphere [46–50], identified via the kinetic operatorsruling the dynamics of suitable overleading components of the asymptotic field. Similarlyto what happens for gravity in D = 4 , the generalised superrotation charges diverge in thelimit r → ∞ and should be properly regularised, in the spirit of [51–53]. Here we do notaddress this issue in its full generality and we limit ourselves to discuss the finiteness ofsuperrotation charges when evaluated on special classes of solutions. In this fashion, for s = 2 , we are at least able to make partial contact with the charges employed in [39] torelate the subleading soft graviton theorem and the superrotation Ward identities.Higher-spin gauge theories have long been supposed to rule the high-energy limit ofstring theory and to provide a symmetric phase of the latter, in a regime where the stringtension may be taken as negligible [54]. Whereas the actual import of this tantalisingconjecture will remain elusive as long as a concrete mechanism for implementing higher-spin gauge symmetry breaking is not found, still one may hope to highlight glimpses of suchhypothetical symmetric phase in possible remnants of higher-spin asymptotic symmetries instring scattering amplitudes. This possibility provides one of the main motivations for theidentification of the proper higher-spin asymptotic symmetry group in higher dimensions.The paper is organised as follows: in Section 2 we focus on supertranslations, presentingthe relevant boundary conditions together with our prescription to associate finite surfacecharges to them. The latter are then used to derive Weinberg’s soft theorems for any spin.Some of the relevant results on the structure of asymptotic symmetries are actually provenin Section 3, where the scope of our analysis widens to include higher-spin superrotations.More technical details can be found in the appendices.– 3 – Higher-spin supertranslations and Weinberg’s soft theorem
We consider free gauge fields of spin s on Minkowski spacetime, obeying the Fronsdalequations in the Bondi-like gauge introduced in [1, 2]: ϕ rµ s − = 0 = γ ij ϕ ijµ s − . (2.1)Assuming the asymptotic expansion in retarded Bondi coordinates ϕ u s − k i k ( r, u, ˆx ) = (cid:88) n r − n U i k ( k,n ) ( u, ˆx ) , (2.2)we investigate the asymptotic structure of the gauge symmetries of the form δϕ µ s = ∇ µ (cid:15) µ s − with g αβ (cid:15) αβµ s − = 0 (2.3)preserving (2.1). In [2] it was shown that, assuming falloffs not weaker than those typical ofradiation, i.e. ϕ u s − k i k ϕ u s − k i k = O ( r − D ) or subleading, the resulting asymptotic symmetriesfor any spin in D > comprise only the global solutions to the Killing tensor equations,with no infinite-dimensional enhancement.For low spins, however, the latter can be recovered upon assuming weaker fall-off con-ditions that, in the radial gauge for s = 1 or in the Bondi gauge for s = 2 , essentiallyamount to accepting asymptotic falloffs as weak as O ( r − ) in any D [16, 17]. Whereas theappropriate choice of falloffs is in itself a gauge-dependent issue, at the physical level whatmatters is how to interpret these additional, low-decaying, configurations from the perspec-tive of observables. In [26] we argued that, for s = 1 , no physical inconsistencies arisein considering such weaker falloffs (of the strength needed in the given gauge) as long asall the overleading contributions above the D − dimensional radiation behaviour are (large)pure-gauge configurations. In the following, we shall adopt the same guiding principle. Inthis fashion, certainly no issues can arise for all gauge-invariant quantities, like the flux ofenergy per unit retarded time carried by the electromagnetic field or quantities dependingon the linearised Weyl tensor for spin two and higher. Nevertheless, the presence of theseoverleading field components may be source of subtleties in general, as the definition ofsuperrotation charges to be discussed in Section 3 testifies.With this proviso, in this section we take the same attitude for any spin: we assumeoverall falloffs as weak as ϕ u s − k i k ϕ u s − k i k = O ( r − ) for any s in any D , we then identify the u − independent residual symmetries preserving (2.1) and we argue that above the radiationorder only pure-gauge configurations survive on shell, while leaving to the next section adetailed derivation of these results. Following [1], we identify such symmetries as higher-spincounterparts of BMS supertranslations. Explicitly, upon imposing ϕ u s − k i k = O ( r k − ) (2.4) For s > , the Bondi-like gauge (2.1) is to be interpreted as an on-shell gauge fixing. Indeed, it fixes anumber of conditions larger than the number of independent components of the gauge parameter. See [55, 56] for some comments on this point. For Maxwell fields in the Lorenz gauge, for instance, inorder to identify an infinite-dimensional asymptotic group it is not sufficient to assume falloffs as weak as O ( r − ) and additional terms proportional to log r are needed in any D ≥ [26]. – 4 –he residual parameters of the Bondi-like gauge are indeed expressed in terms of an arbitraryfunction T ( ˆx ) on the celestial sphere. In particular, one first obtains (cid:15) u s − k − i k = r − k ( − s − k − ( s − k − k !( s − D i · · · D i T ( ˆx ) + γ ii D i k − T ( ˆx ) , (2.5)where the D i l are suitable rank − l differential operators. For instance, for s = 3 , one has (cid:15) uu = T ( ˆx ) , (cid:15) ui = − r ∂ i T ( ˆx ) , (cid:15) ij = 12 r (cid:20) D i D j − D γ ij (∆ − (cid:21) T ( ˆx ) . (2.6)As discussed in Section 3.1 and in Appendix C, one can then express the other componentsof the gauge parameter in terms of those displayed above. Looking at u –independentresidual symmetries allowed us to focus on supertranslations; removing this assumptionwhile keeping the falloffs (2.4) one finds in addition only the global symmetries discussedin [2].We now show how to associate finite surface charges to the symmetries (2.5), to beused in the derivation of Weinberg’s soft theorems [44, 45]. In the Bondi-like gauge (2.1),the surface charge at null infinity associated to a gauge transformation is [2] Q ( u ) = lim r →∞ r D − s − (cid:88) k =0 (cid:18) s − k (cid:19)(cid:73) d Ω D − (cid:26) ( s − k − (cid:15) u s − k − i k ( r∂ r + D − ϕ u s − k i k + ϕ u s − k i k ( r∂ r + D + 2 k − (cid:15) u s − k − i k − s − k − r (cid:15) u s − k − i k D · ϕ u s − k − i k (cid:27) , (2.7)which, for D > , naively diverges as r D − if one evaluates it for the symmetries (2.5)on field configurations decaying at null infinity as (2.4). On the other hand, as discussedin Section 3.2, the equations of motion imply that asymptotically all contributions abovethose of a wave solution be pure gauge. On shell one has indeed ϕ u s − k i k = r k − k ( D + k − s ( D + s − D· ) s − k C i k (1 − s ) ( ˆx ) + O (cid:16) r k +1 − D (cid:17) , (2.8)with the rank– s tensor C (1 − s ) given by C i s (1 − s ) ( ˆx ) = [( s − − D i · · · D i ˜ T ( ˆx ) + · · · , (2.9)where ˜ T ( ˆx ) is an arbitrary function and the omitted terms implement the traceless projec-tion of the symmetrised gradients, as required by the constraints (2.1). Substituting (2.5)and (2.8) in the expression for the surface charge one obtains ( − s − Q T ( u )= lim r →∞ r D − (cid:73) d Ω D − s − (cid:88) k =0 r − k k ! T (cid:104) ( s − k − r∂ r + ( s − k − D − k − (cid:105) ( D· ) k ϕ u s − k = lim r →∞ r D − (cid:32) s − (cid:88) k =1 α k (cid:33) (cid:73) d Ω D − T ( D· ) s C (1 − s ) + O ( r D − ) , (2.10) The charge defined in (2.7) is equal to − ( s − times the charge appearing in Appendix A of [2]. – 5 –ith α k = ( D + k − k − s − k −
2) + ( s − k − D − k − k − D + s − . (2.11)Two types of divergences thus arise if one computes the surface charge by first integrating in(2.7) over a sphere at a given retarded time u and radius r and then taking the limit r → ∞ .However, the divergence O ( r D − ) induced by the overleading, pure-gauge contributionsactually vanishes because (cid:80) s − k =1 α k = 0 for any s . The remaining divergence O ( r D − ) isrelated to the presence of radiation: if one assumes that in a neighbourhood of I + − , sayfor u < u , there is no radiation and the fields attain a stationary configuration, then thesurface charge is finite. A finite charge Q T ( u ) can then be defined for all values of u as theevolution of Q T ( −∞ ) under the equations of motion [26]. In order to compute the supertranslation charges, we thus focus on field configura-tions with the falloffs typical of a stationary solution. Generalising the characterisation ofstationary solutions for fields of spin s ≤ of [58], we consider ϕ u s − k i k = r − D + k U i k ( k ) ( u, ˆx ) + · · · . (2.12)As we shall argue in Appendix D, this choice is tantamount to evaluating the charges onsolutions that satisfy ∂ u U ( k,n ) = 0 in the far past of I + for n ≤ D − k − . Moreover, inthe absence of massless sources, on shell, the rank– k tensors U ( k ) satisfy ( D· ) k U ( k ) = 0 for ≤ k ≤ s , (2.13)as can be checked from (D.3) where U ( k ) = U ( k,D − k − . Taking (2.5), (2.12) and (2.13) intoaccount, the surface charge (2.7) reads Q T ( u ) = ( − s − ( D + s − (cid:73) d Ω D − T ( ˆx ) U (0) ( u, ˆx ) , (2.14)which is closely analogous to the expression for the spin- supertranslation charge in termsof the Bondi mass aspect, Q T ∝ (cid:72) d Ω D − T m B .To summarise, assuming that the fields be on shell up to the falloffs of stationarysolutions and defining the charges according to the prescription of [26], one obtains finitesupertranslation charges for any value of s and in any D . Furthermore, let us note that apure supertranslation configuration carries away no energy to I + per unit retarded time, asdefined via the canonical stress-energy tensor t αβ stemming from the Fronsdal Lagrangian,which in the in the gauge (2.1) takes the Maxwell-like form [59, 60] L = − √− g (cid:0) ∇ α ϕ µ s ∇ α ϕ µ s − s ∇ · ϕ µ s − ∇ · ϕ µ s − (cid:1) . (2.15)Indeed, the canonical stress-energy tensor obtained from this Lagrangian reads t αβ = 12 (cid:0) ∇ α ϕ µ s ∇ β ϕ µ s − s ∇ · ϕ µ s − ∇ α ϕ βµ s − (cid:1) + g αβ ( · · · ) , (2.16) See also [57] for an alternative procedure to define finite charges in
