The asymptotic charges of Curtright dual graviton and Curtright extensions of BMS algebra

The asymptotic cha rges of Curtright dual graviton and Curtright extensions of BMS algeb ra F ederico Manzoni a,b a Mathematics and Physics dep artment, R oma T r e, Via del la V asc a Navale 84, R ome, Italy b INFN R oma T r e Se ction, Physics dep artment, Via del la V asc a Navale 84, R ome, Italy E-mail: federico.manzoni@uniroma3.it, federico13.manzoni97@gmail.com, ORCID ID: 0000-0002-9979-6154 Abstra ct: This pap er studies the asymptotic gauge c harges of the Curtrigh t mixed- symmetry rank-3 field ϕ [ ρσ ] ν in Mink o wski spacetime, interpreted in D = 5 as the dual gra viton. In Bondi co ordinates at future n ull infinity , w e imp ose radiation fall-offs and fix a de Donder-lik e gauge together with an on-shell traceless condition, similarly to what happ ens in linearized gra vit y . Surface c harges asso ciated with the residual gauge transformations are constructed as boundary in tegrals via Nöther’s 2-form. In D = 5 , exploiting Ho dge/Ho dge-lik e decomp ositions on S 3 , the charge splits into a scalar sector Q Φ , a vector sector Q V and a TT sector Q y TT . Q Φ is parametrized by a single arbitrary scalar function Φ (in terpreted as the supertranslation-like parameter), Q V is parametrized b y a v ector field V i ∈ Diff ( S 3 ) and the TT sector Q y TT is parametrized b y a transverse-traceless rank-2 tensor y TT ij ∈ TT ( S 3 ) . The corresp onding charge algebra closes only if V i ∈ o (4) as semidirect sum o (4) + ( C ∞ ( S 3 ) ⊕ TT ( S 3 )) , i.e. an ab elian extension of a BMS -lik e algebra featuring a higher-spin–lik e sup ertranslation sector. Con ten ts 1 In tro duction 1 2 The Curtrigh t three indexes field 4 3 The de Donder-lik e gauge fixing and residual gauge 7 4 Asymptotic c harges 11 5 Connections with the gra viton asymptotic symmetries in D = 5 16 6 Conclusions and outlo ok 19 A Equations 22 A.1 Equations from residual gauge conditions 22 A.2 Equations from fall-off conditions 24 A.3 Equations of motion 26 B Ho dge-lik e decomp osition for ( p, q ) -mixed symmetry tensors 28 C Useful comm utation relations and asymptotic gauge fixing 33 C.1 V anishing of the antisymmetric part of S ij 33 C.2 Asymptotic gauge fixing 36 C.3 Other useful comm utators 36 1 In tro duction The study of asymptotic symmetries has long provided a bridge b etw een gauge redundancy and gen uine ph ysical information enco ded at infinit y [ 1 – 5 ]. In gravit y , the analysis of asymptotically flat space-times led to the discov ery that the sym- metry group at null infinity is larger than the Poincaré group, giving rise to the Bondi-Metzner-Sac hs-v an der Burg (BMS) group and, in particular, to an infinite- dimensional family of angle-dep enden t translations [ 6 – 8 ]. In the last decade this sub ject has acquired a renew ed cen trality , largely due to the realization that asymp- totic symmetries, soft theorems and memory effects are different facets of the same infrared structure of gauge theories [ 9 – 17 ]. These developmen ts ha v e stimulated a systematic re-examination of the precise role pla yed by boundary conditions, gauge c hoices and the definition of conserv ed c harges, not only in four-dimensional gra vity – 1 – [ 18 – 23 ] but also in higher dimensional gravit y [ 24 – 29 ], differen t backgrounds [ 30 – 34 ], higher spin gauge theories [ 13 , 35 – 39 ] and for more exotic gauge systems [ 14 , 40 – 46 ]. A k ey lesson emerging from this mo dern persp ectiv e is that the asymptotic symmetry group is not a purely kinematical datum but dep ends delicately on which fall-off conditions one imp oses on the fields. In D > 4 gravit y , for instance, there is a w ell-kno wn tension b etw een b oundary conditions tailored to accommo date radiative degrees of freedom and those that allo w for an enlargemen t of the asymptotic symmetry group b ey ond P oincaré [ 24 , 25 , 28 , 29 ]. Differen t c hoices can lead to different residual gauge algebras and differen t sets of finite, non-trivial c harges, while still capturing ph ysically meaningful radiation. A ccordingly , one natural strategy is to fix a gauge that reduces the equations of motion to a simple w av e equation, imp ose radiation- compatible fall-offs motiv ated b y flux finiteness, solve explicitly for the residual gauge parameters compatible with these conditions and then construct the asso ciated surface c harges [ 1 , 47 ]. The same circle of ideas extends well beyond the metric formulation of gra vit y . In fact, already at the free-field lev el, massless gauge fields can b e form ulated in terms of Lorentz tensors [ 48 ] of v arious symmetry types, including represen tations described b y non-trivial Y oung diagrams [ 49 – 55 ]. Suc h mixed symmetry fields arise naturally in higher-spin theory [ 56 – 59 ], in the sp ectrum of string theory [ 60 – 63 ] and in duality- co v ariant formulations of gravit y and gauge theories [ 45 , 64 – 66 ]. Their cov ariant description requires a refined notion of gauge symmetry , typically inv olving more than one gauge parameter and, crucially , a hierarc hy of gauge-for-gauge redundancies. These features make the analysis of asymptotic symmetries particularly in teresting and p oten tially subtle. F rom this standp oint, mixed-symmetry fields pro vide a controlled arena in whic h one can test how robust the standard infrared story is under changes of field v ariables and under dualit y transformations. A paradigmatic example is the Curtrigh t field, the simplest “ho ok” mixed- symmetry tensor, whic h in the index conv ention used in this pap er is written as ϕ [ ρσ ] ν and corresp onds to the Y oung tableau λ = (2 , 1) in D = 5 . The Curtrigh t mo del generalizes the notion of gauge field b eyond totally symmetric and totally an tisymmetric tensors and comes equipp ed with t w o gauge parameters, one symmetric and one antisymmetric, together with a gauge-for-gauge vector parameter [ 53 ]. In fiv e space-time dimensions, the Curtright field is of particular interest b ecause it pro vides a dual description of the graviton: on-shell, the tw o formulations enco de the same propagating degrees of freedom, while off-shell their gauge structures lo ok quite differen t [ 67 , 68 ]. Understanding ho w asymptotic c harges b eha ve under such dualities is therefore a natural question, both for clarifying the ph ysical meaning of dual observ ables and for exploring whether the infrared symmetry algebra is in v ariant under c hanges of field v ariables and dual descriptions. The aim of this paper is to carry out a detailed analysis of asymptotic c harges for the Curtright field at n ull infinit y , in a framework designed to mak e con tact with – 2 – the gra vitational BMS story in D = 5 . The starting p oint is considering Bondi patch of Mink o wski space-time with co ordinates adapted to future null infinit y , together with radiation fall-offs motiv ated b y finiteness and non-v anishing of the energy flux. W e then fix a de Donder-like gauge condition, supplemen ted by an on-shell traceless condition in order to isolate the irreducible SO ( D − 1 , 1) represen tation, and thereby reduces the equations of motion to a free w a v e equation for ϕ [ ρσ ] ν . This is v ery similar to what happ ens in linearized gra vity and this prescription could b e extended to all mixed symmetry tensors. With these gauge conditions in place, the residual gauge parameters are constrained b y differen tial equations and by the requiremen t that they preserv e the fall-offs. T o solv e this coupled system in a uniform w a y across dimensions, the pap er assumes p olyhomogeneous expansions for the gauge parameters, allowing b oth half-in teger p ow ers and logarithmic terms. Ha ving identified the relev an t residual transformations, the next step is the construction of the corresp onding asymptotic c harges. The resulting Nö ether