D > for angle-dependent asymp-totic symmetries in Maxwell’s electrodynamics and [22] for a discussion of supertranslation charges inhigher-dimensional gravity. For the scalar case, the charges considered in [19] formally coincide with (2.14) evaluated for s = 0 . – 6 –hile the energy flux at a given retarded time u is given by P ( u ) = lim r →∞ (cid:73) ( t uu − t ur ) d Ω D − . (2.17)In the latter expression, the term of the stress-energy tensor (2.16) proportional to thebackground metric g αβ drops out, while the remaining ones involve derivatives with respectto u . Pure supertranslations however are u -independent, and therefore eventually providea vanishing contribution.Let us also rewrite the surface charge evaluated at I + − in terms of an integral over I + according to Q T (cid:12)(cid:12) I + − = Q T (cid:12)(cid:12) I ++ − (cid:90) + ∞−∞ d Q T ( u ) du du , (2.18)where the first contribution accounts for the presence of stable massive particles in thetheory. In their absence, making use of (2.14), one finds Q T (cid:12)(cid:12) I + − = ( − s ( D + s − (cid:90) + ∞−∞ du (cid:73) d Ω D − T ( ˆx ) ∂ u U (0) ( u, ˆx ) . (2.19)We can now connect the charge (2.19) to Weinberg’s soft theorem in even D . As usual, thestrategy is to express the Coulombic contributions appearing in the charge in terms of theradiative contributions making use of the equations of motion, so as to make contact withthe free field oscillators naturally contained in the radiation components. The soft theoremcan then be retrieved by simplifying the insertions of these operators in the correspondingWard identities so as to highlight the factorisation of S -matrix elements that takes place inthe soft limitLet us consider the spin-three case first. The equations of motion in the Bondi-likegauge allow one to express the charge (2.19) in terms of the spin-three generalisation of theBondi news tensor via ∂ D − u U (0) = D ( D· ) C (cid:16) D − (cid:17) ( D − D − D − , (2.20)where the operator D is defined as D = D − (cid:89) l = D D l , with D l = ∆ − ( l − D − l − D − l − . (2.21)One can therefore rewrite the charge (2.19) as follows Q T (cid:12)(cid:12) I + − = − D − D − (cid:90) + ∞−∞ du (cid:73) d Ω D − T ( ˆx ) ∂ − D u D D i D j D k C (cid:16) D − (cid:17) ijk ( u, ˆx )= 14( D − D −
3) lim ω → + (cid:88) λ (cid:73) d Ω D − (2 π ) D − T ( ˆx ) D D i D j D k (cid:15) ( λ ) ijk ( ˆx ) ωa λ ( ω ˆx ) + H.c. , (2.22)– 7 –here in the last equality we inserted the expansion in oscillators of the leading radiationcontribution to ϕ ijk , C (cid:16) D − (cid:17) ijk ( u, ˆx ) = 12(2 iπ ) D − (cid:90) ∞ dω π ω D − e − iωu (cid:88) λ (cid:15) ( λ ) ijk ( ˆx ) ωa λ ( ω ˆx ) + H.c. , (2.23)and we used the relations ( i∂ u ) − D (cid:90) ∞ ω D − e − iωu f ( ω ) dω = (cid:90) ∞ ωe − iωu f ( ω ) dω , (2.24) π (cid:90) + ∞−∞ du (cid:90) ∞ ωe − iωu f ( ω ) dω = 12 lim ω → + [ ωf ( ω )] . (2.25)The charge (2.22) enters the Ward identity (cid:104) out | (cid:16) Q I + − S − S Q I − + (cid:17) | in (cid:105) = (cid:88) (cid:96) g (3) (cid:96) E (cid:96) T ( ˆx (cid:96) ) (cid:104) out | S | in (cid:105) , (2.26)under the assumption that higher-spin supertranslations are symmetries of a putative scat-tering matrix involving particles with arbitrary spins. More precisely, we follow the pro-cedure detailed in [1, 61] for connecting the soft portion of the asymptotic charge to theWard identity (2.26), which avoids the need to explicitly discuss external currents. In orderto highlight the relation to Weinberg’s soft theorem it is useful to choose a specific form forthe function T ( ˆx ) : T ˆw ( ˆx ) = ( D ˆw ) i ( D ˆw ) j ( D ˆw ) k (cid:15) ( ijk ) lmn ( ˆw )( ˆx ) l ( ˆx ) m ( ˆx ) n − ˆx · ˆw , (2.27)where the choice of polarisations is discussed in Appendix B.3. Inserting (2.27) in (2.22)one finds Q T ˆw (cid:12)(cid:12) I + − = −
12 lim ω → + D i ˆw D j ˆw D k ˆw (cid:104) ωa ijk ( ω ˆw ) + ωa † ijk ( ω ˆw ) (cid:105) . (2.28)Substituting this relation into the Ward identity (2.26) then yields the 3-divergence ofWeinberg’s theorem, lim ω → + (cid:104) out | ωa ijk ( ω ˆw ) S | in (cid:105) = − (cid:88) (cid:96) g (3) (cid:96) E (cid:96) (cid:15) ( ijk ) lmn ( ˆw )( ˆx (cid:96) ) l ( ˆx (cid:96) ) m ( ˆx (cid:96) ) n − ˆw · ˆx (cid:96) (cid:104) out | S | in (cid:105) . (2.29)This argument holds for any values of the couplings g (3) (cid:96) , thus showing that the relationbetween the Ward identity and the soft theorem is actually universal and does not rely onthe actual possible dynamical incarnations of the theory itself.The proof extends verbatim to the spin- s case. One starts with the charge Q T (cid:12)(cid:12) I + − = ( − s ( D − s ) (cid:90) + ∞−∞ du (cid:73) d Ω D − T ∂ u U (0) , (2.30)and makes repeated use of the equations of motion, using in particular ∂ D − u U (0) = ( D − D + s − D ( D· ) s C ( D − s − ) , (2.31)– 8 –o put it in the form Q T (cid:12)(cid:12) I + − = ( − s ( D − D + s − (cid:90) + ∞−∞ du (cid:73) d Ω D − T ∂ − D − u D ( D· ) s C ( D − s − )= ( − s − ( D − D + s − ω → + (cid:88) λ (cid:73) d Ω D − (2 π ) D − T D ( D· ) s (cid:15) ( λ ) ( ˆx ) ωa λ ( ω ˆx ) + H.c. , (2.32)where in the last equality we substituted the asymptotic limit of the free field near I + while the operator D is defined as in (2.21). In order to connect the Ward identity ofhigher-spin supertranslations to the soft theorem it is useful once again to make use of aspecific form of T ( ˆx ) , T ˆw ( ˆx ) = ( D ˆw ) i s (cid:15) ( i s ) j s ( ˆw )( ˆx ) j s − ˆx · ˆw , (2.33)in terms of which the charge reads Q T ˆw (cid:12)(cid:12) I + − = −
12 lim ω → + D i s ˆw (cid:104) ωa i s ( ω ˆw ) + ωa † i s ( ω ˆw ) (cid:105) . (2.34)Substituting this relation into the spin- s version of the Ward identity (2.26) then yields the s –divergence of Weinberg’s theorem. The reverse implication, on the other hand, namelythat Weinberg’s theorem yields the Ward identity (2.26) as well as its spin- s counterpart,is of less relevance in the context of higher spins given that Weinberg’s result also impliesthe vanishing of the soft couplings for s > . In this section we classify all residual symmetries of the Bondi-like gauge (2.1) and we showthat they comprise, in any dimension and for any value of the spin, suitable generalisationsof the superrotations introduced for s = 2 and D = 4 in [34]. In particular, within thelimits of our linearised analysis, for s = 2 we find extended BMS symmetries comprisingboth supertranslations and Diff( S D − ) transformations as in [39]. For arbitrary values ofthe spin we find instead asymptotic symmetries generated by a set of traceless tensors onthe celestial sphere of rank , , . . . , s − , that turn out to be in one-to-one correspondencewith the partially massless representations of spin s , with supertranslations corresponding inparticular to the representations with maximal depth. To keep all such residual symmetriesof the Bondi-like gauge, the non-vanishing components of the fields must scale as ϕ u s − k i k = O ( r s + k − ) , (3.1)although, eventually, only pure-gauge contributions are allowed on shell above the ordertypical of a radiative solution, ϕ u s − k i k = O ( r k +1 − D/ ) . Still, the definition of surfacecharges for (higher-spin) superrotations entails a number of subtleties that here we are ableto face only to a partial extent and that require further investigations.– 9 – .1 Symmetries of the Bondi-like gauge We begin by identifying the residual symmetries allowed by the Bondi-like gauge (2.1),without any further specifications on the falloffs of the components ϕ u s − k i k . To this end,it is convenient to split the components of the gauge parameter in two groups: thosewithout any index u , that we denote by (cid:15) i k ( k ) ≡ (cid:15) r s − k − i k , and the rest. Notice that notall components are independent because the gauge parameter is traceless: here we chose toexpress those with at least one index r and one index u in terms of the others.The elements of the first group are constrained by δϕ r s − k = 1 r (cid:110) ( s − k ) ( r∂ r − k ) (cid:15) ( k ) − γ (cid:15) ( k ) (cid:48) (cid:111) + D (cid:15) ( k − + r γ (cid:15) ( k − = 0 , (3.2)where a prime denotes a contraction with γ ij and where we omitted all sets of symmetrisedangular indices. These equations are solved by (cid:15) ( k ) ( r, u, ˆx ) = r k ρ ( k ) ( u, ˆx ) + k − (cid:88) l = k r l (cid:15) ( k,l ) ( u, ˆx ) , (3.3)where, at this stage, ρ ( k ) ( u, ˆx ) is an arbitrary traceless tensor because r k belongs to the ker-nel of ( r∂ r − k ) . It is however bound to be traceless because of Fronsdal’s trace constraint.The (cid:15) ( k,l ) are instead determined recursively (and algebraically) in terms of the ρ ( l ) with l < k . The precise form of the tensors (cid:15) ( k,l ) is not relevant for the ensuing considerations;we thus refer to Appendix C for more details.One can express the remaining components (cid:15) u s − k − i k in terms