charges are surface integrals ov er S D − 2 at fixed retarded time and are expressed in terms of the leading b oundary data of the (partial) field strength comp onents with r u - indices, giving us an electric-like c harge. In this resp ect, the mechanism is closely analogous to what happ ens in p -form gauge theories in Lorenz gauge [ 46 ]: a gauge condition remo v es p otentially div ergent comp onen ts, the charge becomes electric-like in that it in v olves radial-n ull comp onen ts of the field strength and only angular comp onen ts of the gauge parameters surviv e in the final expression. This parallel is conceptually useful because it highlights how mixed-symmetry gauge fields fit into the broader pattern of infrared charges for massless gauge systems while at the same time stressing the new ingredien ts brough t in b y Y oung-pro jected tensors and b y complex gauge-for-gauge redundancy hierarc h y . In D = 5 we can directly compare to the graviton because of duality . Exploiting Ho dge and Hodge-like decompositions on S 3 for the symmetric and an tisymmetric gauge parameters, the Curtrigh t charge naturally splits into a scalar sector Q Φ , a vector sector Q V and a TT sector Q y TT . The scalar sector is parametrized b y a single arbitrary function Φ on S 3 , which is in terpreted as the analogue of the sup ertranslation parameter while the v ector sector is parametrized b y a vector field V i that could pla y the role of sup errotations in this setting. Ho w ever, the presence of the TT sector, interpretable as a higher spin supertranslation imp oses V i ∈ o (4) , i.e. a Killing vector, to ensure closure of the algebra. The resulting c harge algebra closes as the semidirect sum Diff ( S 3 ) + ( C ∞ ( S 3 ) ⊕ TT ( S 3 )) forming an abelian extension of a BMS -lik e algebra: the Curtright extension CBMS ( S 3 ) . Notably , within the b oundary conditions and asymptotic gauge-fixing adopted here, one finds only one indep enden t scalar sup ertranslation-lik e parameter, rather than t w o scalar parameters as suggested b y certain Hamiltonian treatmen ts [ 29 ]. Beyond the sp ecific results for D = 5 , the analysis presen ted here fits in to a broader programme of clarifying asymptotic symmetries for general massless fields in arbitrary dimensions. – 3 – The paper is structured as follows. Section 2 reviews the Curtright (2 , 1) field, its gauge symmetries and gauge-for-gauge redundancy , and sets up the conv entions used throughout. Section 4 imp oses radiation fall-offs in Bondi co ordinates, implemen ts the de Donder-like and traceless gauge fixing and solves the resulting residual-gauge system via p olyhomogeneous expansions, isolating the gauge-parameter comp onen ts relev an t for finite charges. Section 4 constructs the asymptotic charges from Nöether’s 2-form and deriv es a general expression v alid in arbitrary dimension. Section 5 specializes to D = 5 , p erforms the Ho dge/Ho dge-like decomp ositions of gauge parameters on S 3 , derives the split Q = Q Φ + Q V + Q y TT , computes the c harge algebra and discusses the comparison with gra viton asymptotic c harges. The app endices collect complemen tary material, including the explicit residual-gauge and fall-off equations, the Ho dge-like decomp osition technology for ( p, q ) -mixed symmetry tensors and tec hnical comm utation relations and asymptotic gauge-fixing steps. Notation. W e adopt the notations ˆ ϕ ν := ϕ αν α = η αβ ϕ [ αβ ] ν , ¯ ϕ ρ := ϕ ρα α = η αβ ϕ [ αρ ] β ; ˆ ∂ · ϕ ρν := ∂ α ϕ αρν = η αβ ∂ α ϕ [ β ρ ] ν , ¯ ∂ · ϕ ρσ := ∂ α ϕ ρσ α = η αβ ∂ α ϕ [ ρσ ] β , (1.1) and similar for lo w er indexes and other tensors. 2 The Curtrigh t three indexes field In order to describe gauge fields the standard w a y is to use both totally symmetric and totally antisymmetric Loren tz tensors of arbitrary rank. Both t yp es of gauge field tensors app ear quite naturally in theoretical analyses of some interesting physical problems suc h as the description of massless particle fields of spin s ( s + 1 2 ) using symmetric gauge tensors (spinor-tensors) of rank s or gauge fields naturally coupled to strings described by an tisymmetric tensors. How ever, in 1980 Thomas Curtrigh t generalize the concept of a gauge field to include higher rank Loren tz tensors whic h are neither totally symmetric nor totally an tisymmetric under spacetime index p er- m utations [ 49 – 51 ]. The basic simplest example is the ho ok field: a rank 3 tensor whose index p erm utation symmetry corresp onds to the Y oung diagram (2 , 1) . (2.1) The fundamen tal step is to construct irreducible represen tations of the p ermutation group using symmetrizers and an tisymmetrizers; in the following w e assume to hav e applied the an tisymmetrizer last, obtaining a field with manifest antisymmetry on the first t w o indexes ϕ [ ρσ ] ν . F rom the Y oung diagram of the Curtrigh t three indexes – 4 – field w e can extract the gauge parameter remo ving one b ox so that w e still hav e a Y oung diagram ⇒ , ; (2.2) so we ha v e one totally symmetric gauge parameter λ ( µν ) ≡ λ µν and one totally an tisymmetric gauge parameter Λ [ µν ] ≡ Λ µν . W e can note that we can again eliminate a b o x and still get a Y oung diagram ⇒ ; ⇒ ; (2.3) this is called gauge-for-gauge redundancy: essen tially the gauge parameters of a gauge field ha v e their own gauge redundancy , here parametrized by a v ector Θ µ . The most ob vious gauge transformation for the Curtrigh t three indexes field is giv en by ϕ ′ [ ρσ ] ν = ϕ [ ρσ ] ν + ∂ ρ λ ( σ ν ) − ∂ σ λ ( ρν ) − ∂ ν Λ ( ρσ ) , (2.4) with Λ ′ [ µν ] = Λ [ µν ] + ∂ µ Θ ν − ∂ ν Θ µ , λ ′ ( µν ) = λ ( µν ) + 2 ∂ µ Θ ν + 2 ∂ ν Θ µ , (2.5) but the irreducibilit y condition under GL( D − 1 , 1) , namely ϕ [ ρσ ] ν + ϕ [ ν ρ ] σ + ϕ [ σ ν ] ρ = 0 , (2.6) requires adding the term − 2 ∂ ν Λ σ ρ to the gauge redundancy of the field ϕ ( ρσ ) ν ; therefore w e ha ve ϕ ′ [ ρσ ] ν = ϕ [ ρσ ] ν + ∂ ρ λ ( σ ν ) − ∂ σ λ ( ρν ) + ∂ ρ Λ [ σ ν ] − ∂ σ Λ [ ρν ] + 2 ∂ ν Λ [ σ ρ ] . (2.7) W e define a field strength H [ αβ γ ] µ := ∂ α ϕ [ β γ ] µ + ∂ β ϕ [ γ α ] µ + ∂ γ ϕ [ αβ ] µ , (2.8) that is more precisely a "partial field strength" using the nomenclature of [ 45 ] since it is not completely gauge in v arian t δ λ, Λ H [ αβ γ ] µ = − 2 ∂ ν [ ∂ α Λ [ β γ ] + ∂ β Λ [ γ α ] + ∂ γ Λ [ αβ ] ] . (2.9) Ho w ever the contractions H [ αβ γ ] µ H [ αβ γ ] µ and H αβ H αβ , where H αβ := η γ µ H αβ γ µ , can b e com bined into a gauge inv ariat lagrangian density L := − 1 6 ( H αβ γ µ H αβ γ µ − 3 H αβ H αβ ) . (2.10) – 5 – The Euler-Lagrange equations obtained b y v arying the lagrangian densit y are E [ αβ ] γ + 1 2 [ η αγ E β − η β γ E α ] = 0 , (2.11) where E [ αβ ] γ := η µν ∂ µ H [ ν αβ ] γ − η µν ∂ γ H [ αβ µ ] ν , E α := η µν E [ αµ ] ν = 2 η µν ∂ ν H µα . (2.12) Here we pro v e an off-shell and an on-shell iden tit y satisfied b y the partial field strength ( 2.8 ). Prop osition 2.1 (Off-shell identities of H [ αβ γ ] µ ) . The p artial field str ength H [ αβ γ ] µ define d in ( 2.8 ) satisfies the off-shel l identities (a) H [ αβ γ ] µ − H [ µαβ ] γ + H [ γ µα ] β − H [ β γ µ ] α = 0 , (2.13) (b) ∂ [ δ H αβ γ ] µ = 0 , (2.14) (c) ∂ µ H αβ γ µ − ∂ α H β γ − ∂ β H γ α − ∂ γ H αβ = 0; (2.15) Pr o of. Let us start with the off-shell iden tities. F or the iden tity (a) , let us use the irreducibilit y condition for ϕ [ β γ ] µ , ϕ [ γ α ] µ , ϕ [ αβ ] µ , namely ϕ [ β γ ] µ + ϕ [ µβ ] γ + ϕ [ γ µ ] β = 0 , ϕ [ γ α ] µ + ϕ [ µγ ] α + ϕ [ αµ ] γ = 0 , ϕ [ αβ ] µ + ϕ [ µα ] β + ϕ [ β µ ] α = 0 , to rewrite ( 2.8 ) as H [ αβ γ ] µ = ∂ α ( − ϕ [ γ µ ] β − ϕ [ µβ ] γ ) + ∂ β ( − ϕ [ αµ ] γ − ϕ [ µγ ] α ) + ∂ γ ( − ϕ [ β µ ] α − ϕ [ µα ] β ) . Using the definition of H [ γ µα ] β and H [ αµβ ] γ w e ha ve H [ αβ γ ] µ = − H [ γ µα ] β − H [ αµβ ] γ + ∂ µ ( ϕ [ β α ] γ + ϕ [ αγ ] β ) − ∂ β ϕ [ µγ ] α − ∂ γ ϕ [ β µ ] α . Thanks to the irreducibilit y condition of ϕ [ β α ] γ and the definition of H [ γ β µ ] α w e get H [ αβ γ ] µ = − H [ γ µα ] β − H [ αµβ ] γ − H [ γ β µ ] α , rearranging and using an tisymmetry prop erties we get ( 2.13 ) . The iden tit y (b) is a consequence of the an tisymmetry prop erty of ϕ [ αβ ] µ and the comm utation of partial deriv ativ es while the iden tit y (c) follo ws from identit y (b) b y con tracting on ( δ, µ ). – 6 – In order to discuss asymptotic symmetries at future n ull infinit y of Bondi patc h of Mink o wski space-time, we in tro duce Bondi co ordinates ( u, r, { x i } ) space-time where u = t − r and { x i } is a set o D − 2 angular v ariables parameterizing the n ull infinity ( D − 2) -dimensional sphere S D − 2 . Mink owski line elemen t reads ds 2 = − du 2 − 2 dudr + r 2 γ ij dx i dx j i, j = 1 , ..., D − 2; (2.16) metric and non v anishing Christoffel sym b ols are giv en by g µν =   − 1 − 1 0 − 1 0 0 0 0 r 2 γ ij   , g µν =   0 − 1 0 − 1 1 0 0 0 1 r 2 γ − 1 ij   ; (2.17) Γ i j r = Γ i rj = 1 r δ i j , Γ u ij = − Γ r ij = r γ ij , Γ k ij = 1 2 γ kl [ − ∂ l γ ij + ∂ j γ li + ∂ i γ j l ] . (2.18) 3 The de Donder-lik e gauge fixing and residual gauge Let us enumerate the independent field comp onen ts. First of all, note that when the three indexes are all equal the field comp onen t is v anishing due to the irreducibility condition; moreov er, since the an tisymmetry prop erty , field comp onents suc h that the first tw o indexes are equal turn out to b e v anishing. F urthermore some field comp onen ts are related to others b y antisymmetry or irreducubilit y prop ert y . Below the coun ting of the in tep endent field components. Comp onen ts with ( u, r ) in the antisymmetric pair. By antisymmetry , the non-v anishing comp onen ts are ϕ [ ur ] u , ϕ [ ur ] r , ϕ [ ur ] i . (3.1) This giv es 1 + 1 + ( D − 2) = D indep endent components in this sector. Comp onen ts with one angular index in the an tisymmetric pair. F or eac h i = 1 , . . . , D − 2 , w e hav e ϕ [ ui ] u , ϕ [ ui ] r , ϕ [ ui ] j , ϕ [ ri ] u , ϕ [ ri ] r , ϕ [ ri ] j . (3.2) F or fixed i , eac h row giv es 2 + ( D − 2) = D comp onen ts; therefore 2( D − 2)[2 + ( D − 2)] = 2 D ( D − 2) indep enden t comp onents in this sector. – 7 – Comp onen ts with tw o angular indices in the antisymmetric pair. F or i < j w e ha ve ϕ [ ij ] u , ϕ [ ij ] r , ϕ [ ij ] k . (3.3) The comp onen ts ϕ [ ij ] u and ϕ [ ij ] r con tribute 2  D − 2 2  indep enden t comp onen ts. The purely angular comp onen ts ϕ [ ij ] k are sub ject to the restriction obtained from the irreducibilit y condition, ϕ [ ij ] k + ϕ [ ki ] j + ϕ [ j k ] i = 0 , (3.4) whic h remo ves the totally antisymmetric part. Hence their n umber is ( D − 2)  D − 2 2  −  D − 2 3  . Altogether, w e ha ve ( D − 2)( D − 3) 2 ( D + 1) indep enden t comp onents in this sector. T otal n um b er of comp onents. Summing all con tributions, one finds D + 2 D ( D − 2) + ( D − 2)( D − 3) 2 ( D + 1) = D ( D 2 − 1) 3 , (3.5) in agreement with the dimension of the irreducible GL ( D − 1 , 1) representation asso ciated with the Y oung diagram (2 , 1) . Since w e are interested in irreducible represen tation of SO ( D − 1 , 1) w e need to require the v anishing of the unique trace of the Curtrigh t field. Ho w ev er, the traceless condition is not gauge in v ariant, therefore, as in the graviton case, w e are going to require it on-shell ensuring a compatibilit y condition with other gauge fixing. Let us discuss the gauge and gauge-for-gauge fixing. In order to reduce the equations of motion ( 2.12 ) to a massless w a v e equation we ha ve to require that D β α := ˆ ∂ · ϕ β α − 1 2 ∂ α ¯ ϕ β = 0 , (3.6) whic h is a de Donder-like fixing. Indeed, calling E (  □ ) [ αβ ] γ the non-Box term in the equations of motion and assuming ( 3.6 ) to hold, we ha ve E (  □ ) [ αβ ] γ = ∂ γ ( ∂ α ¯ ϕ β − ∂ β ¯ ϕ α ) + ∂ β ˆ ∂ · ϕ αγ − ∂ α ˆ ∂ · ϕ β γ − ∂ γ ¯ ∂ · ϕ [ αβ ] = = ∂ γ ( ∂ α ¯ ϕ β − ∂ β ¯ ϕ α ) + 1 2 ∂ β ∂ γ ¯ ϕ α − 1 2 ∂ α ∂ γ ¯ ϕ β − ∂ γ ¯ ∂ · ϕ [ αβ ] = = 1 2 ∂ α ∂ γ ¯ ϕ β − 1 2 ∂ β ∂ γ ¯ ϕ α − ∂ γ ˆ ∂ · ( ϕ β α − ϕ αβ ) = 0 , (3.7) where in the last line w e used the irriducibility condition. More generally , the equations of motion ( 2.12 ) can b e written as □ ϕ [ αβ ] γ + ∂ γ ( D αβ − D β α ) + ∂ β D αγ − ∂ α D β γ = 0 , (3.8) hence, in de Donder-lik e gauge the equation of motion are □ ϕ [ αβ ] γ = 0; (3.9) – 8 – whose explicit writing for the indep endent field comp onen ts is in Appendix A , Section A.3 . The residual gauge after the de Donder gauge fixing is giv en by gauge parameters that satisfy □ ( λ β α + Λ β α ) − ∂ β  ∂ · ( λ α + Λ α )  − 1 2 ∂ α ∂ β λ µ µ + 1 2 ∂ α  ∂ · λ β  − 5 2 ∂ α  ∂ · Λ β  = 0 . (3.10) As for the graviton field, w e no w use the residual gauge to fix the traceless condition in order to reco v er the irreducible representation of SO ( D − 1 , 1) . The traceless condition can b e imp osed requiring ∂ β λ µ µ − ∂ · λ β + ∂ · Λ β = − ¯ ϕ β . (3.11) The compatibilit y condition with the de Donder-lik e gauge is giv en by □  ∂ · Λ α  = 1 4 ∂ α  ∂ β ¯ ϕ β  ; (3.12) Requiring ( 3.11 ) and ( 3.12 ) it is enough to reac h de Donder-like plus traceless gauge. In de Donder gauge, the equations of motion for the (2 , 1) field tak e the simple form ( 3.9 ). Con tracting indices immediately yields □ ¯ ϕ β = 0 , (3.13) and taking an additional div ergence giv es □  ∂ β ¯ ϕ β  = ∂ β  □ ¯ ϕ β  = 0 . (3.14) Therefore, the source term on the righ t-hand side of ( 3.12 ) is the gradient of a harmonic function, i.e. w ell defined. Hence, the compatibility equation do es not require ∂ β ¯ ϕ β to v anish identically . It is sufficient that the field satisfies the equations of motion ( 3.9 ) , whic h ensure that the source term in ( 3.12 ) is w ell defined. In this sense, the compatibilit y condition is solv able on-shell, although it does not collapse to an identit y , in contrast to what happ ens in the gra viton case. Moreov er, from the irreducibili t y condition ( 2.6 ) implies that the ¯ • -div ergence is related to the ˆ • - div ergencies. Since those ones are v anishing due to the de Donder-lik e gauge plus the traceless gauge w e hav e also that the field is b oth ¯ • and ˆ • div ergenceless. Therefore, also the trace is div ergenceless. This is the analogue of what happ ens in the graviton case. After this gauge fixing the equations of motion ( 2.12 ) can b e written as ∂ ν H [ ν αβ ] γ = 0 , H ν α = 0 . (3.15) The residual gauge is then parameterized b y ∂ β λ µ µ − ∂ · λ β + ∂ · Λ β = 0 , □  ∂ · Λ α  = 0 . (3.16) – 9 – Regarding the gauge for gauge, w e note that whatever gauge-for-gauge w e wan t to fix, it m ust b e such that the new gauge parameters m ust b e in the residual gauge in order to not spoil de Dender-lik e plus traceless gauge and so to b e compatible with our gauge fixing. Therefore Θ µ has to satisfy the following system in order to fix a p ossible the gauge-for-gauge without pro ducing gauge parameter that are not in the residual gauge giv en b y ( 3.16 ) □ Θ β − ∂ β ( ∂ · Θ) = 0 , □  □ Θ α − ∂ α ( ∂ · Θ)  = 0 , (3.17) whic h are redundan t, so only the first one is necessary . The gauge-for-gauge fixing w e require is λ ui = λ ri = Λ ui = Λ ri = 0; (3.18) lo oking the gauge-for-gauge transformations ( 2.5 ) this means ∂ r Θ i = − 1 4 λ ri − 1 2 Λ ri , (3.19) ∂ i Θ r = − 1 4 λ ri + 1 2 Λ ri , (3.20) ∂ u Θ i = − 1 4 λ ui − 1 2 Λ ui , (3.21) ∂ i Θ u = − 1 4 λ ui + 1 2 Λ ui , (3.22) with fix gradien ts of some comp onen ts of Θ µ . No w w e need to c heck compatibilit y with the residual gauge, i.e. with ( 3.17 ) , that can b e rewritten in terms of an auxiliary ob ject F µν := ∂ µ Θ ν − ∂ ν Θ µ as ∂ µ F µν = 0 ⇒        ∂ µ F µr = ∂ u F ur + ∂ i F ir = 0 , ∂ µ F µu = ∂ r F ru + ∂ i F iu = 0 , ∂ µ F µi = ∂ u F ui + ∂ r F ri + ∂ j F j i = 0 . (3.23) Equations ( 3.19 ) – ( 3.22 ) fix some comp onen ts of F µν but we ha ve enough free functions to ensure compatibilit y . T o studdy the residual gauge we ha ve