of the ρ ( k ) by impos-ing that all traces of the fields be gauge invariant, i.e. γ mn δϕ u s − k i k − mn = 0 . Imposing δϕ r s − k u l i k = 0 for k + l < s leads instead to a constraint on the free tensors in (3.3): ∂ u ρ ( k ) + s − k − D + s + k − D · ρ ( k +1) = 0 (3.4)for any k < s − (see Appendix C).For a field of spin s , we thus obtain residual symmetries parameterised by the s − traceless tensors on the celestial sphere ρ (0) , ρ (1) i , . . . , ρ ( s − i s − , where the tensor of highestrank still admits an arbitrary dependence on u . As we shall see in the next subsection, onecan eliminate the u –dependence by demanding that ϕ i s falloffs as fast as the δϕ i s inducedby (3.3) and imposing the equations of motion above the radiation order. Under theseassumptions, one obtains the falloffs (3.1), while the differential equation (3.4) holds forany value of k , so that ∂ u ρ ( s − = 0 .When s = 2 , the residual symmetries of the Bondi gauge h rr = h ru = h ri = γ ij h ij = 0 are generated by (cid:15) r = f , (cid:15) i = r v i + r ∂ i f , (cid:15) u = (cid:15) r + r − D − D · (cid:15) , (3.5)with the constraint ∂ u f + 1 D − D · v = 0 . (3.6)– 10 –mposing the falloffs (3.1), that is h ij = O ( r ) , h ui = O ( r ) and h uu = O (1) , one obtainsthe additional condition ∂ u v i = 0 ⇒ f ( u, ˆx ) = T ( ˆx ) − uD − D · v ( ˆx ) . (3.7)For any value of the space-time dimension, we thus recovered the supertranslations discussedin the previous section, together with a transformation generated by a free vector on thecelestial sphere. To leading order, the latter acts on h ij as the traceless projection of alinearised diffeomorphism, δh ij = r (cid:18) D ( i v j ) − D − γ ij D · v (cid:19) + O ( r ) . (3.8)In a full, non-linear theory this transformation corresponds to the superrotations of [34, 39](see also [62] for a related discussion).This pattern continues for arbitrary values of the spin. For instance, for s = 3 theresidual symmetries of the Bondi-like gauge are generated by (cid:15) rr = f , (3.9a) (cid:15) ri = r v i + r ∂ i f , (3.9b) (cid:15) ij = r K ij + r (cid:18) D ( i v j ) − D − γ ij D · v (cid:19) + r (cid:18) D i D j − D γ ij (∆ − (cid:19) f, (3.9c)where K ij must be traceless to fulfil the constraint g µν (cid:15) µν = 0 , while ∂ u f + 2 D − D · v = 0 , ∂ u v i + 1 D D · K i = 0 . (3.10)Out of the remaining components of the gauge parameter one finds (cid:15) ru = (cid:15) rr + r − (cid:15) (cid:48) , whilethe conditions γ jk δϕ ijk = 0 and γ ij δϕ uij = 0 imply, respectively, (cid:15) ui = (cid:15) ri + r − D (cid:0) D · (cid:15) i + D i (cid:15) (cid:48) (cid:1) , (cid:15) uu = (cid:15) ru + r − D − (cid:0) D · (cid:15) u + ∂ u (cid:15) (cid:48) (cid:1) . (3.11a)Imposing the boundary conditions (3.1) then selects the following solution for (3.10): K ij = K ij ( ˆx ) , (3.12a) v i = ρ i ( ˆx ) − uD D · K i ( ˆx ) , (3.12b) f = T ( ˆx ) − uD − D · ρ ( ˆx ) + u D ( D − D · D · K ( ˆx ) . (3.12c)As expected, keeping only the u –independent contributions forces ρ i = 0 and K ij = 0 sothat one recovers (2.6).In Bondi coordinates, the solutions of the Killing tensor equation δϕ µ s = ∇ µ (cid:15) µ s − = 0 take the same form, but the tensors K ij , ρ i and T are bound to satisfy the following– 11 –dditional (traceless) differential constraints, that only leave a finite number of solutionsfor D > [2]: D ( i K jk ) − D γ ( ij D · K k ) = 0 , (3.13a) D ( i D j ρ k ) − D γ ( ij (cid:2) (∆ + D − ρ k ) + 2 D k ) D · ρ (cid:3) = 0 , (3.13b) D ( i D j D k ) T − D γ ( ij D k ) (3 ∆ + 2( D − T = 0 . (3.13c)With the boundary conditions (3.1) we thus observe an infinite-dimensional enhancement ofall classes of higher-spin symmetries appearing in (3.9), but of a different kind compared tothe higher-spin superrotations introduced for D = 4 where, following the spin-2 proposal of[15], we showed that the first two constraints in (3.13) admit locally an infinite-dimensionalsolution space [1]. We now show that on shell only local pure-gauge field configurations are allowed above theradiation order for fields that admit the asymptotic expansion (2.2). Let us stress thatmost of the conclusions in this section apply to both even and odd values of the space-timedimension D , with the proviso that in the latter case one also has to consider half-integervalues of n . For simplicity, however, in the following we focus on the case of even D , andthus consider n ∈ Z . See also [25] for the corresponding analysis in D = 5 .We study the equations of motion above the falloffs typical of radiation discussed in[2], and in this range matter sources cannot contribute. Furthermore, since the number ofangular indices carried by each tensor U ( k,n ) appearing in the radial expansion (2.2) is equalto k , from now on we shall omit them altogether. Introducing the shorthand C ( n ) ≡ U ( s,n ) ,the source-free Fronsdal equations in the Bondi-like gauge imply U ( k,n ) = ( n + 2 k − D − n − n + s + k − D − n + s − k − D· ) s − k C ( n − s + k ) (3.14)for − s − k ≤ n ≤ D − , and ( D − n − s − ∂ u C ( n ) = [∆ − ( n − D − n − s − − s ( D − s − C ( n − − D + 2( s − n + 2 s − D − n − (cid:18) DD · C ( n − − D + 2( s − γ D · D · C ( n − (cid:19) (3.15)for − s ≤ n ≤ D − . Out of the specified ranges of n , some of the U ( k,n ) may not beexpressed solely in terms of the C ( n ) and they satisfy differential equations in u similar to(3.15) (see Appendix D).The last equation shows that C ( D − s − )( u, ˆx ) is an arbitrary function, correspondingto the “radiation order”. For n = D − s − one thus obtains (cid:20) ∆ − ( D − s − D − s − − s ( D − s − (cid:21) C ( D − s − ) + · · · , (3.16) For s = 1 we thus show that the pure-gauge configurations above the usual radiation falloffs that weintroduced in [26] exhaust all solutions of the equations of motions. See also [58] for a similar analysis ofthe equations of motion for s ≤ . – 12 –hat, on a compact manifold like the celestial sphere, implies C ( D − s − ) = 0 . One canreach this conclusion by first eliminating the divergences of the tensor via the divergencesof (3.16), and then by noticing that the differential operator (∆ − λ ) entering (3.16) isinvertible. This is so because the eigenvalues of the Laplacian acting on a traceless anddivergenceless tensor of rank s are always negative (see (B.7)), while λ > s .The previous procedure can be iterated to get C ( n ) = 0 for − s < n < D − s − , (3.17)where the two extrema correspond to the radiation order and to the order at which su-pertranslations act on the purely angular component ϕ i s , respectively. Notice that theycoincide when D = 4 for any value of the spin: in this case supertranslations act at theradiation order, that in the Bondi-like gauge encodes information about the local degreesof freedom of a propagating wave packet [1]. To prove (3.17) it is useful to compute thedivergences of (3.15): ( D − n − s − ∂ u ( D· ) k C ( n ) = ( n + 2 s − k − D − n − k − n + 2 s − D − n − ×× [∆ − ( n + k − D − n − s + k − − ( s − k )( D − s + k − D· ) k C ( n − − D + 2( s − k − n + 2 s − D − n − (cid:18) D − D + 2( s − k − γ D· (cid:19) ( D· ) k +1 C ( n − . (3.18)For D − s − ≤ n ≤ − s these equations set to zero recursively all divergences of C ( n − and eventually the whole tensor itself since all operators in the second line are invertible.To make this analysis more transparent it is convenient to let n = D − s − (cid:96) , so that(3.18) takes the form (2 − D − (cid:96) ) ∂ u ( D· ) k C ( D − s − (cid:96) ) = ( D − s + (cid:96) − k )( s − − (cid:96) − k )( D − s + (cid:96) )( s − − (cid:96) ) [∆ + (cid:96) ( (cid:96) + D − − ( s − k )] ( D· ) k C ( D − s − (cid:96) ) − D + 2( s − k − D − s + (cid:96) )( s − − (cid:96) ) (cid:18) D − D + 2( s − k − γ D· (cid:19) ( D· ) k +1 C ( D − s − (cid:96) ) . (3.19)The values of (cid:96) at which the operator appearing in the second line fails to be invertible are (cid:96) = 4 − D + k − s , (cid:96) = s − − k , (3.20)where the overall coefficient vanishes, or (cid:96) = s − k, s − k + 1 , s − k + 2 , . . . , (3.21)as dictated by the eigenvalues of the Laplacian on divergence-free tensors (see (B.7)).The iterative procedure that sets the C ( n ) to zero thus stops at n = 2 − s , since for k = s the overall coefficient in the second line of (3.18) vanishes and one does not obtainany information on ( D· ) s C (1 − s ) . This result agrees with those of Section 3.1: one cannotconclude C (1 − s ) = 0 because under a supertranslation this tensor transforms as δC (1 − s ) = [( s − − D s T + · · · , (3.22)– 13 –here the omitted terms implement a traceless projection. This can be easily verified for s = 2 and s = 3 by substituting (3.5) in δh ij and (3.9) in δϕ ijk .This phenomenon extends to all instances of (3.15) in the range − s ≤ n ≤ − s .To make this manifest, let us relabel n → − s − t : the rhs of (3.15) then becomes M ( s,t ) ≡ [∆ − ( D + s −
4) + t ( D + t − C (1 − s − t ) − D + 2( s − s − t )( D + s + t − (cid:18) DD · C (1 − s − t ) − D + 2( s − γ D · D · C (1 − s − t ) (cid:19) , (3.23)and in the range ≤ t ≤ s − eq. (3.18) implies ( D· ) s − t M ( s,t ) = − D + 2( t − s − t )( D + s + t − (cid:18) D − D + 2( t − γ D· (cid:19) ( D· ) s − t +1 C (1 − s − t ) . (3.24)The latter can be interpreted as a Bianchi identity for the operator M ( s,t ) and, indeed, itallows one to prove that it is invariant under δC (1 − s − t ) = D s − t λ ( t ) with D · λ ( t ) = λ ( t ) (cid:48) = 0 . (3.25)In our context, these transformations can be identified with the portion of the asymptoticsymmetries generated by the u –independent and divergence-free part of the parameters(3.3). The other contributions to the residual symmetries of the Bondi-like gauge arereinstated by the sources on the lhs of (3.18), while their action on the other non-vanishingcomponents of the field, i.e. δϕ u s − k i k , can be recovered from (3.14) since gauge symmetriesmap solutions of the eom into other solutions. For instance, for s = 2 one obtains δC ij ( − = D ( i v j ) − D − γ ij D · v , (3.26a) δC ij ( − = 2 (cid:18) D i D j − D − γ ij ∆ (cid:19) f, (3.26b)and, correspondingly, δh ui = D · δC i ( − D −