also to require some fall-offs condition to preserv e. T o this goal w e study The energy-momen tum tensor obtained b y v arying the action with resp ect to the metric. Since the lagrangian dep ends on the metric only through index con tractions, the result can b e written as T µν = 4 3 H µαβ γ H αβ γ ν − 2 H µα H α ν − 1 6 η µν  H αβ γ δ H αβ γ δ − 3 H αβ H αβ  . (3.24) All comp onents of T µν are therefore quadratic in the partial field strength or its unique trace. Requiring the energy flux to b e finite and non-v anishing implies the asymptotic conditions H uriu ∼ O ( r − ( D − 4 2 ) ) , H urir ∼ O ( r − ( D − 4 2 ) ) , H urij ∼ O ( r − ( D − 6 2 ) ) . (3.25) – 10 – where w e ha v e tak en in to account that in general, for eac h angular index i w e ha v e one r more than the cartesian comp onen t since in Bondi co ordinates the field carries additional p o wers of r due to the jacobian of the co ordinate transformation. In Bondi co ordinates the gauge transformations has to preserv e the radiation fall-offs on field comp onen ts δ λ, Λ ( ϕ [ αβ ] γ ) ∼            O  r − ( D − 2) 2  if α, β , γ / ∈ { x i } ; O  r − ( D − 4) 2  if α or β or γ ∈ { x i } ; O  r − ( D − 6) 2  if α, β or β , γ or α, γ ∈ { x i } ; O  r − ( D − 8) 2  if α, β , γ ∈ { x i } . (3.26) W e now assume a p olyhomogeneous p ow er law expansion for the gauge parameters of the form 1 λ µν = X l ∈ 1 2 Z λ ( l ) µν ( u, { x i } ) r l + ¯ λ ( l ) µν ( u, { x i } ) r l l n ( r ) , Λ µν = X l ∈ 1 2 Z Λ ( l ) µν ( u, { x i } ) r l + ¯ Λ ( l ) µν ( u, { x i } ) r l l n ( r ) . (3.27) Lo oking at the residual gauge condition equations in A.1 , preserv ation of fall-off condition equations in A.2 and equation of motions in A.3 w e find the expansions of the gauge parameter comp onen ts en tering in the asymptotic charge λ ij = λ ( D − 8 2 ) ij ( { x i } ) r D − 8 2 + X l> D − 8 2 λ ( l ) µν ( u, { x i } ) r l + ¯ λ ( l ) µν ( u, { x i } ) r l l n ( r ) , Λ ij = Λ ( D − 8 2 ) ij ( { x i } ) r D − 8 2 + X l> D − 8 2 Λ ( l ) µν ( u, { x i } ) r l + ¯ Λ ( l ) µν ( u, { x i } ) r l l n ( r ) . (3.28) 4 Asymptotic c harges The asymptotic c harge relev ant for our discussion has its ro ots in the co v ariant phase space formalism à la W ald [ 1 , 69 ]; we consider the Nöther c harge Q Λ ,λ [ S D − 2 u ] = lim r → + ∞ I S D − 2 u k ur Λ ,λ r D − 2 d Ω , (4.1) where w e already sp ecialize to the co dimension-2 celestial sphere at Mink owski n ull infinit y . In ( 4.1 ) , k µν Λ ,λ is the Nöther t wo-form, S D − 2 u is the celestial sphere at Mink o wski 1 W e will b e interested in l of the form D − n 2 where n ∈ Z ; we ha ve D − n 2 < D − m 2 if and only if m < n and D − n 2 > D − m 2 if and only if m > n . Given 1 r D − n 2 the pow er 1 r D − m 2 is subleading if and only if m < n . – 11 – n ull infinit y , and r D − 2 d Ω is its in tegration measure. T o compute the Nöther t wo-form k µν Λ ,λ w e need to compute the v ariation of the lagrangian densit y ( 2.10 ) δ L = − 1 3 [ δ ( H αβ γ µ ) H αβ γ µ − 3 δ ( H αβ ) H αβ ] =: δ L 1 + δ L 2 , (4.2) V ariation of the first term δ L 1 . The v ariation of the first term reads δ L 1 = − 1 3 H αβ γ µ δ H αβ γ µ . (4.3) Using the definition ( 2.8 ) and the an tisymmeetry prop erties of H αβ γ µ w e get H αβ γ µ δ H αβ γ µ = 3 H αβ γ µ ∂ α δ ϕ [ β γ ] µ , (4.4) and hence δ L 1 = − H αβ γ µ ∂ α δ ϕ [ β γ ] µ . (4.5) In tegrating b y parts, δ L 1 =  ∂ α H αβ γ µ  δ ϕ [ β γ ] µ − ∂ α  H αβ γ µ δ ϕ [ β γ ] µ  . (4.6) V ariation of the trace term δ L 2 . The v ariation of the second term of the Lagrangian reads δ L 2 = H αβ δ H αβ = H αβ ∂ γ δ ϕ [ αβ ] γ + H αβ  ∂ α δ ¯ ϕ β − ∂ β δ ¯ ϕ α  . (4.7) Since H αβ is an tisymmetric, the second term reduces to H αβ  ∂ α δ ¯ ϕ β − ∂ β δ ¯ ϕ α  = 2 H αβ ∂ α δ ¯ ϕ β , (4.8) and, in tegrating b y parts, we get δ L 2 = −  ∂ γ H αβ  δ ϕ [ αβ ] γ − 2  ∂ α H αβ  δ ϕ [ β γ ] γ + ∂ µ  H αβ δ ϕ [ αβ ] µ + 2 H µβ δ ϕ [ β γ ] γ  , (4.9) T otal v ariation. Com bining δ L 1 and δ L 2 , the total v ariation tak es the form δ L =  ∂ α H αβ γ µ − ∂ µ H β γ  δ ϕ [ β γ ] µ − 2  ∂ α H αβ  δ ϕ [ β γ ] γ + ∂ µ j µ , (4.10) where j µ = − H µβ γ ν δ ϕ [ β γ ] ν + H β γ δ ϕ [ β γ ] µ + 2 H µβ δ ϕ [ β γ ] γ . (4.11) – 12 – Nö ether t wo-form. The Nö ether can be computed by inserting the gauge v ariation δ λ, Λ ϕ [ ρσ ] ν = ∂ ρ λ ( σ ν ) − ∂ σ λ ( ρν ) | {z } δ λ ϕ [ ρσ ] ν + ∂ ρ Λ [ σ ν ] − ∂ σ Λ [ ρν ] + 2 ∂ ν Λ [ σ ρ ] | {z } δ Λ ϕ [ ρσ ] ν , (4.12) in to j µ = − H µβ γ ν δ ϕ [ β γ ] ν , (4.13) where we hav e assumed the de Donder-like plus traceless gauge fixing. W e note that since H µβ γ ν is con tracted with a (2,1) irreducible tensor, therefore all other represen tations different from (2,1) on the con tracted indices do not enter. This forces a further iden tit y on H µβ γ ν in the determination of j µ H µβ γ ν + H µγ ν β + H µν β γ = 0 . (4.14) W e get, using antisymmetry property on ( β , γ ) and identit y ( 4.14 ) j µ = − 2 H µβ γ ν ∂ β λ γ ν − 2 H µβ γ ν ∂ β Λ γ ν + 2 H µβ γ ν ∂ ν Λ β γ = = − 2 H µβ γ ν ∂ β λ γ ν − 6 H µν β γ ∂ ν Λ β γ . (4.15) In tegrating b y parts w e get j µ = ∂ α k µα + R µ , (4.16) where k µα := − 2 H µαγ β λ γ β − 6 H µαβ γ Λ β γ (4.17) while the residual term R µ is R µ := 2 ( ∂ β H µβ γ ν ) λ γ ν + 6 ( ∂ ν H µν β γ ) Λ β γ , (4.18) that is v anishing due to equations of motion ( 3.15 ). In particular, on-shell, w e get j µ ≈ ∂ α k [ µα ] . (4.19) The c harge is Q Λ ,λ [ S D − 2 u ] = − 2 lim r → + ∞ I S D − 2 u k ur Λ ,λ r D − 2 d Ω = − 2 lim r → + ∞ I S D − 2 u H urij  λ ( ij ) + 3 Λ [ ij ]  r D − 2 d Ω = = − 2 lim r → + ∞ I S D − 2 u g uα g rβ g iγ g j δ H αβ γ δ  λ ( ij ) + 3 Λ [ ij ]  r D − 2 d Ω = = − 2 lim r → + ∞ I S D − 2 u g ur g rβ g i ˜ i g j ˜ j H rβ ˜ i ˜ j  λ ( ij ) + 3 Λ [ ij ]  r D − 2 d Ω = = − 2 lim r → + ∞ I S D − 2 u γ i ˜ i γ j ˜ j H ru ˜ i ˜ j  λ ( ij ) + 3 Λ [ ij ]  r D − 6 d Ω , (4.20) – 13 – since g ij = γ ij r 2 . Let us note here that the gauge condition ˆ ∇ · ϕ β α = 0 , whic h is the result of D αβ = 0 plus the traceless gauge, implies that ∂ u ϕ ( D − 6 2 ) rij = 0 ; therefore the leading order of H ruij is O ( r − ( D − 4) / 2 ) . Inserting the leading orders ( 3.28 ) w e get a finite O (1) , non-v anishing c harge Q Λ ,λ [ S D − 2 u ] = − 2 I S D − 2 u γ i ˜ i γ j ˜ j H  D − 4 2  ru ˜ i ˜ j  λ  D − 8 2  ( ij ) + 3 Λ  D − 8 2  [ ij ]  d Ω (4.21) since − D − 4 2 − D − 8 2 + D − 6 = 0 . This form of the charge resem ble the one of p -form gauge fields and, in fact, the mec hanisms leading to charge ( 4.21 ) are v ery similar to those entering in the p -form c harge [ 44 , 46 ]: gauge condition that sh ut off the field components that can the c harge div ergen t, the entering of the partial field strength 2 comp onen ts with r u -indices (hence an electric-lik e charge), only angular comp onents of the gauge parameters en tering in the c harge. Moreo v er, let us note that the gauge condition fixed here is essen tially the same fixed for p -forms once we tak e into accoun t the indices structure of the field. Indeed, for a p -form, there is no non-trivial trace and the de Donder-lik e gauge fixing reduces to Lorenz-lik e gauge fixing since there is only one independent div ergence for a p -form field. There are so at least tw o p ossible guesses w e can made with a relative degree of safet y: the form of the c harge for a ho ok field ( p, 1 ) with p > 1 and the path to follows for deriv e finite O (1) , non-v anishing c harge for a generic mixed symmetry tensor gauge field. Asymptotic c harges for ( p, 1 )-ho ok field. The guess follo ws from a straightfor- w ard merging of the tec hnicalities in deriving the c harge ( 4.21 ) and in generalizing Maxw ell field to the p -form case [ 17 , 40 , 44 , 46 ]. The ( p, 1 ) ho ok field ϕ [ µ 1 ,...µ p ] ν has only one trace (by contracting the last index with one of the first p an tisymmetric ones) and tw o indep enden t divergencies (the one where the deriv ative contacts with the last index and the one where contracts with one of the antisymmetric indices). Therefore, w e are in a situation very similar to the case of the (2 , 1) field with p -an tisymmetric indices lik e the case of a p -form field. The guess is that the c harge is of the form Q ( p, 1) [ S D − 2 u ] = # 1 I S D − 2 u γ i 1 ˜ i 1 ...γ i p ˜ i p H  a ( D,p )  ru ˜ i 1 ... ˜ i p  λ  b ( D,p )  [ i 1 ...i p − 1 ] i p + # 2 Λ  b ( D,p )  [ i 1 ...i p ]  d Ω , (4.22) where a ( D , p ) := D − 2( p +1)+2 2 , b ( D , p ) := D − (2( p +1)+2) 2 and # 1 , # 2 are tw o constants whic h dep ends critically form the co efficients in the lagrangian. Asymptotic c harges for general mixed symmetry tensor fields. The asymp- totic charges of a general mixed symmetry tensor gauge field with N -families of indices 2 In the case of p -forms is the true field strength that enter in the c harge. – 14 – that carry the irreducible representation ( p 1 , ..., p N ) could b e computed follo wing the same fo otsteps. A tensor field of this t yp e will b e denoted by ϕ A (1) A (2) ··· A ( N ) , A ( i ) := [ a ( i ) 1 · · · a ( i ) p i ] , (4.23) meaning ϕ A (1) ··· A ( N ) = ϕ [ a (1) 1 ··· a (1) p 1 ] ··· [ a ( N ) 1 ··· a ( N ) p N ] . (4.24) Besides an tisymmetry in eac h column, Y oung irreducibility has to b e imposed: for eac h i = 1 , . . . , N − 1 and for any c hoice of one index j tak en from the ( i + 1) -th column, the total antisymmetrization o ver the whole i -th column together with j v anishes ϕ A (1) ... [ A ( i ) j ] , A ( i +1) \ j , A ( i +2) ··· A ( N ) = 0 . (4.25) Here A ( i +1) \ j denotes the ( i + 1) -th family with the index j remo v ed and the an tisymmetrization brack et in runs o v er p i + 1 indices (the whole i -th family plus j ). The key p oint is that w e can expresses div ergences div i +1 ϕ on column ( i + 1) as linear com binations of divergences div i ϕ on column i . Iterating from the righ tmost column to the left, one obtains a c hain of reductions div N ϕ − → div N − 1 ϕ − → · · · − → div 1 ϕ . (4.26) T o see the mec hanism, fix an adjacent pair ( i, i + 1) and select an index j ap- p earing in A ( i +1) . The irreducibility condition will con tains a term of the form ϕ A (1) ... [ a ( i ) 1 ··· a ( i ) p i ][ j ··· a ( i +1) p i +1 ] ...A ( N ) and terms of the form ϕ A (1) ... [ a ( i ) 1 ··· j ][ a ( i ) p i ··· a ( i +1) p i +1 ] ...A ( N ) . T aking the div ergence with respect to j will pro duce a relation betw een div i ϕ and div i +1 ϕ . In this precise sense, for a Y oung–irreducible tensor in column con ven tion, the diver- gences tak en on later families are not indep enden t: they can b e rewritten in terms of div ergences tak en on earlier families, ultimately in terms of div 1 ϕ . Similarly , for the traces tr ( i,j ) ϕ − → tr ( i,j − 1) ϕ − → · · · − → tr ( i,i +1) ϕ , (4.27) namely any trace b etw een non-adjacent columns can be rewritten as a linear com bina- tion of traces b et w een closer columns, ultimately reducing to traces b et w een adjacen t columns (and, if desired, to traces in v olving a preferred column). The crucial step no w will b e to reduce the equations of motion to an homogeneous w a ve equation rewriting all the non- □ terms in terms of a single divergence and a single trace, for example div N ϕ and tr ( N − 1 ,N ) ϕ , and fixing a de Donder-like gauge to cancel them. This pro duce residual gauge conditions and w e use the residual gauge to eliminate the trace tr ( N − 1 ,N ) ϕ so that all the traces will v anish and the field will b e divergences-free. The next step is the gauge-for-gauge fixing such that only gauge parameters with angular indices en ter in the c harge. The t wo paragraphs abov e are mainly sp eculative and a deep er understanding of these tec hnicalities and the effectiv eness of these guesses is one of the next point in the agenda for future w orks. – 15 – 5 Connections with the gra viton asymptotic symme- tries in D = 5 Since in D = 5 the graviton and the Curtrigh t field are dual, our scop e is to connect the Curtrigh t field asymptotic charge to supertranslation and sup errotation c harges of the gra viton. The D = 5 c harge for the Curtright field reads Q Λ ,λ [ S 3 u ] = I S 3 u γ i ˜ i γ j ˜ j H  1 2  ru ˜ i ˜ j  λ  − 3 2  ( ij ) + 3 Λ  − 3 2  [ ij ]  d Ω; (5.1) the t wo gauge parameters can b e decomp osed using Ho dge and Ho dge-like decomp o- sitions: • An tisymmetric parameter. The antisymmetric parameter can be consider as a 2-form on S 3 ; since H 2 dR ( S 3 ) = 0 we ha ve no harmonic part Λ  − 3 2  [ ij ] = D [ i W j ] + ϵ ij k D k Φ; (5.2) • Symmetric parameter. The symmetric parameter can be considered as a (1,1) mixed symmetry tensor obtained by the Y oung pro jector Π (1 , 1) acting of the bi-form space on S 3 . Since H 1 dR ( S 3 ) = 0 we ha ve λ  − 3 2  ( ij ) = − D k B k ( ij ) + D ( i C j ) + D i D j ϕ − D k D ( i Ξ j ) k + D k D ℓ Y kiℓj . (5.3) where Y lisj carries the same irreducible represen tation of the Riemann tensor and S ij := D k D ℓ Y kiℓj is symmetric since its an tisymmetric part is proportional to the an tisymmetric part of the "Ricci" tensor associated to Y kil s that is v anishing (see Appendix C , Section C.1 ). Moreo v er, B has to be v anishing. Indeed the (2) ⊗ (1) = (2 , 1) ⊕ (3) and b oth terms v anishes 3 requiring a simmetrization on ( i, j ) . Let us define H ij := γ i ˜ i γ j ˜ j H  1 2  ru ˜ i ˜ j that represen ts the leading-order dynamical comp o- nen ts of the Curtrigh t field at the b oundary; p erforming an asymptotic gauge fixing (see App endix C , Section C.2 ) to get D ˜ i H  1 2  ru ˜ i ˜ j = 0 , w e hav e D i H ij = D i γ i ˜ i γ j ˜ j H  1 2  ru ˜ i ˜ j = γ j ˜ j D ˜ i H  1 2  ru ˜ i ˜ j = 0 . (5.4) 3 F or the 3-form is obvious, while for the (2 , 1) we would hav e B kij = B kj i = − B j k i . Substituting this into the irreducibility condition yields 0 = B kij + B ij k + B j k i = B kij + B ij k − B kij = B ij k . – 16 – Moreo v er by the equations of motion we also ha ve that H := γ ij H ij ≈ 0 . (5.5) The c harge reduces, up on in tegration by parts, to Q Λ ,λ [ S 3 u ] = I S 3  Φ ϵ ij k D k H ij + Y kiℓj D k D ℓ H ij  d Ω =: Q Φ + Q Y . (5.6) Let us b etter study the c harge Q Y . T o this goal we define v i := D j H ij and the "Ricci" tensor and scalar asso ciated to Y kiℓj y ij := γ kℓ Y kiℓj , y := γ ij y ij . (5.7) A tensor with the same symmetries of the Riemann tensor is completely determined b y its "Ricci" tensor and scalar in 3 dimensions, hence Q Y can b e completely rewritten it terms of y ij and y as Q Y = Z S 3 h y ij ∆ H ij − y iℓ D j D ℓ H ij + y 2 D i v i i d Ω . (5.8) Using the comm utation relation in App endix C , ( C.7 ) and ( C.9 ) with K = 1 and the fact that y ij is symmetric w e get Q Y = Z S 3 h y ij (∆ − 3) H ij − y iℓ D ℓ v i i d Ω = Z S 3 y ij h (∆ − 3) H ij − D j v i i d Ω . (5.9) A con v enient parametrization of a y ij on S 3 is the Y ork decomp osition y ij = 1 3 γ ij A +  D i D j − 1 3 γ ij ∆  B +  D ( i V j ) − 1 3 γ ij D k V k  + y TT ij , (5.10) where V i is a v ector field on S 3 and y TT ij is transverse-traceless. Using ( 5.10 ) in to ( 5.9 ) and in tegrate b y parts on S 3 w e ha ve Q Y = Z S 3  A J A + B J B + V i J i V + y TT ij J ij y TT  d Ω , (5.11) with curren ts J A := 1 3 γ ij  (∆ − 3) H ij − D j v i  , J B := −  D i D j − 1 3 γ ij ∆   (∆ − 3) H ij − D j v i  , J i V := − D j  (∆ − 3) H ( ij ) − D ( j v i )  − 1 3 D i  γ kℓ  (∆ − 3) H kℓ − D ℓ v k  ! , J ij y TT := (∆ − 3) H ( ij ) (5.12) – 17 – Under relations ( 5.5 ), ( 5.4 ) and ( C.9 ) J A ≈ 0 , J B ≈ 0 , J i V ≈ − D j  (∆ − 3) H ( ij )  + ∆ v i + v i . (5.13) In the end, the Curtrigh t c harge in D = 5 is Q Λ ,λ [ S 3 u ] = Q Φ + Q V + Q y TT , (5.14) where Q Φ := I S 3 Φ ϵ ij k D k H [ ij ] d Ω , Q V := − I S 3 V i  D j  (∆ − 3) H ( ij )  − ∆ v i − v i  d Ω , Q y TT := I S 3 y TT ij (∆ − 3) H ( ij ) d Ω . (5.15) W e hav e so a c harge whose parameter is the scalar arbitrary function Φ ∈ C ∞ ( S 3 ) , a c harge whose parameter is a vector field V i ∈ Diff ( S 3 ) on S 3 and a c harge whose parameter is a rank-2 transv erse-traceless tensor y TT ij ∈ TT ( S 3 ) on S 3 . Moreov er, the c harge Q Φ can b e rewritten using exterior calculus as Q Φ = 2 I S 3 Φ ∧ d H (5.16) where w e used the relation 4 2 d H = ε ij k D k H [ ij ] d Ω . Let us no w compute the c harge algebra whic h is defined by the relations { Q Φ 1 , Q Φ 2 } = δ Φ 1 Q Φ 2 , { Q V , Q W } = δ V Q W , { Q V , Q Φ } = δ V Q Φ , { Q y TT , Q Φ } = δ y TT Q Φ , { Q V , Q y TT } = δ V Q y TT , { Q y TT 1 , Q y TT 2 } = δ y TT 1 Q y TT 2 . (5.17) 1. The computation of δ Φ 1 Q Φ 2 reads δ Φ 1 Q Φ 2 = 2 I S 3 Φ 2 ∧ d ( δ Φ 1 H ) = 0 , (5.18) since the gauge v ariation of H ru ˜ i ˜ j con tains only Λ ru so it do es not con tain Φ . F or similar reasons also δ y TT Q Φ = 0 and δ y TT 1 Q y TT 2 = 0 since the gauge v ariation do es not con tains these pieces of the parameters. 