3) = D i (∆ + D − fD − , (3.27a) δh uu = − D · D · δC ( − ( D − D −
3) = − D − D · v ( D − . (3.27b)Let us also observe that the differential operator (∆ − m s,t ) in (3.23) identifies themass shell of a partially-massless field of spin s and depth t (see e.g. [49]). Moreover, for t = s − one recovers in (3.23) the Maxwell-like kinetic operator for a massless field of spin s propagating on a constant curvature background [60], while for the other values of t oneobtains kinetic operators describing more complicated spectra. In particular, for s = 2 and Let us note that the operator implicitly defined in (3.22) provides the spin- s counterpart of the differ-ential operator computing the linear memory effect in terms of the supertranslation parameter for spin twoin D = 4 , where it indeed acts at the correct Coulombic order [63]. – 14 – = 0 eq. (3.18) gives the conformally-invariant equation of motion introduced in [64], thatdoes not describe only a partially-massless spin-2 field.We now impose the additional condition that the field components above the order atwhich asymptotic symmetries act be zero, that is C ( n ) = 0 for n < − s , (3.28)or more generally U ( k,n ) = 0 for n < − s − k . This corresponds to the boundary conditions(3.1). Thanks to (3.17), under this assumption the only non-vanishing C ( n ) above theradiation order are those with − s ≤ n ≤ − s and we wish to argue that only thepure-gauge configurations that we discussed above satisfy the equations of motion.In order to support this statement, let us examine in detail the low-spin examples. Forspin one, the only nontrivial overleading component is C i (0) and it satisfies the free Maxwellequation on the Euclidean sphere [∆ − D + 3] C i (0) − D i D · C (0) = 0 . (3.29)We can separate C i (0) into a divergence-free part, ˜ C i with D · ˜ C = 0 , and a pure gradientpart according to C i (0) = ˜ C i + ∂ i T . (3.30)Furthermore, since C i ( − = 0 , the equations of motion also imply ∂ u C i (0) = 0 , so that ˜ C i and T can be chosen to be u -independent. Equation (3.29) thus reduces to [∆ − D + 3] ˜ C i = 0 . (3.31)This implies ˜ C i = 0 because [∆ − D + 3] is invertible and hence that C i (0) is a pure-gaugeconfiguration, C i (0) = ∂ i T .Moving to spin two, we need to discuss − D + 2] C ij ( − − D − D − (cid:20) D ( i D · C j )( − − D − γ ij D · D · C ( − (cid:21) , (3.32) ( D − ∂ u C ij ( − = [∆ − C ij ( − − (cid:20) D ( i D · C j )( − − D − γ ij ( D· ) C ( − (cid:21) , (3.33)which are the only two instances of (3.15) above the radiation order that are not identicallysatisfied on account of (3.17) and (3.28). Note also that, in view of (3.28), ∂ u C ij ( − = 0 ⇒ ∂ u C ij ( − = 0 . (3.34)That is, C ij ( − is u –independent while C ij ( − is at most linear in u : C ij ( − ( ˆx ) = H ij ( ˆx ) , C ij ( − ( u, ˆx ) = F ij ( ˆx ) + u G ij ( ˆx ) . (3.35)– 15 –e then have − D + 2] F ij − D − D − (cid:20) D ( i D · F j ) − D − γ ij ( D· ) F (cid:21) , (3.36) − D + 2] G ij − D − D − (cid:20) D ( i D · G j ) − D − γ ij ( D· ) G (cid:21) , (3.37) ( D − G ij = [∆ − H ij − (cid:20) D ( i D · H j ) − D − γ ij ( D· ) H (cid:21) . (3.38)The first two relations imply F ij ( ˆx ) = 2 (cid:18) D i D j − D − γ ij ∆ (cid:19) T ( ˆx ) , G ij ( ˆx ) = (cid:18) D i D j − D − γ ij ∆ (cid:19) S ( ˆx ) . (3.39)To see why this is the case, let us focus on the second one and use the decomposition G ij = ˜ G ij + D ( i ˜ v j ) + (cid:18) D i D j − D − γ ij ∆ (cid:19) S , (3.40)where
D · ˜ G i = 0 , γ ij ˜ G ij = 0 , D · ˜ v = 0 , (3.41)which is tantamount to the decomposition of the irreducible so ( D − tensor G ij in irre-ducible so ( D − components. Substituting into the divergence of (3.37), we find [∆ − D + 3] [∆ + D −
3] ˜ v i = 0 . (3.42)This implies that ˜ v i belongs to the kernel of [∆ + D − , i.e., ˜ v i is an irreducible harmonicwith (cid:96) = 1 as discussed in Appendix B.2. Such vectors give zero contribution to (3.40).Then, from (3.37), [∆ − D + 2] ˜ G ij = 0 ⇒ ˜ G ij = 0 . (3.43)This proves (3.39) given (3.37).The condition in (3.38) can be then regarded as an equation for H ij ( ˆx ) given the sourceterm G ij ( ˆx ) , or equivalently S ( ˆx ) in view of (3.39). To solve it, it is convenient to resortto the following decomposition for the traceless tensor H ij , H ij = ˜ H ij + D ( i v j ) − D − γ ij D · v , (3.44)where
D · ˜ H i = 0 , while v i is now a generic vector. Substituting (3.39) and (3.44) into (3.38)and taking divergences eventually allows one to show that ˜ H ij = 0 and that S = − D − D · v (3.45)up to a constant terms and (cid:96) = 1 scalar harmonics. Substituting into (3.35), we find thatthe most general solution for the overleading terms is C ( − ij ( ˆx ) = D ( i v j ) ( ˆx ) − D − γ ij D · v ( ˆx ) , (3.46) C ( − ij ( u, ˆx ) = 2 (cid:18) D i D j − D − γ ij ∆ (cid:19) (cid:18) T ( ˆx ) − uD − D · v ( ˆx ) (cid:19) , (3.47)– 16 –here we recognise a supertranslation and a superrotation, parametrised by T ( ˆx ) and v i ( ˆx ) respectively.A very similar chain of arguments allows one to prove explicitly that, even for spin-three fields, the only nontrivial overleading components C ( − ijk , C ( − ijk , C ( − ijk allowed by theequations of motion take precisely the form of the asymptotic symmetries identified in theprevious sections. It is natural to expect that this pattern actually holds for all spins sothat, in this setup, only the structures trivially allowed by the gauge symmetry can appearto overleading orders. We now discuss the structure of surface charges that could be associated to all higher-spinsuperrotations. To identify it we shall follow, at least in some steps, a strategy similarto that we employed in Section 2 to define finite supertranslation charges. Let us stress,however, that the setup is not completely equivalent, as it is manifest already for spin-twofields. Indeed, evaluating the surface charge (2.7) for all residual symmetries (3.5) of theBondi gauge on field configurations with overleading, pure-gauge terms h uu = − D − D − D · ˜ v ( ˆx ) + O ( r − D ) , (3.48) h ui = D i (∆ + D − D − (cid:18) ˜ T ( ˆx ) − uD − D · ˜ v ( ˆx ) (cid:19) + O ( r − D ) (3.49)one obtains Q ( u ) = lim r →∞ r D − D − (cid:73) d Ω D − (cid:110) T ( ˆx ) (∆ + D − D · ˜ v ( ˆx ) − D · v ( ˆx ) (∆ + D −
2) ˜ T ( ˆx ) (cid:111) + O ( r D − ) . (3.50)Contrary to the discussion in Section 2, based on the more restrictive boundary conditions(2.4) only allowing for asymptotic supertranslations, here the pure-gauge overleading fieldconfigurations give a divergent contribution to the charge of O ( r D − ) . Notice, however, thatthe surface charge (3.50) already diverges linearly in four space-time dimensions. In thiscontext, the charge has been regularised in [51, 52] (see also [35, 65]) and in the followingwe assume that a similar regularisation is possible also for higher values of the space-timedimension D . This conjecture will be cross-checked by comparing with the charges thathave been used to derive subleading soft theorems from asymptotic symmetries [34, 39].Assuming that the divergence of O ( r D − ) associated to the overleading terms can becancelled by adding suitable counterterms to the surface charge (3.50), one is left with adivergence of O ( r D − ) related to radiation. In analogy with Section 2, the latter could beeliminated by defining the charge as the evolution under the equations of motion of Q ( −∞ ) and assuming that no radiation is present for u smaller than a given u . This would amountto computing the surface charge (2.7) for u < u on stationary field configurations, that is– 17 –n [58] h uu = r − D U (0 ,D − ( ˆx ) + O ( r − D ) , (3.51a) h ui = r − D U i (1 ,D − ( ˆx ) + r − D U i (1 ,D − ( u, ˆx ) + O ( r − D ) . (3.51b)Notice that U i (1 ,D − ( ˆx ) is restricted to be u -independent and divergence-free on stationarysolutions, but it need not vanish. Substituting these solutions into the charge (2.14) andincluding both supertranslations and superrotations for completeness, one gets Q ( u ) = − ( D − (cid:73) d Ω D − T U (0 ,D − + lim r →∞ r ( D − (cid:73) d Ω D − v i U i (1 ,D − − (cid:73) d Ω D − v i (cid:110) u ∂ i U (0 ,D − − ( D − U i (1 ,D − (cid:111) . (3.52)In the first line we recovered the (finite) supertranslation charge (2.14) already discussedin Section 2. The second line exhibits instead a linear divergence in r involving the su-perrotation vector v i . In other words, for superrotations, restricting to stationary solutionsdoes not completely solve the issue of the O ( r D − ) contributions related to radiation.Actually, this singular behaviour in r is not the only puzzling feature of (3.52). Evenits last line, which is finite in the limit r → ∞ for fixed u , seems to diverge as u is then sentto −∞ , i.e. as one approaches I + − . Indeed the equations of motion for stationary solutionsrequire u ∂ i U (0 ,D − ( ˆx ) − ( D − (cid:104) U i (1 ,D − ( u, ˆx ) − q i ( ˆx ) (cid:105) = u [∆ − U (1 ,D − i ( ˆx ) , (3.53)where q i ( ˆx ) is an arbitrary u -independent integration function, so that (3.52) can be recastas Q ( u ) = − ( D − (cid:73) d Ω D − T U (0 ,D − + ( D − (cid:73) d Ω D − v i q i + lim r →∞ r ( D − (cid:73) d Ω D − v i U i (1 ,D − − u (cid:73) d Ω D − v i [∆ − U (1 ,D − i . (3.54)In view of these observations, we note that further restricting to the set of Coulombic stationary solutions already considered in [2], for which the divergence-free component U (1 ,D − i is zero, namely h uu = r − D U (0 ,D − ( ˆx ) + O ( r − D ) , (3.55a) h ui = r − D U i (1 ,D − ( u, ˆx ) + O ( r − D ) , (3.55b) The linear divergence vanishes on shell (as it should) if v i generates a global symmetry: indeed in thiscase U i (1 ,D − ∼ D · C i ( D − (see eq. (3.14)), while v i satisfies the conformal Killing equation. – 18 –olves both problems at once. The second line of (3.54) indeed vanishes identically and weretrieve a well-behaved expression for the charge near I + − , Q (cid:12)(cid:12) I + − = − ( D − (cid:73) d Ω D − T ( ˆx ) U (0 ,D − ( ˆx ) + ( D − (cid:73) d Ω D − v i ( ˆx ) q i ( ˆx ) , (3.56)where q i ( ˆx ) is defined by q i ( ˆx ) = U i (1 ,D − ( u, ˆx ) − uD − ∂ i U (0 ,D − ( ˆx ) . (3.57)Relabelling U (0 ,D − ≡ M and q i ≡ N i , the expression (3.56) for the charge thus agreeswith the one presented in [2] for global symmetries.The behaviour of Q ( u ) for generic u can be then retrieved from (3.56) by consider-ing its evolution under the equations of motion, in the same way as we discussed for thesupertranslation charge in Section 2. In particular, considering that both U (0 ,D − and U (1 ,D − will acquire a non-trivial u -dependence as dictated by the equations of motion inthe presence of radiation, rewriting the charge as in (2.18) we find Q| I ++ − Q| I + − = − ( D − (cid:90) + ∞−∞ du (cid:73) d Ω D − T ∂ u U (0 ,D − − (cid:90) + ∞−∞ du (cid:73) d Ω D − u D · v ∂ u U (0 ,D − + (cid:90) + ∞−∞ du (cid:73) d Ω D − v i (cid:110) ∂ i U (0 ,D − − ( D − ∂ u U i (1 ,D − (cid:111) . (3.58)If one considers a global, Poincaré transformation the right-hand side of this relation van-ishes identically, and indeed the surface charge Q ( u ) must be independent of u in this case,even in the presence of radiation [2]. The same result can be achieved by instead restrictingthe calculation to static (i.e. u -independent), rather than stationary or Coulombic solutions(see Appendix D). The first line is just the soft supertranslation charge that has been em-ployed in Section 2 to derive the leading soft theorem in any even dimension. The secondline exhibits a term linear in u and corresponds to the superrotation charge that has beenused in [39] to derive the subleading soft-graviton theorem in any even dimension [33]. Re-covering it in our approach supports our conjecture that the regularisation of [51, 52] canbe extended also to higher space-time dimensions and successfully applied to (2.7).We now move to the higher-spin case. All types of divergences encountered above forspin two continue to be present and they are even more severe for s > . Let us observe,however, that for any value of the spin all divergences in r can be eliminated by evaluatingthe charge on static, rather than stationary, solutions for u < u and that this operationgives a consistent surface charge even if we are not dealing with global symmetries. Staticsolutions are indeed defined by ∂ u U ( k,n ) = 0 ∀ n , (3.59) The overleading pure-gauge contributions to the boundary conditions (3.1) bring a divergence of O ( r D +2 s − ) , radiation brings a divergence of O ( r D +2 s − ) , while stationary configurations bring a divergenceof O ( r s − ) . – 19 –nd we argue in Appendix D that this condition implies U ( k,n ) = 0 for n < D − (3.60)together with K ( k ) ≡ D U ( k,D − − D + 2( k − γ D · U ( k,D − = 0 . (3.61)The latter traceless combination is the conformal Killing equation for the rank– k tensor U ( k,D − ( ˆx ) on the celestial sphere [66].Given (3.60), with this prescription the surface charge is manifestly finite in the limit r → ∞ and we now show that the property (3.61) of the leading contributions of anystatic solution guarantees that the charge thus defined is also conserved in u . Notice thatthis is not obvious a priori, since the gauge parameters generating superrotations bring apolynomial dependence in u (cf. (3.3) or, e.g., (3.5) and (3.9)): Q static ( u ) = (cid:73) d Ω D − s − (cid:88) k =0 ( − s + k (cid:18) s − k (cid:19) ( D + s + k − ρ ( k ) ( u, ˆx ) U ( k,D − ( ˆx ) . (3.62)A dependence on u could imply that the charge is not conserved even in the absence ofradiation and would also create problems in evaluating it for u → ±∞ . The u -dependenceof the rank– k traceless tensors ρ ( k ) is however fixed by (3.4) as ρ ( k ) ( u, ˆx ) = K ( k ) ( ˆx ) + s − k − (cid:88) m =1 ( − m u m ( D + s + k − m (cid:18) s − k − m (cid:19) ( D· ) m K ( k + m ) ( ˆx ) , (3.63)where, for a field of spin s , we introduced the s − traceless tensors on the celestial sphere K (0) , K (1) i ,. . . , K ( s − i s − (for s = 3 they correspond to K (0) = T , K (1) i = ρ i and K (2) ij = K ij inthe notation of (3.12)). When the field configuration is static, each term in the charge (3.62)which depends on u thus contains at least one divergence of the tensors K ( l ) . Integrating itby parts one reconstructs the conformal Killing equation for the tensors U ( k,D − and thusthe contribution vanishes on shell on account of (3.61). The surface charge associated tohigher-spin superrotations thus becomes Q static = (cid:73) d Ω D − s − (cid:88) k =0 ( − s + k (cid:18) s − k (cid:19) ( D + s + k − K ( k ) ( ˆx ) U ( k,D − ( ˆx ) . (3.64)The result has the same form as the charge associated to global higher-spin symmetries[2], with the difference that now the traceless tensors K ( k ) do not satisfy any differentialequation (e.g., for s = 3 they need not obey (3.13)). In analogy with the discussionin Section 2, a contribution from radiation might be obtained by substituting in (3.64)the U ( k,D − ( u, ˆx ) obtained via the evolution under the equations of motion of a fieldconfiguration that is static for u smaller than a given u . It would be obviously preferable,however, to implement a procedure leading to finite charges for more general configurations.We hope to come back to this issue in future work, with the goal of finding a properrenormalisation scheme valid in higher dimensions and for any value of the spin.– 20 – cknowledgments We are grateful to G. Barnich, D. Grumiller and M. M. Sheikh-Jabbari for discussions.The work of A.C. was supported by the Fonds de la Recherche Scientifique - FNRS underGrants No. F.4503.20 (“HighSpinSymm”) and T.0022.19 (“Fundamental issues in extendedgravitational theories”). D.F. gratefully acknowledges the Asian-Pacific Center for Theo-retical Physics (APCTP) in Pohang for the kind hospitality extended to him during thepreparation of this work. The work of C.H. is supported by the Knut and Alice WallenbergFoundation under grant KAW 2018.0116.
A Notation and conventions
Throughout the paper we employ retarded Bondi coordinates ( x µ ) = ( u, r, x i ) , where x i ,for i = 1 , , . . . , n , denotes the n := D − angular coordinates on the sphere at null infinity.In these coordinates, the Minkowski metric reads ds = − du − dudr + r γ ij dx i dx j , (A.1)where γ ij is the metric of the Euclidean n -sphere S n . We denote by D the covariant deriva-tive on the sphere while D· and ∆ stand for the corresponding divergence and Laplacian,respectively.We use a symbol with a subscript, like µ k or i k , to denote a group of symmetrisedindices, whose number is specified by the subscript k . Repeated indices denote insteada symmetrisation without any overall factor. For instance, ∇ µ (cid:15) µ s − is a shorthand for ∇ µ (cid:15) µ ··· µ s + ∇ µ (cid:15) µ ··· µ s µ + · · · . In Section 3 and in Appendices C and D, we also employan index-free notation, where all symmetrised indices are omitted and the trace is denotedby a prime. In this case, the previous expression is denoted by ∇ (cid:15) . B Geometry of the sphere and polarisations
B.1 Properties of the n -sphere Let us recall here some properties of the embedding of the unit n -sphere S n in the Eu-clidean space R n +1 . Changing coordinates according to x I = r ˆ x I where x I are Cartesiancoordinates on R n +1 and ˆ x I is a parametrisation of unit vectors in terms of the angles x i ,the Euclidean metric reads ds = dx I dx I = dr + r γ ij dx i dx j , (B.1)where γ ij = e Ii e Ij , e Ii = ∂ i ˆ x I . (B.2)The induced metric on the unit sphere γ ij also defines a covariant derivative D i thereon.As a consequence of ∂ I ∂ J x K = 0 , one can show the useful identity D i D j ˆ x I + γ ij ˆ x I = 0 . (B.3)– 21 –he metric γ ij can be also represented by the rank- n matrix γ IJ = δ IJ − ˆ x I ˆ x J (B.4)which projects any vector to its component tangent to the sphere. Therefore, for any unitvector ˆ q I , its projection on the sphere ˆ q i = ˆ q I e Ii obeys ˆ q i γ ij ˆ q j = 1 − (ˆ x I ˆ q I ) . (B.5) B.2 Spectrum of ∆ The eigenvalues λ of the operator ∆ = D i D i acting on symmetric, traceless, divergence-freetensors K i i ··· i s , i.e. irreducible tensors of rank s , (∆ − λ ) K i i ··· i s = 0 , (B.6)are λ = − (cid:96) ( (cid:96) + n −