2. F or the computation of δ V Q W w e ha ve δ V Q W = − I S 3 W i δ V ( J i d Ω) = I S 3 W i ( L V ( J i d Ω)) = − I S 3 ( L V W i ) J i d Ω = Q [ V ,W ] . (5.19) 4 This relation can b e shown starting from de i + ω i k ∧ e k = 0 and using that H := 1 2 H ij e i ∧ e j since in Q Φ only the antisymmetric part of H ij is relev ant. By explicit v aluation of d H and using that on D = 3 we hav e e k ∧ e i ∧ e j = ε kij d Ω , w e find d H = 1 2 ε kij D k H ij d Ω . – 18 – where L V is the Lie deriv ative with resp ect to V . By a similar computation we get also δ V Q y TT = Q L V ( y TT ) , (5.20) ho w ev er, to ensure that L V ( y TT ) is itself transverse-traceless we need V to be a Killing v ector, i.e. an isometry of S 3 . Therefore, to close the algebra w e need V i ∈ o (4) . 3. Let us compute δ V Q Φ : δ V Q Φ = I S 3 Φ ∧ d ( δ V H ) = I S 3 Φ ∧ d ( −L V H ) = I S 3 L V (Φ) ∧ d ( H ) = Q L V (Φ) (5.21) where we used that the v ariation along a v ector field of a tensor is its Lie deriv ativ e with resp ect to the vector field, that Lie deriv ative and exterior deriv ativ e comm ute and integrating by parts. Hence, { Q V , Q Φ } = Q L V (Φ) (5.22) and w e ha ve a semi-direct sum algebra. The final algebra is CBMS ( S 3 ) = o (4) + ( C ∞ ( S 3 ) ⊕ TT ( S 3 )) , (5.23) that resem ble BMS algebra where supertranslation-like transformations are pa- rameterized by Φ while rotation transformations are parameterized b y V i . The Curtrigth-BMS algebra ( 5.23 ) forms an ab elian extension of a MBS -lik e algebra; ho w ever by Lie algebra cohomology this extension can be only trivial, in the sense that the extension splits as a semidirect sum. The TT ( S 3 ) term can b e in terpreted as a higher spin-like supertranslation since giv e rise to an ab elian ideal that is a mo dule for o (4) . Despite this, we are able to find only one scalar parameterizing sup ertranslations instead of the tw o scalar functions exp ected from hamiltonian analysis. The missing second scalar could be suppressed b y the specific Curtright gauge-fixing conditions used to define H ij . A smarter c hoice of gauge fixing or gauge-for-gauge fixing, as w ell as an analysis con taining subleading or logarithmic expansion terms could b e the righ t path to un v eil the sup ertranslations structure. In order to obtain extended or generalized Curtrigth versions of BMS algebra w e probably need to relax b oundary constrain ts, m uch like of what happens in D = 4 . 6 Conclusions and outlo ok This w ork studied the asymptotic charge structure of the Curtright field, a proto- t ypical gauge system with mixed index symmetry . Mixed-symmetry fields bring – 19 – in genuinely new ingredien ts: there are multiple gauge parameters with different symmetry structure and there is complex gauge-for-gauge redundancy . The analysis dev elop ed a concrete and work able framework in whic h all these features can b e con trolled sim ultaneously , leading to finite charges and a closed asymptotic symmetry algebra. A first ma jor step w as the selection of gauge fixing and the imp osition of radiation- compatible fall-off conditions. These fall-offs w ere not c hosen arbitrarily: they w ere motiv ated by the requiremen t that the radiative degrees of freedom ha v e a w ell-defined flux through n ull infinit y . In this setting, the equations of motion were simplified b y imp osing a de Donder-lik e gauge condition, supplemen ted by an on-shell condition that remo ves trace components. Conceptually , these conditions play the same role as the familiar gauge choices in linearized gra vity: they isolate the propagating conten t, reduce the dynamics to a tractable w av e-like form and turn the problem of asymp- totic symmetries into a problem of c haracterizing the residual gauge transformations compatible with b oth the gauge c hoice and the b oundary b ehavior. A tec hnically imp ortan t part of the work was that to capture all potentially c harge- relev an t mo des, the analysis allow ed for more general asymptotic expansions, including half-in teger p ow ers and p ossible logarithmic branches. This p olyhomogeneous ap- proac h is a natural generalization of what app ears in higher-dimensional analyses of other gauge theories. With the residual transformations under control, the w ork constructed the asso ciated surface charges using a co v ariant No ether-based strategy; c harges are "electric-lik e" in the sense that they are built from radial–null compo- nen ts of the field strength contracted with asymptotic gauge data on the celestial sphere. This parallels the structure familiar from other gauge theories like p -forms one, while also reflecting the extra algebraic pro jections and redundancies intrinsic to mixed-symmetry fields. A sp ecial focus of the w ork w as the fiv e-dimensional case, where the Curtrigh t field is on-shell dual to the graviton. Using appropriate decomp ositions, the residual parameters can b e organized into three sectors: one con trolled b y an arbitrary function on the three-sphere, whic h pla ys the role of a sup ertranslation-like parameter, a second controlled b y a vector field on the three-sphere and a third controlled b y TT mo des. After an additional asymptotic gauge fixing that enforces transversalit y and remo ves trace pieces in the leading b oundary tensor, the charge correspondingly splits into three con tributions asso ciated with these three sectors. This decomp osition is v aluable not only b ecause it yields explicit expressions for the charges but also b ecause it makes the symmetry algebra transparen t forming an ab elian extension of a BMS -lik e algebra. The scalar and TT sectors form ab elian ideals, while the v ector sector repro duces the standard comm utator structure of v ector fields on the three-sphere generating the o (4) part of the algebra. The mixed brack et is the natural action of sphere diffeomorphisms on tensors. Altogether, the algebra closes as a semidirect sum b et ween the Killing v ectors of the celestial sphere and an ab elian ideal – 20 – giv en b y the direct sum of scalar and TT sectors interpreted as sup ertranslations and higher spin sup ertranslations. This is the mixed-symmetry analogue of the familiar BMS-t yp e structure, we call Curtright extension of BMS algebra: CBMS . One conceptual tension that emerged concerns the scalar sector in fiv e dimensions. Within the adopted boundary conditions and asymptotic gauge fixing, only one indep enden t supertranslation-like function survives in the c harge. This con trasts with some exp ectations in the literature, where tw o indep endent scalar functions ha ve b een argued to arise at leading order in related Hamiltonian analyses. In the present framew ork, the "missing" scalar mo de ma y b e remo ved by the specific asymptotic gauge c hoice used to define the b oundary tensor, or it ma y reside in subleading or logarithmic branches that w ere consisten tly truncated when isolating the minimal data