1) + s , (cid:96) = s, s + 1 , s + 2 , . . . . (B.7)Explicit eigenfunctions are provided by the irreducible tensor spherical harmonics. Theycan be constructed starting from tensors with constant Cartesian components C [ A I ][ A I ] ··· [ A s I s ] I s +1 ··· I (cid:96) for (cid:96) ≥ s , (B.8)which are assumed to be completely traceless, symmetric under permutations of I s +1 , . . . , I (cid:96) ,symmetric under permutations of the pairs [ A k I k ] , and antisymmetric under exchanges ofan A k with its corresponding I k index. It then follows that the tensors K A A ··· A s = C [ A I ][ A I ] ··· [ A s I s ] I s +1 ··· I (cid:96) x I x I · · · x I (cid:96) (B.9)are symmetric, harmonic ∆ R n +1 K A A ··· A s = 0 and homogeneous of degree (cid:96) under rescal-ings of x I . They are also tangent to the sphere x A K AA ··· A s = 0 and the correspondingtensors K i i ··· i s on the sphere, defined by K A A ··· A s = r (cid:96) e A i e A i · · · e A s i s K i i ··· i s , (B.10)are divergence-free and trace-free, as can be checked using (B.2), (B.3) and (B.4), togetherwith the above properties. Using also ∆ R n +1 f = 1 r n ∂ r ( r n ∂ r f ) + 1 r ∆ f , (B.11)together with the properties of K i i ··· i s and the action (B.3), one then obtains R n +1 K A A ··· A s = r (cid:96) − e A i e A i · · · e A s i s [∆ + (cid:96) ( (cid:96) + n − − s ] K i i ··· i s , (B.12)thus retrieving (B.6) with the eigenvalues (B.7). The uniqueness of these eigenvalues canbe inferred from the density of the homogeneous polynomials in the domain of ∆ underconsideration. – 22 –he spectrum on reducible tensors can be obtained from the above one by first decom-posing the desired tensor in terms of symmetrised gradients of irreducible tensors and thenusing the commutation relation [ D i , D j ] v k = R klij v l , R ijkl = γ ik γ jl − γ il γ jk . (B.13)For instance, for a traceless but generically not divergence free tensor C ( s ) i i ··· i s , the desireddecomposition can be cast in the form C ( s ) i i ··· i s = K ( s ) i i ··· i s + D ( i K ( s − i ··· i s ) + D ( i D i K ( s − i ··· i s ) + · · · + D ( i D i · · · D i s ) K (0) + · · · , (B.14)where the dots implement a traceless projection. More details for s = 0 , , are availablein [67] for irreducible case and in [68] for the reducible case. B.3 Polarisation Tensors
In this appendix we sketch the construction of a useful set of polarization tensors. We applythe notation of Appendix B.1 for quantities defined on the n = D − sphere.To construct the physical polarisations for the electromagnetic potential A µ , we startfrom the Fierz system written in Minkowski coordinates, (cid:3) A µ = 0 , ∂ · A = 0 , (cid:3) Λ = 0 , (B.15)where Λ is the gauge parameter. According to the first equation, the Fourier transform (cid:15) µ of A µ has support restricted to null vectors q µ = ω (1 , ˆ x I ) (B.16)and satisfies, letting (cid:15) µ = ( (cid:15) , (cid:15) I ) , (cid:15) = − ˆ x I (cid:15) I . (B.17)The residual gauge parameters must also have support on vectors of the form (B.16) andthus (cid:15) µ is equivalent, up to a gauge transformation, to (cid:15) µ ∼ (cid:15) µ − iω ( − , ˆ x I ) λ , (B.18)where λ stands for the Fourier transform of Λ . Taking (B.17) into account and choosing agauge parameter such that iωλ = ˆ x I (cid:15) I thus leads to (cid:15) µ ∼ (0 , γ IJ (cid:15) J ) (B.19)with γ IJ as in (B.4). Therefore, (cid:15) µ can be parametrised by the projection on the unit sphereof a generic vector (cid:15) I and has D − n independent components, up to gauge equivalence.Given a set of coordinates x i on the sphere, we can choose the following basis forphysical polarisations, (cid:15) ( i ) µ (ˆ x ) = (0 , D i ˆ x I ) . (B.20)In retarded components, one then finds (cid:15) ( i ) u = 0 = (cid:15) ( i ) r , (cid:15) ( i ) j = δ ij . (B.21)– 23 –ith this choice all sums over polarisations can be understood as contractions with themetric γ ij , while the projection over the space of physical polarisations is just the projectionon the sphere.The spin-two Fierz system, (cid:3) h µν = 0 , (cid:3) Λ µ = 0 , ∂ · h µ = 0 ,∂ · Λ = 0 h (cid:48) = 0 , (B.22)implies that the Fourier transforms (cid:15) µν and λ µ of h µν and Λ µ have support on the nullvectors (B.16), and λ µ = ( λ , λ I ) satisfies λ = − ˆ x I λ I . (B.23)Furthermore (cid:15) µν satisfies (cid:15) I = − ˆ x I (cid:15) IJ , (cid:15) = ˆ x I (cid:15) IJ ˆ x J = δ IJ (cid:15) IJ . (B.24)The latter condition implies that (cid:15) IJ has a traceless projection on the sphere, γ IJ (cid:15) IJ = 0 .Up to gauge equivalence, the polarization tensor reads (cid:15) ∼ ˆ x I (cid:15) IJ ˆ x J − iω ˆ x I λ I , (B.25) (cid:15) I ∼ − (cid:15) IJ ˆ x J + iω ( λ I + ˆ x I ˆ x J λ J ) , (B.26) (cid:15) IJ ∼ (cid:15) IJ − iω (ˆ x I λ J + ˆ x J λ I ) . (B.27)We parametrise the gauge vector as follows, λ I = A ˆ x + B i e Ii , (B.28)with e Ii as in (B.2). Imposing that the right-hand sides of (B.25) and (B.26) vanish, wehave ˆ x I (cid:15) IJ ˆ x J − iωA = 0 , − (cid:15) IJ ˆ x J + 2 iω ˆ x I A + iωe iI B i = 0 , (B.29)where the index i is raised and lowered using γ ij . This fixes the coefficients A and B i tobe iωA = ˆ x I (cid:15) IJ ˆ x J , iωB i = e Ii (cid:15) IJ ˆ x J , (B.30)upon taking suitable projections of the second equation in (B.29). One thus finally arrivesat the expression (cid:15) ∼ , (cid:15) I ∼ , (cid:15) IJ ∼ γ IK γ JL (cid:15) KL (B.31)substituting (B.28) and (B.30) into (B.27). Recalling the constraint γ IJ (cid:15) IJ = 0 , thisimplies that the most general polarization tensor, up to gauge equivalence, is identified bya symmetric traceless tensor on the ( D − -sphere and thus characterises ( D − D − − degrees of freedom.A convenient basis for symmetric tensors on the sphere is furnished by E ( ij ) kl = δ ik δ jl + δ il δ jk , (B.32)– 24 –hose traceless projection reads (cid:15) ( ij ) kl = δ ik δ jl + δ il δ jk − D − γ ij γ kl . (B.33)We therefore adopt polarization tensors (cid:15) ( ij ) µν such that, in retarded components, (cid:15) ( ij ) uu = 0 , (cid:15) ( ij ) ur = 0 , (cid:15) ( ij ) uk = 0 , (cid:15) ( ij ) rr = 0 , (cid:15) ( ij ) rk = 0 and with angular components (cid:15) ( ij ) kl specified byequation (B.33).We consider the spin-three Fierz system (cid:3) ϕ µνρ = 0 , (cid:3) Λ µν = 0 , ∂ · ϕ µν = 0 ,∂ · Λ µ = 0 ϕ (cid:48) µ = 0 , Λ (cid:48) = 0 . (B.34)The Fourier transforms (cid:15) µνρ and λ µν of ϕ µνρ and Λ µν have support on the null vectors(B.16). The gauge parameter satisfies λ I = − ˆ x I λ IJ , λ = ˆ x I λ IJ ˆ x J = δ IJ λ IJ , (B.35)so that in particular γ IJ λ IJ = 0 . The polarization is instead constrained by (cid:15) IJ = − (cid:15) IJK ˆ x K ,(cid:15) I = (cid:15) IJK ˆ x I ˆ x K ,(cid:15) = − (cid:15) IJK ˆ x I ˆ x J ˆ x K , (B.36)and satisfies the trace conditions γ IJ (cid:15) IJ = 0 , γ IJ (cid:15) IJK = 0 . (B.37)Up to gauge equivalence, the polarization tensor reads (cid:15) ∼ − (cid:15) IJK ˆ x I ˆ x J ˆ x K + 3 iω ˆ x I ˆ x J λ IJ , (B.38) (cid:15) K ∼ ˆ x I ˆ x J (cid:15) IJK − iω ˆ x I λ IK − iω ˆ x K (cid:0) ˆ x I (cid:15) IJ ˆ x J (cid:1) , (B.39) (cid:15) IJ ∼ − (cid:15) IJK ˆ x K + iωλ IJ + iω (ˆ x I λ JK + ˆ x J λ IK )ˆ x K , (B.40) (cid:15) IJK ∼ (cid:15) IJK − iω (ˆ x I λ JK + ˆ x J λ KI + ˆ x K λ IJ ) . (B.41)We parametrise the gauge tensor as λ IJ = A ˆ x I ˆ x J + B i (ˆ x I e iJ + ˆ x J e iI ) + C ij e iI e jJ , (B.42)where C ij are symmetric traceless coefficients. Imposing that the right-hand sides of (B.38),(B.39) and (B.40) vanish then fixes the coefficients A , B i and C ij to be iωA = (cid:15) IJK ˆ x I ˆ x J ˆ x K , iωB i = e Ii (cid:15) IJK ˆ x J ˆ x K , iωC ij = e Ii e Jj (cid:15) IJK ˆ x K . (B.43)One thus arrives at (cid:15) ∼ , (cid:15) I ∼ , (cid:15) IJ ∼ , (cid:15) IJK ∼ γ IL γ JM γ KN (cid:15) LMN . (B.44)– 25 –ubstituting into (B.41). Recalling the constraint γ IJ (cid:15) IJK = 0 , this means that the mostgeneral polarization tensor, up to gauge equivalence, is identified by a symmetric tracelesstensor on the n -sphere and thus characterises ( D − D − D − ( D − degrees of freedom.A convenient basis for such tensors is furnished by (cid:15) ( ijk ) lmn = δ i ( l δ jm δ kn ) − D γ ( lm γ ( ij δ k ) n ) . (B.45)We therefore adopt polarization tensors whose only nonzero components are the one withindices on the sphere (cid:15) ( ij ) kl specified by equation (B.45).For spin- s , the Fierz system reads (cid:3) ϕ µ s = 0 , (cid:3) Λ µ s − = 0 , ∂ · ϕ µ s − = 0 ,∂ · Λ µ s − = 0 ϕ (cid:48) µ s − = 0 , Λ (cid:48) µ s − = 0 . (B.46)As in the previous cases, the transversality conditions in Fourier space imply that one canalways trade a index for a projection along ˆ x , namely (cid:15) m I s − m = − (cid:15) m − I s − m J ˆ x J . (B.47)This fact allows us to take the independent components of the gauge parameter and of thepolarization tensor to be contained in λ I s − and (cid:15) I s and the trace conditions imply thatsuch tensors have traceless projections on the sphere. Parametrising λ I s − as λ IJ ··· K = A ˆ x I ˆ x J · · · ˆ x K + B i e i ( I ˆ x J · · · ˆ x K ) + · · · + C ij ··· k e i ( I e jJ · · · e kK ) , (B.48)with suitable traceless coefficients, one can then solve for (cid:15) m I s − m ∼ , m ≥ , (B.49)which maps (cid:15) I s to its projection on the sphere, (cid:15) IJ ··· K = γ IL γ JM · · · γ KN (cid:15) LM ··· N . (B.50)Symmetric traceless tensors on the n -sphere indeed possess (cid:0) D − ss (cid:1) − (cid:0) D − ss − (cid:1) degrees offreedom. C Symmetries of the Bondi-like gauge
In this appendix we provide an algorithm to fix the structure of the subleading terms inthe radial expansion of the components (cid:15) r s − k − i k of the gauge parameter, while checkingthe consistency of the solution with all constraints coming from the Bondi-like gauge.In Bondi coordinates, the gauge variation of a generic field component reads δϕ r s − k − l u l = l ∂ u (cid:15) r s − k − l u l − + s − k − lr ( r∂ r − k ) (cid:15) r s − k − l − u l + D (cid:15) r s − k − l u l + 2 r γ (cid:0) (cid:15) r s − k − l +1 u l − (cid:15) r s − k − l u l +1 (cid:1) , (C.1)where, as in Section 3.1, we omitted all sets of symmetrised angular indices.– 26 –ocussing on the variation of the components without any index u and using the traceconstraint on the gauge parameter, implying (cid:15) r s − k u = 12 (cid:18) (cid:15) r s − k +1 + 1 r (cid:15) (cid:48) r s − k − (cid:19) , (C.2)one finds (3.2): δϕ r s − k = 1 r (cid:110) ( s − k ) ( r∂ r − k ) (cid:15) ( k ) − γ (cid:15) ( k ) (cid:48) (cid:111) + D (cid:15) ( k − + r γ (cid:15) ( k − = 0 . (C.3)Let us recall that we denoted by (cid:15) ( k ) the components (cid:15) r s − k − i k of the gauge parameter,while a prime denotes a contraction with γ ij . Looking at (C.3), it is clear that the generalsolution of this equation has the form (3.3), (cid:15) ( k ) ( r, u, ˆx ) = r k ρ ( k ) ( u, ˆx ) + k − (cid:88) l = k r l α ( k,l ) ( u, ˆx ) , (C.4)where the α ( k,l ) are determined recursively (and algebraically) in terms of the ρ ( l ) with l < k . Indeed, (cid:15) ( k − and (cid:15) ( k − are fixed by the previous iterations, while the traces of each α ( k,l ) can be eliminated by computing the traces of (C.3): r (cid:110) [( k − s ) ( r∂ r − k ) + m ( D + 2( k − m − (cid:15) ( k )[ m ] + γ (cid:15) ( k )[ m +1] (cid:111) (C.5) = 2 m D · (cid:15) ( k − m − + D (cid:15) ( k − m ] + r (cid:104) m ( D + 2( k − m − (cid:15) ( k − m − + γ (cid:15) ( k − m ] (cid:105) , where (cid:15) ( k )[ l +1] ≡ γ · (cid:15) ( k )[ l ] . For instance, for k even, introducing for simplicity α ( k, k ) ≡ ρ ( k ) ,the last available trace gives (cid:20) (2 k − l )( s − k ) + k ( D + k − (cid:21) α ( k,l )[ k ] = k D · α ( k − ,l − k − ] + k ( D + k − α ( k − ,l − k − ] , (C.6)and one can substitute the result in (C.5) to determine α ( k,l )[ k − ] and proceed recursively.The recursion relations become rather cumbersome after the first few values of k , but,as an example, we can provide their explicit solution up to k = 3 (with s generic), that– 27 –uffices to capture all residual symmetries of the Bondi-like gauge for fields of spin s ≤ : (cid:15) r s − = ρ (0) , (C.7) (cid:15) r s − = r ρ (1) + rs − ∂ρ (0) , (C.8) (cid:15) r s − = r ρ (2) + r s − (cid:18) D ρ (1) − D + s − γ D · ρ (1) (cid:19) + ( s − r s − (cid:18) D ρ (0) − D + 2 s − γ [∆ − ( s − s − ρ (0) (cid:19) , (C.9) (cid:15) r s − = r ρ (3) + r s − (cid:18) D ρ (2) − D + s − γ D · ρ (2) (cid:19) + ( s − r s − (cid:18) D ρ (1) − D + 2 s − γ (cid:104) (∆ + D − s ( s − − ρ (1) + 2( D + 2 s − D + s − DD · ρ (1) (cid:105)(cid:19) + ( s − r s − (cid:18) D ρ (0) − D + 2 s − γ D (3∆ + 2 D − s (3 s − − ρ (0) (cid:19) . (C.10)Setting to zero all ρ ( k ) with k > , one recovers the full structure of the supertranslationparameters (cf. (2.5)) up to k = 3 for any value of s .The (cid:15) ( k ) are the only components of the gauge parameter that enter the surface charge(2.7). Still, we imposed only part of the conditions necessary to preserve the Bondi-likegauge (2.1) and we have to check if the rest imposes additional constraints on the tensors ρ ( k ) . Preserving the vanishing of the traces of the non-zero components of the field yields δϕ (cid:48) u s − k = − r (cid:2) ( D + 2( k − (cid:15) u s − k − + γ (cid:15) (cid:48) u s − k +1 (cid:3) + ( s − k ) ∂ u (cid:15) (cid:48) u s − k − + 2 D · (cid:15) u s − k + D (cid:15) (cid:48) u s − k + 2 r (cid:2) ( D + 2( k − (cid:15) ru s − k + γ (cid:15) (cid:48) ru s − k (cid:3) = 0 , (C.11)which fixes the components (cid:15) u s − k − using the same strategy we employed in the analysisof (3.2). Indeed, all components with at least one index r and one index u , like thoseappearing in the last line of (C.11), can be rewritten in terms of the (cid:15) ( k ) using the traceconstraint on the gauge parameter: ξ r α u β = 12 (cid:18) ξ r α +1 u β − + 1 r ξ (cid:48) r α − u β − (cid:19) . (C.12)At this stage we have fixed all components of the gauge parameter in terms of the tensors ρ (0) ( u, ˆx ) , . . . , ρ ( s − ( u, ˆx ) , but we still have to check if the vanishing of the variations (C.1)with l ≥ and k + l < s imposes additional constraints on them. This question has beenalready addressed in Section 5.2 of [2], where it has been shown that preserving the Bondi-like gauge requires (3.4), that is ∂ u ρ ( k ) + s − k − D + s + k − D · ρ ( k +1) = 0 for k < s − . (C.13)– 28 – Stationary and static solutions of Fronsdal’s equations
In this appendix we characterise the behaviour near I + of stationary and static solutions ofFronsdal’s equations, assuming that asymptotically the fields can be expanded in powers ofthe radial coordinate as in (2.2). Under this hypothesis, the source-free Fronsdal equations F r s − k i k = 0 imply ( n + 2 k )( D − n − U ( k,n ) = ( n + 2 k − D · U ( k +1 ,n − , (D.1)while the equations F u s − k i k = 0 give [( D − n − s − k −
1) + ( n + 2 k )] ∂ u U ( k,n ) = ( s − k ) ∂ u D · U ( k +1 ,n − − [∆ − ( n − D − n − k − − k ( D − k − U ( k,n − + DD · U ( k,n − − ( D − n − D U ( k − ,n ) − γ D · U ( k − ,n ) + 2( D − n − γ U ( k − ,n +1) . (D.2)All other equations of motion are identically satisfied in the Bondi-like gauge. Combining(D.1) with the traces of (D.2) one obtains (3.14), and using the latter in (D.2) with k = s one eventually obtains (3.15). In the following, instead, we shall also have to consider theequations above for values of n for which some of these substitutions are not possible. Forinstance, for n = D − the lhs of (D.1) vanishes and this implies ( D· ) k − l U ( k,D − k + l − = 0 , (D.3)that for l = 0 gives the constraint (2.13) that we use in the evaluation of supertranslationcharges. D.1 Stationary solutions
In analogy with [58], we identify stationary solutions of Fronsdal’s equations as those sat-isfying ∂ u U ( k,n ) = 0 for n ≤ D − k − , (D.4)and we now prove that, up to pure-gauge contributions, this definition implies U ( k,n ) = 0 for n < D − k − . (D.5)The latter condition corresponds to eq. (2.12), that we used in the evaluation of higher-spinsupertranslations charges.To begin with, we now prove that (D.4) implies U ( k,D − k − = 0 . To this end, let usfirst consider (D.2) for k = 0 and n = D − . Taking (D.4) into account, this gives − D + 4] U (0 ,D − ⇒ U (0 ,D − . (D.6)We can now proceed by induction on k . Assuming U ( l,D − l − = 0 for l < k one first obtains D · U ( k,D − k − from (D.1) and, then, evaluating (D.2) at n = D − k − , − ( D + k − U ( k,D − k − ⇒ U ( k,D − k − = 0 . (D.7)– 29 –he last implication follows from (B.7), implying, in particular, that the eigenvalues of theLaplacian on an irreducible tensor of any rank are always negative. The same argument canbe extended, modulo pure-gauge contributions, also to all n < D − k − with an inductionin n . Assuming U (0 ,n ) = 0 eq. (D.2) gives indeed − ( n − D − n − U (0 ,n − = [∆ + (cid:96) ( (cid:96) + D − U (0 ,D − (cid:96) ) , (D.8)which implies U (0 ,D − (cid:96) ) = 0 for (cid:96) ≤ − or, equivalently, U (0 ,n ) = 0 for n ≤ D − . Alltensors U ( k,n + k − can then be set to zero with the same strategy that led to (D.7). D.2 Static solutions
Static solutions are defined by ∂ u U ( k,n ) = 0 ∀ n , (D.9)and we now argue that this condition implies, up to pure-gauge configurations, U ( k,n ) = 0 for n < D − (D.10)together with K ( k ) ≡ D U ( k,D − − D + 2( k − γ D · U ( k,D − = 0 . (D.11)For the U ( k,n ) with − k < n < D − k − the same considerations as in the previoussubsection apply, so that (D.5) is valid also for static solutions. Outside of this range, wehave to go back to equations (D.1) and (D.2). For n = D − , they imply k − D − − k ( D − k − U ( k,D − − K ( k − , (D.12)where K ( k ) denotes the conformal Killing equation for U ( k,D − as in (D.11). For n = D − one can use again (D.1) for all tensors in the second line of (D.2) to obtain D · K ( k ) . (D.13)For static fields, the equations of motion thus set to zero the divergence of the conformalKilling tensor equation (3.61) and on a compact space like the celestial sphere this sufficesto set it to zero altogether. To support this statement let us notice that, assuming that thedivergence of U ( k,D − satisfies (cid:0) D + 2 γ (cid:1) D· U ( k,D − − D + 2 k − γ (cid:18) D − D + 2 k − γ D· (cid:19) D·D· U ( k,D − = 0 , (D.14)the symmetrised gradient of (D.13) can be rewritten as DD · K ( k ) = [∆ + ( k − k + D − − ( k + 1)] K ( k ) + 2 D + 2( k − γ D · D · K ( k ) . (D.15)The second term vanishes on shell, while the invertibility on S D − of the differential operatorin the first term guarantees K ( k ) = 0 . (D.16)– 30 –n the previous step we used the identity (D.15) that one can prove with a similar approachstarting from the double symmetrised gradient of D · D · K ( k ) = 0 and progressing byrecursion to identify a series of identities of the type D k +1 ( D· ) k U ( k,D − = γ ( · · · ) . We donot have a complete proof that (D.13) implies (D.16), but we checked that this is true upto s = 3 and we shall assume this implication.Eq. (D.12) then allows one to set to zero all U ( k,D − and, via (D.1), also all U ( k,n ) inthe range D − k − ≤ n ≤ D − , thus providing the missing instances of (D.10). References [1] A. Campoleoni, D. Francia and C. Heissenberg, “
On higher-spin supertranslations andsuperrotations ,” JHEP (2017) 120 [arXiv:1703.01351 [hep-th]].[2] A. Campoleoni, D. Francia and C. Heissenberg, “
Asymptotic Charges at Null Infinity in AnyDimension ,” Universe (2018) 47 [arXiv:1712.09591 [hep-th]].[3] H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, “ Gravitational waves in generalrelativity. VII. Waves from axisymmetric isolated systems ,” Proc. Roy. Soc. Lond. A (1962) 21.[4] R. K. Sachs, “
Gravitational waves in general relativity. VIII. Waves in asymptotically flatspace-times ,” Proc. Roy. Soc. Lond. A (1962) 103.[5] R. Sachs, “
Asymptotic symmetries in gravitational theory ,” Phys. Rev. (1962) 2851.[6] A. Strominger, “
Asymptotic Symmetries of Yang-Mills Theory ,” JHEP (2014) 151[arXiv:1308.0589 [hep-th]].[7] G. Barnich and P. H. Lambert, “
Einstein-Yang-Mills theory: Asymptotic symmetries ,” Phys.Rev. D (2013) 103006 [arXiv:1310.2698 [hep-th]].[8] T. He, P. Mitra, A. P. Porfyriadis and A. Strominger, “ New Symmetries of Massless QED ,”JHEP (2014) 112 [arXiv:1407.3789 [hep-th]].[9] M. Campiglia and A. Laddha, “
Asymptotic symmetries of QED and Weinberg’s soft photontheorem ,” JHEP (2015) 115 [arXiv:1505.05346 [hep-th]].[10] A. Strominger,
Lectures on the Infrared Structure of Gravity and Gauge Theory , PrincetonUniv. Press, Princeton, New Jersey, USA (2018) [arXiv:1703.05448 [hep-th]].[11] S. Hollands and A. Ishibashi, “
Asymptotic flatness and Bondi energy in higher dimensionalgravity ,” J. Math. Phys. (2005) 022503 [gr-qc/0304054].[12] K. Tanabe, S. Kinoshita and T. Shiromizu, “ Asymptotic flatness at null infinity in arbitrarydimensions ,” Phys. Rev. D (2011) 044055 [arXiv:1104.0303 [gr-qc]].[13] S. Hollands, A. Ishibashi and R. M. Wald, “ BMS Supertranslations and Memory in Four andHigher Dimensions ,” Class. Quant. Grav. (2017) 155005 [arXiv:1612.03290 [gr-qc]].[14] G. Barnich and C. Troessaert, “ Symmetries of asymptotically flat 4 dimensional spacetimesat null infinity revisited ,” Phys. Rev. Lett. (2010) 111103 [arXiv:0909.2617 [gr-qc]].[15] G. Barnich and C. Troessaert, “
Aspects of the BMS/CFT correspondence ,” JHEP (2010) 062 [arXiv:1001.1541 [hep-th]].[16] D. Kapec, V. Lysov and A. Strominger, “
Asymptotic Symmetries of Massless QED in EvenDimensions ,” Adv. Theor. Math. Phys. (2017), 1747-1767 [arXiv:1412.2763 [hep-th]]. – 31 –
17] D. Kapec, V. Lysov, S. Pasterski and A. Strominger, “
Higher-Dimensional Supertranslationsand Weinberg’s Soft Graviton Theorem ,” Ann. Math. Sci. Appl. (2017) 69[arXiv:1502.07644 [gr-qc]].[18] P. Mao and H. Ouyang, “ Note on soft theorems and memories in even dimensions ,” Phys.Lett. B (2017), 715-722 [arXiv:1707.07118 [hep-th]].[19] M. Campiglia and L. Coito, “
Asymptotic charges from soft scalars in even dimensions ,”Phys. Rev. D (2018) 066009 [arXiv:1711.05773 [hep-th]].[20] M. Pate, A. M. Raclariu and A. Strominger, “ Gravitational Memory in Higher Dimensions ,”JHEP (2018) 138 [arXiv:1712.01204 [hep-th]].[21] H. Afshar, E. Esmaeili and M. M. Sheikh-Jabbari, “
Asymptotic Symmetries in p -FormTheories ,” JHEP (2018) 042 [arXiv:1801.07752 [hep-th]].[22] A. Aggarwal, “ Supertranslations in Higher Dimensions Revisited ,” Phys. Rev. D (2019)026015 [arXiv:1811.00093 [hep-th]].[23] T. He and P. Mitra, “ Asymptotic symmetries and Weinberg’s soft photon theorem inMink d +2 ,” JHEP (2019) 213 [arXiv:1903.02608 [hep-th]].[24] T. He and P. Mitra, “ Asymptotic symmetries in (d + 2)-dimensional gauge theories ,” JHEP (2019) 277 [arXiv:1903.03607 [hep-th]].[25] F. Capone and M. Taylor, “ Cosmic branes and asymptotic structure ,” JHEP (2019) 138[arXiv:1904.04265 [hep-th]].[26] A. Campoleoni, D. Francia and C. Heissenberg, “ Electromagnetic and color memory in evendimensions ,” Phys. Rev. D (2019) 085015 [arXiv:1907.05187 [hep-th]].[27] C. Heissenberg, “
Topics in Asymptotic Symmetries and Infrared Effects ,” arXiv:1911.12203[hep-th].[28] F. Capone, “
BMS Symmetries and Holography: An Introductory Overview ,” in
EinsteinEquations: Physical and Mathematical Aspects of General Relativity , Cacciatori S., GüneysuB., Pigola S. (eds.), DOMOSCHOOL 2018. Birkhäuser, Cham, Switzerland (2019).[29] E. Esmaeili, “ p -form gauge fields: charges and memories ,” arXiv:2010.13922 [hep-th].[30] E. Esmaeili, “ Asymptotic Symmetries of Maxwell Theory in Arbitrary Dimensions at SpatialInfinity ,” JHEP (2019) 224 [arXiv:1902.02769 [hep-th]].[31] M. Henneaux and C. Troessaert, “ Asymptotic structure of electromagnetism in higherspacetime dimensions ,” Phys. Rev. D (2019) 125006 [arXiv:1903.04437 [hep-th]].[32] F. Cachazo and A. Strominger, “ Evidence for a New Soft Graviton Theorem ,”[arXiv:1404.4091 [hep-th]].[33] N. Afkhami-Jeddi, “
Soft Graviton Theorem in Arbitrary Dimensions ,” [arXiv:1405.3533[hep-th]].[34] M. Campiglia and A. Laddha, “
Asymptotic symmetries and subleading soft gravitontheorem ,” Phys. Rev. D (2014) 124028 [arXiv:1408.2228 [hep-th]].[35] M. Campiglia and A. Laddha, “ New symmetries for the Gravitational S-matrix ,” JHEP (2015) 076 [arXiv:1502.02318 [hep-th]].[36] D. Kapec, V. Lysov, S. Pasterski and A. Strominger, “ Semiclassical Virasoro symmetry ofthe quantum gravity S -matrix ,” JHEP (2014) 058 [arXiv:1406.3312 [hep-th]]. – 32 –
37] E. Conde and P. Mao, “
BMS Supertranslations and Not So Soft Gravitons ,” JHEP (2017)060 [arXiv:1612.08294 [hep-th]].[38] S. G. Avery and B. U. W. Schwab, “ Burg-Metzner-Sachs symmetry, string theory, and softtheorems ,” Phys. Rev. D (2016) 026003 [arXiv:1506.05789 [hep-th]].[39] D. Colferai and S. Lionetti, “ Asymptotic symmetries and subleading soft graviton theorem inhigher dimensions ,” arXiv:2005.03439 [hep-th].[40] A. Campoleoni, D. Francia and C. Heissenberg, “
Asymptotic symmetries and charges at nullinfinity: from low to high spins ,” EPJ Web Conf. (2018) 06011 [arXiv:1808.01542[hep-th]].[41] A. Campoleoni, M. Henneaux, S. Hörtner and A. Leonard, “
Higher-spin charges inHamiltonian form. I. Bose fields ,” JHEP (2016) 146 [arXiv:1608.04663 [hep-th]].[42] A. Campoleoni, M. Henneaux, S. Hörtner and A. Leonard, “ Higher-spin charges inHamiltonian form. II. Fermi fields ,” JHEP (2017) 058 [arXiv:1701.05526 [hep-th]].[43] D. Grumiller, A. Pérez, M. M. Sheikh-Jabbari, R. Troncoso and C. Zwikel, “ Spacetimestructure near generic horizons and soft hair ,” Phys. Rev. Lett. (2020) 041601[arXiv:1908.09833 [hep-th]].[44] S. Weinberg, “
Photons and Gravitons in S -Matrix Theory: Derivation of ChargeConservation and Equality of Gravitational and Inertial Mass ,” Phys. Rev. (1964)B1049.[45] S. Weinberg, “ Infrared photons and gravitons ,” Phys. Rev. (1965) B516.[46] S. Deser and R. I. Nepomechie, “
Gauge Invariance Versus Masslessness in De Sitter Space, ”Annals Phys. (1984) 396.[47] A. Higuchi, “
Symmetric Tensor Spherical Harmonics on the N Sphere and Their Applicationto the De Sitter Group SO( N ,1) ,” J. Math. Phys. (1987) 1553 [erratum: J. Math. Phys. (2002) 6385].[48] S. Deser and A. Waldron, “ Partial masslessness of higher spins in (A)dS ,” Nucl. Phys. B (2001), 577-604 [arXiv:hep-th/0103198 [hep-th]].[49] Y. M. Zinoviev, “
On massive high spin particles in AdS ,” arXiv:hep-th/0108192 [hep-th].[50] L. Dolan, C. R. Nappi and E. Witten, “
Conformal operators for partially massless states ,”JHEP (2001) 016 [arXiv:hep-th/0109096 [hep-th]].[51] G. Compère, A. Fiorucci and R. Ruzziconi, “ Superboost transitions, refraction memory andsuper-Lorentz charge algebra ,” JHEP (2018) 200 [arXiv:1810.00377 [hep-th]].[52] G. Compère, A. Fiorucci and R. Ruzziconi, “ The Λ -BMS group of dS and new boundaryconditions for AdS ,” Class. Quant. Grav. (2019) 195017 [arXiv:1905.00971 [gr-qc]].[53] A. Fiorucci and R. Ruzziconi, “ Charge Algebra in Al(A)dS n Spacetimes ,” arXiv:2011.02002[hep-th].[54] A. Sagnotti, “
Notes on Strings and Higher Spins ,” J. Phys. A (2013) 214006[arXiv:1112.4285 [hep-th]].[55] D. Francia and C. Heissenberg, “ Two-Form Asymptotic Symmetries and Scalar SoftTheorems ,” Phys. Rev. D (2018) 105003 [arXiv:1810.05634 [hep-th]]. – 33 –
56] M. Henneaux and C. Troessaert, “
Asymptotic structure of a massless scalar field and its dualtwo-form field at spatial infinity ,” JHEP (2019) 147 [arXiv:1812.07445 [hep-th]].[57] L. Freidel, F. Hopfmüller and A. Riello, “ Asymptotic Renormalization in Flat Space:Symplectic Potential and Charges of Electromagnetism ,” JHEP (2019) 126[arXiv:1904.04384 [hep-th]].[58] G. Satishchandran and R. M. Wald, “ Asymptotic behavior of massless fields and the memoryeffect ,” Phys. Rev. D (2019) 084007 [arXiv:1901.05942 [gr-qc]].[59] D. Francia, “ Generalised connections and higher-spin equations ,” Class. Quant. Grav. (2012) 245003 [arXiv:1209.4885 [hep-th]].[60] A. Campoleoni and D. Francia, “ Maxwell-like Lagrangians for higher spins ,” JHEP (2013)168 [arXiv:1206.5877 [hep-th]].[61] S. G. Avery and B. U. W. Schwab, “ Noether’s second theorem and Ward identities for gaugesymmetries ,” JHEP (2016) 031 [arXiv:1510.07038 [hep-th]].[62] H. Adami, M. M. Sheikh-Jabbari, V. Taghiloo, H. Yavartanoo and C. Zwikel, “ Symmetries atnull boundaries: two and three dimensional gravity cases ,” JHEP (2020) 107[arXiv:2007.12759 [hep-th]].[63] A. Strominger and A. Zhiboedov, “ Gravitational Memory, BMS Supertranslations and SoftTheorems ,” JHEP (2016) 086 [arXiv:1411.5745 [hep-th]].[64] M. S. Drew and J. D. Gegenberg, “ Conformally Covariant Massless Spin-2 Field Equations ,”Nuovo Cim. A (1980), 41-56.[65] M. Campiglia and J. Peraza, “ Generalized BMS charge algebra ,” Phys. Rev. D (2020)104039 [arXiv:2002.06691 [gr-qc]].[66] M. G. Eastwood, “
Higher symmetries of the Laplacian ,” Annals Math. (2005), 1645-1665[arXiv:hep-th/0206233].[67] M. A. Rubin and C. R. Ordonez, “
Eigenvalues and degeneracies for n -dimensional tensorspherical harmonics ,” J. Math. Phys. (1984) 2888.[68] M. A. Rubin and C. R. Ordonez, “ Symmetric Tensor Eigen Spectrum of the Laplacian on n Spheres ,” J. Math. Phys. (1985) 65.(1985) 65.