needed for finite leading c harges. Sev eral natural directions for future work follo w from these results. A first di- rection is to revisit the b oundary conditions and asymptotic gauge c hoices in five dimensions with the explicit goal of clarifying the status of the second scalar mode. This would in volv e relaxing some of the asymptotic constrain ts imposed on the leading b oundary tensor, exploring alternative gauge-for-gauge fixings and systematically re- taining a larger set of subleading terms in the polyhomogeneous expansions. Moreo ver, the study of p ossible Curtrigh t extension of extended or generalized BMS-structure is a an in teresting p ossibilit y: lik e in D = 4 , these extensions could app ear relaxing fall-of conditions on fields. A second direction is to extend the construction of asymptotic c harges to broader families of mixed-symmetry gauge fields. The Curtright field is the simplest non triv- ial example but the metho ds developed here suggest a general blueprint: c ho ose a gauge that reduces the dynamics to a w a v e-lik e form, imp ose irreducibilit y conditions that isolate the correct Y oung-pro jected degrees of freedom and classify residual parameters on the celestial sphere while carefully accounting for gauge-for-gauge redundancies. In that con text, one exp ects surface charges to b e built from the appropriate radial–n ull curv ature comp onents contracted with the leading sphere data of the residual parameters, generalizing the "electric-lik e" pattern found here and in p -forms con texts. Carrying this out for other Y oung pro jected tensors would pro vide a systematic answ er to how asymptotic symmetry algebras depend on the represen tation conten t of the field and it would establish a general theory of large gauge charges for tensors with mixed symmetries, as already anticipated and discussed in the main text. A third direction concerns dualit y at null infinit y . While the Curtright field and the gra viton are dual on-shell in fiv e dimensions, their gauge structures differ off-shell and their b oundary descriptions need not coincide automatically . A detailed dictionary b et w een metric b oundary data and the dual mixed-symmetry b oundary tensor would clarify whic h asp ects of the infrared s tructure are in v arian t under duality and whic h are resh uffled. This includes not only the mapping of charges and symmetry param- – 21 – eters, but also the mapping of fluxes, memory observ ables, and the in terpretation of radiative data. In particular, it w ould b e v aluable to understand whether the semidirect-sum algebra obtained here is exactly the dual image of the gra vitational asymptotic algebra for the same class of fall-offs, or whether additional b oundary coun terterms and corner contributions are needed to match the t wo descriptions fully . A fourth direction is to in vestigate related soft theorem and memory effects. Establish- ing the existence and the precise form of soft theorems and memory effects asso ciated with these mixed-symmetry gauge systems. As in other gauge theories, large gauge transformations at null infinit y are exp ected to yield non-trivial W ard iden tities for the scattering op erator, whic h in turn should con trol univ ersal soft limits of amplitudes with an additional soft particle. Suc h effects could b e relev ant b oth conceptually and tec hnically: conceptually , they generalize the IR triangle relating symmetries, soft limits, and memory b ey ond symmetric tensor gauge fields; tec hnically , they pro vide new constrain ts on consisten t couplings, b oundary conditions, and the definition of conserv ed quantities for mixed-symmetry fields. Moreov er, since ho ok-type fields app ear in higher-spin theories, dualit y-co v arian t form ulations and in certain stringy regimes, understanding their IR sector may unco ver univ ersal features shared across these framew orks. A Equations A.1 Equations from residual gauge conditions Assume the r p olyhomogeneous expansions λ µν = X l ∈ 1 2 Z r − l  λ ( l ) µν ( u, x )+ ¯ λ ( l ) µν ( u, x ) ln r  , Λ µν = X l ∈ 1 2 Z r − l  Λ ( l ) µν ( u, x )+ ¯ Λ ( l ) µν ( u, x ) ln r  . (A.1) The residual gauge conditions are G β := ∇ β λ µ µ − ∇ α λ αβ + ∇ α Λ αβ = 0 , S α := □  ∇ β Λ β α  = 0 , (A.2) where □ := ∇ µ ∇ µ . Notation. W e adopt the following notations where n := D − 2 ( ∂ r λ µν ) ( l ) = − ( l − 1) λ ( l − 1) µν + ¯ λ ( l − 1) µν , ( ∂ r λ µν ) ( l ) = − ( l − 1) ¯ λ ( l − 1) µν , (A.3) ( ∂ r Λ µν ) ( l ) = − ( l − 1)Λ ( l − 1) µν + ¯ Λ ( l − 1) µν , ( ∂ r Λ µν ) ( l ) = − ( l − 1) ¯ Λ ( l − 1) µν , (A.4) ( λ µ µ ) ( l ) = − 2 λ ( l ) ur + λ ( l ) rr + γ ij λ ( l − 2) ij , ( λ µ µ ) ( l ) = − 2 ¯ λ ( l ) ur + ¯ λ ( l ) rr + γ ij ¯ λ ( l − 2) ij . (A.5) ( ∂ r λ µ µ ) ( l ) = − ( l − 1)( λ µ µ ) ( l − 1) + ( λ µ µ ) ( l − 1) , ( ∂ r λ µ µ ) ( l ) = − ( l − 1)( λ µ µ ) ( l − 1) (A.6) – 22 – ( ∇ β Λ β u ) ( l ) = ∂ u Λ ( l ) ur + ( l − 1 − n )Λ ( l − 1) ur − ¯ Λ ( l − 1) ur , (A.7) ( ∇ β Λ β u ) ( l ) = ∂ u ¯ Λ ( l ) ur + ( l − 1 − n ) ¯ Λ ( l − 1) ur . (A.8) ( ∇ β Λ β r ) ( l ) = ( l − 1 − n )Λ ( l − 1) ur − ¯ Λ ( l − 1) ur , (A.9) ( ∇ β Λ β r ) ( l ) = ( l − 1 − n ) ¯ Λ ( l − 1) ur . (A.10) ( ∇ β Λ β i ) ( l ) = D j Λ ( l − 2) j i , (A.11) ( ∇ β Λ β i ) ( l ) = D j ¯ Λ ( l − 2) j i , (A.12) Moreo v er, w e use D i to indicate the cov ariant deriv ative on S D − 2 , so ∆ := D i D i is the Laplace-Beltrami op erator on S D − 2 . Equation G u = 0 . + ∂ u ( λ µ µ ) ( l ) + ∂ u λ ( l ) ur + ∂ u Λ ( l ) ur + ( n − l + 1) λ ( l − 1) uu + ( l − 1 − n ) λ ( l − 1) ur + + ( l − 1 − n )Λ ( l − 1) ur + ¯ λ ( l − 1) uu − ¯ λ ( l − 1) ur − ¯ Λ ( l − 1) ur = 0 . (A.13) + ∂ u ( λ µ µ ) ( l ) + ∂ u ¯ λ ( l ) ur + ∂ u ¯ Λ ( l ) ur + ( n − l + 1) ¯ λ ( l − 1) uu + ( l − 1 − n ) ¯ λ ( l − 1) ur + + ( l − 1 − n ) ¯ Λ ( l − 1) ur = 0 . (A.14) Equation G r = 0 . − ( l − 1)( λ µ µ ) ( l − 1) + ( λ µ µ ) ( l − 1) + ∂ u λ ( l ) rr + ( n − l + 1) λ ( l − 1) ur + ( l − 1 − n ) λ ( l − 1) rr + + ( l − 1 − n )Λ ( l − 1) ur + γ ij λ ( l − 3) ij + + ¯ λ ( l − 1) ur − ¯ λ ( l − 1) rr − ¯ Λ ( l − 1) ur = 0 . (A.15) − ( l − 1)( λ µ µ ) ( l − 1) + ∂ u ¯ λ ( l ) rr + ( n − l + 1) ¯ λ ( l − 1) ur + ( l − 1 − n ) ¯ λ ( l − 1) rr + + ( l − 1 − n ) ¯ Λ ( l − 1) ur + γ ij ¯ λ ( l − 3) ij = 0 . (A.16) Equations G i = 0 . D i ( λ µ µ ) ( l ) − D j λ ( l − 2) j i + D j Λ ( l − 2) j i = 0 . (A.17) D i ( λ µ µ ) ( l ) − D j ¯ λ ( l − 2) j i + D j ¯ Λ ( l − 2) j i = 0 . (A.18) Equation S u = 0 . +  2( l − 1) − n  ∂ u ( ∇ β Λ β u ) ( l − 1) − 2 ∂ u ( ∇ β Λ β u ) ( l − 1) + (A.19) + ( l − 2)  ( l − 1) − n  ( ∇ β Λ β u ) ( l − 2) + (A.20) +  n − 2 l + 3  ( ∇ β Λ β u ) ( l − 2) + ∆( ∇ β Λ β u ) ( l − 2) = 0 , (A.21) +  2( l − 1) − n  ∂ u ( ∇ β Λ β u ) ( l − 1) + ( l − 2)  ( l − 1) − n  ( ∇ β Λ β u ) ( l − 2) + + ∆ ( ∇ β Λ β u ) ( l − 2) = 0 . (A.22) – 23 – Equation S r = 0 . +  2( l − 1) − n  ∂ u ( ∇ β Λ β r ) ( l − 1) − 2 ∂ u ( ∇ β Λ β r ) ( l − 1) + + ( l − 2)  ( l − 1) − n  ( ∇ β Λ β r ) ( l − 2) +  n − 2 l + 3  ( ∇ β Λ β r ) ( l − 2) + ∆( ∇ β Λ β r ) ( l − 2) + − 2 D i ( ∇ β Λ β i ) ( l − 3) + n h ( ∇ β Λ β u ) ( l − 2) − ( ∇ β Λ β r ) ( l − 2) i = 0 , (A.23) +  2( l − 1) − n  ∂ u ( ∇ β Λ β r ) ( l − 1) + ( l − 2)  ( l − 1) − n  ( ∇ β Λ β r ) ( l − 2) + + ∆ ( ∇ β Λ β r ) ( l − 2) − 2 D i ( ∇ β Λ β i ) ( l − 3) + n h ( ∇ β Λ β u ) ( l − 2) − ( ∇ β Λ β r ) ( l − 2) i = 0 . (A.24) Equation S i = 0 . +  2( l − 1) − ( n − 2)  ∂ u ( ∇ β Λ β i ) ( l − 1) − 2 ∂ u ( ∇ β Λ β i ) ( l − 1) + + ( l − 2)  ( l − 1) − ( n − 2)  ( ∇ β Λ β i ) ( l − 2) +  ( n − 2) − 2 l + 3  ( ∇ β Λ β i ) ( l − 2) + + D j D j ( ∇ β Λ β i ) ( l − 2) − ( n − 1)( ∇ β Λ β i ) ( l − 2) − 2 D i ( ∇ β Λ β u ) ( l − 1) + (A.25) − 2 D i ( ∇ β Λ β r ) ( l − 1) = 0 , (A.26) +  2( l − 1) − ( n − 2)  ∂ u ( ∇ β Λ β i ) ( l − 1) + ( l − 2)  ( l − 1) − ( n − 2)  ( ∇ β Λ β i ) ( l − 2) + + D j D j ( ∇ β Λ β i ) ( l − 2) − ( n − 1)( ∇ β Λ β i ) ( l − 2) + − 2 D i ( ∇ β Λ β u ) ( l − 1) + 2 D i ( ∇ β Λ β r ) ( l − 1) = 0 . (A.27) A.2 Equations from fall-off conditions The fall-off conditions in our gauge and gauge-for-gauge fixing reads as Case I δ  ϕ [ ur ] u  = ∂ u λ ( ru ) − ∂ r λ ( uu ) + 3 ∂ u Λ [ ru ] ∼ O  r − D − 2 2  , δ  ϕ [ ur ] r  = ∂ u λ ( rr ) − ∂ r λ ( ur ) − 3 ∂ r Λ [ ur ] ∼ O  r − D − 2 2  . (A.28) Case I I δ  ϕ [ ui ] u  = − ∂ i λ ( uu ) ∼ O  r − D − 4 2  , δ  ϕ [ ui ] r  = − ∂ i λ ( ur ) − ∂ i Λ [ ur ] ∼ O  r − D − 4 2  , δ  ϕ [ ri ] u  = − ∂ i λ ( ru ) − ∂ i Λ [ ru ] ∼ O  r − D − 4 2  , δ  ϕ [ ri ] r  = − ∂ i λ ( rr ) ∼ O  r − D − 4 2  . (A.29) – 24 – Case I I I δ  ϕ [ ui ] j  = + ∂ u λ ( ij ) + ∂ u Λ [ ij ] ∼ O  r − D − 6 2  , δ  ϕ [ ri ] j  = + ∂ r λ ( ij ) + ∂ r Λ [ ij ] ∼ O  r − D − 6 2  , δ  ϕ [ ij ] u  = +2 ∂ u Λ [ j i ] ∼ O  r − D − 6 2  , δ  ϕ [ ij ] r  = +2 ∂ r Λ [ j i ] ∼ O  r − D − 6 2  . (A.30) Case IV δ  ϕ [ ij ] k  = ∇ i λ ( j k ) − ∇ j λ ( ik ) + ∇ i Λ [ j k ] − ∇ j Λ [ ik ] + 2 ∇ k Λ [ j i ] ∼ O  r − D − 8 2  . (A.31) Assuming the p olyhomogeneous r expansions for the gauge parameters λ µν = X l ∈ 1 2 Z r − l  λ ( l ) µν + ¯ λ ( l ) µν ln r  , Λ µν = X l ∈ 1 2 Z r − l  Λ ( l ) µν + ¯ Λ ( l ) µν ln r  . (A.32) and defining p 1 = D − 2 2 , p 2 = D − 4 2 , p 3 = D − 6 2 , p 4 = D − 8 2 , the recursiv e equations are Case I ∂ u ¯ λ ( l ) ( ru ) + 3 ∂ u ¯ Λ ( l ) [ ru ] + ( l − 1) ¯ λ ( l − 1) ( uu ) = 0 , ∀ l < p 1 . (A.33) ∂ u λ ( l ) ( ru ) + 3 ∂ u Λ ( l ) [ ru ] + ( l − 1) λ ( l − 1) ( uu ) − ¯ λ ( l ) ( uu ) = 0 , ∀ l < p 1 . (A.34) ∂ u ¯ λ ( l ) ( rr ) + ( l − 1) ¯ λ ( l − 1) ( ur ) + 3( l − 1) ¯ Λ ( l − 1) [ ur ] = 0 , ∀ l < p 1 . (A.35) ∂ u λ ( l ) ( rr ) + l λ ( l ) ( ur ) + 3 l Λ ( l ) [ ur ] − ¯ λ ( l ) ( ur ) − 3 ¯ Λ ( l ) [ ur ] = 0 , ∀ l < p 1 . (A.36) Case I I ∂ i λ ( l ) ( uu ) = 0 , ∂ i ¯ λ ( l ) ( uu ) = 0 , ∀ l < p 2 . (A.37) ∂ i  λ ( l ) ( ur ) + Λ ( l ) [ ur ]  = 0 , ∂ i  ¯ λ ( l ) ( ur ) + ¯ Λ ( l ) [ ur ]  = 0 , ∀ l < p 2 . (A.38) ∂ i  λ ( l ) ( ru ) + Λ ( l ) [ ru ]  = 0 , ∂ i  ¯ λ ( l ) ( ru ) + ¯ Λ ( l ) [ ru ]  = 0 , ∀ l < p 2 . (A.39) ∂ i λ ( l ) ( rr ) = 0 , ∂ i ¯ λ ( l ) ( rr ) = 0 , ∀ l < p 2 . (A.40) – 25 – Case I I I ∂ u  λ ( l ) ( ij ) + Λ ( l ) [ ij ]  = 0 , ∂ u  ¯ λ ( l ) ( ij ) + ¯ Λ ( l ) [ ij ]  = 0 , ∀ l < p 3 . (A.41) − l  ¯ λ ( l ) ( ij ) + ¯ Λ ( l ) [ ij ]  = 0 , ∀ l + 1 < p 3 . (A.42) − l  λ ( l ) ( ij ) + Λ ( l ) [ ij ]  +  ¯ λ ( l ) ( ij ) + ¯ Λ ( l ) [ ij ]  = 0 , ∀ l + 1 < p 3 . (A.43) ∂ u Λ ( l ) [ j i ] = 0 , ∂ u ¯ Λ ( l ) [ j i ] = 0 , ∀ l < p 3 . (A.44) − 2 l ¯ Λ ( l ) [ j i ] = 0 , ∀ l + 1 < p 3 . (A.45) − 2 l Λ ( l ) [ j i ] + 2 ¯ Λ ( l ) [ j i ] = 0 , ∀ l + 1 < p 3 . (A.46) Case IV ∇ i ¯ λ ( l ) ( j k ) − ∇ j ¯ λ ( l ) ( ik ) + ∇ i ¯ Λ ( l ) [ j k ] − ∇ j ¯ Λ ( l ) [ ik ] + 2 ∇ k ¯ Λ ( l ) [ j i ] = 0 , ∀ l < p 4 . (A.47) ∇ i λ ( l ) ( j k ) − ∇ j λ ( l ) ( ik ) + ∇ i Λ ( l ) [ j k ] − ∇ j Λ ( l ) [ ik ] + 2 ∇ k Λ ( l ) [ j i ] = 0 , ∀ l < p 4 . (A.48) A.3 Equations of motion Making explicit the equations of motion for the indep endent field comp onen ts, we ha v e ∂ 2 r ϕ uru − 2 ∂ u ∂ r ϕ uru + D − 2 r ( ∂ r − ∂ u ) ϕ uru + 1 r 2 D 2 ϕ uru − 2 r 3 D i ϕ uiu − D − 2 r 2 ϕ uru = 0 , (A.49) ∂ 2 r ϕ [ ur ] r − 2 ∂ u ∂ r ϕ [ ur ] r + 1 r 2 D 2 ϕ [ ur ] r − 1 r 3 D i  ϕ [ ui ] r + ϕ [ ur ] i  + 2 r 4 γ ij ϕ [ ui ] j = 0 . (A.50) ∂ 2 r ϕ [ ur ] i − 2 ∂ u ∂ r ϕ [ ur ] i − 2 r ∂ r ϕ [ ur ] i + 2 r ∂ u ϕ [ ur ] i + 1 r 2 D 2 ϕ [ ur ] i − 1 r 3 D k ϕ [ uk ] i − D − 4 r 2 ϕ [ ur ] i = 0 . (A.51) ∂ 2 r ϕ [ ui ] u − 2 ∂ u ∂ r ϕ [ ui ] u + D − 4 r ∂ r ϕ [ ui ] u − D − 4 r ∂ u ϕ [ ui ] u + 1 r 2 D 2 ϕ [ ui ] u − 1 r 3 D k ϕ [ uk ] u − D − 3 r 2 ϕ [ ui ] u = 0 . (A.52) – 26 – ∂ 2 r ϕ [ ui ] r − 2 ∂ u ∂ r ϕ [ ui ] r + D − 2 r ∂ r ϕ [ ui ] r − D − 4 r ∂ u ϕ [ ui ] r + 1 r 2 D 2 ϕ [ ui ] r − 1 r 3 D k ϕ [ uk ] r − D − 2 r 2 ϕ [ ui ] r + 1 r 3 D i ϕ [ ur ] r = 0 . (A.53) ∂ 2 r ϕ [ ui ] j − 2 ∂ u ∂ r ϕ [ ui ] j + D − 4 r ∂ r ϕ [ ui ] j − D − 6 r ∂ u ϕ [ ui ] j + 1 r 2 D 2 ϕ [ ui ] j − D − 3 r 2 ϕ [ ui ] j − 1 r 3  D j ϕ [ ui ] r + D i ϕ [ uj ] r − γ ij D k ϕ [ uk ] r  + 2 r 3 D [ i ϕ [ u | j ] u = 0 . (A.54) ∂ 2 r ϕ [ ri ] u − 2 ∂ u ∂ r ϕ [ ri ] u + D − 2 r ∂ r ϕ [ ri ] u − D − 2 r ∂ u ϕ [ ri ] u + 1 r 2 D 2 ϕ [ ri ] u − D − 2 r 2 ϕ [ ri ] u − 1 r 3  D i ϕ [ ru ] u + D k ϕ [ rk ] u  + 2 r 3 D i ϕ [ ur ] u = 0 . (A.55) ∂ 2 r ϕ [ ri ] r − 2 ∂ u ∂ r ϕ [ ri ] r + D r ∂ r ϕ [ ri ] r − D − 2 r ∂ u ϕ [ ri ] r + 1 r 2 D 2 ϕ [ ri ] r − D r 2 ϕ [ ri ] r − 1 r 3 D k ϕ [ rk ] r + 1 r 3 D i ϕ [ ur ] r − 2 r 3 D i ϕ [ rr ] u = 0 . (A.56) ∂ 2 r ϕ [ ri ] j − 2 ∂ u ∂ r ϕ [ ri ] j + D − 2 r ∂ r ϕ [ ri ] j − D − 4 r ∂ u ϕ [ ri ] j + 1 r 2 D 2 ϕ [ ri ] j − D − 2 r 2 ϕ [ ri ] j − 1 r 3  D j ϕ [ ri ] r + D i ϕ [ rj ] r − γ ij D k ϕ [ rk ] r  + 1 r 3 D j ϕ [ ui ] r − 1 r 3 D i ϕ [ uj ] r = 0 . (A.57) ∂ 2 r ϕ [ ij ] u − 2 ∂ u ∂ r ϕ [ ij ] u + D − 4 r ∂ r ϕ [ ij ] u − D − 6 r ∂ u ϕ [ ij ] u + 1 r 2 D 2 ϕ [ ij ] u − 2( D − 3) r 2 ϕ [ ij ] u − 1 r 3  D i ϕ [ j u ] r − D j ϕ [ iu ] r  + 2 r 3  D i ϕ [ j u ] u − D j ϕ [ iu ] u  = 0 . (A.58) ∂ 2 r ϕ [ ij ] r − 2 ∂ u ∂ r ϕ [ ij ] r + D − 2 r ∂ r ϕ [ ij ] r − D − 4 r ∂ u ϕ [ ij ] r + 1 r 2 D 2 ϕ [ ij ] r − 2( D − 2) r 2 ϕ [ ij ] r − 1 r 3  D i ϕ [ j r ] r − D j ϕ [ ir ] r  + 1 r 3  D i ϕ [ j u ] r − D j ϕ [ iu ] r  = 0 . (A.59) – 27 – ∂ 2 r ϕ [ ij ] k − 2 ∂ u ∂ r ϕ [ ij ] k + D − 4 r ∂ r ϕ [ ij ] k − D − 6 r ∂ u ϕ [ ij ] k + 1 r 2 D 2 ϕ [ ij ] k − 2( D − 3) r 2 ϕ [ ij ] k − 1 r 3  D k ϕ [ ij ] r + D i ϕ [ j k ] r − D j ϕ [ ik ] r  + 1 r 3  D k ϕ [ ij ] u + D i ϕ [ j k ] u − D j ϕ [ ik ] u  = 0 . (A.60) B Ho dge-lik e decomp osition for ( p, q ) -mixed symme- try tensors In this section w e are going to generalize the Ho dge decomp osition to a Y oung pro jected ( p, q ) -biform, i.e. a mixed symmetry tensors assoaiced with Y oung diagram λ = ( p, q ) . W e refer to [ 45 ] for the basic definitions. The context is the generalization of the de Rham complex (for differen tial forms) to multi-forms and, after Y oung pro jection, to differen tial tensors of mixed symmetry . Let M b e a smo oth D -dimensional manifold (with pseudo-riemannian structure and metric g when Ho dge op erators are used). F or integers p 1 , . . . , p N ∈ { 0 , . . . , D } , the N -multi-form space is Ω p 1 ⊗···⊗ p N ( M ) := Ω p 1 ( M ) ⊗ · · · ⊗ Ω p N ( M ) . (B.1) An element T ∈ Ω p 1 ⊗···⊗ p N ( M ) has lo cal comp onen ts T [ µ (1) 1 ··· µ (1) p 1 ] ··· [ µ ( N ) 1 ··· µ ( N ) p N ] an tisym- metric within each blo c k of indices. In general, Ω p 1 ⊗···⊗ p N ( M ) carries a reducible represen tation of GL ( D ) . Given a Y oung diagram λ compatible with the degrees ( p 1 , . . . , p N ) , one extracts the corresp onding irreducible mixed-symmetry sector via the Y oung pro jector. F or eac h slot i ∈ { 1 , . . . , N } , the i -th differ ential is the ordinary de Rham differen tial acting only on the i -th factor: d ( i ) : Ω p 1 ⊗···⊗ p i ⊗···⊗ p N ( M ) − → Ω p 1 ⊗···⊗ ( p i +1) ⊗···⊗ p N ( M ) . (B.2) They satisfy the standard relations d ( i ) ◦ d ( i ) = 0 ∀ i , d ( i ) ◦ d ( j ) = d ( j ) ◦ d ( i ) ∀ i  = j . (B.3) Giv en a mixed-symmetry gauge field B (understo o d as Y oung pro jected), one defines the i -th (partial) field strength H ( i ) := Π λ ( i )  d ( i ) B  , (B.4) whenev er the Y oung pro jection yields a w ell-defined tableau λ ( i ) (obtained b y adding one b ox in the i -th blo ck). These partial field strengths are generally not fu lly gauge – 28 – in v ariant but are useful intermediate ob jects whose appropriate combination with traces can giv e rise to gauge in v ariant ob jects. The unam biguous gauge-in v ariant field strength is obtained by comp osing all slot differen tials. Define the de Rham-like differen tial δ ( N ) := d ( N ) ◦ d ( N − 1) ◦ · · · ◦ d (2) ◦ d (1) δ ( N ) ◦ δ ( N ) = 0 . (B.5) The field strength of B is then H := Π λ +  δ ( N ) B  ∈ Ω ( p 1 +1) ⊗···⊗ ( p N +1) ( M ) , (B.6) where λ + denotes the tableau obtained b y increasing each block degree b y one (when meaningful). F or N = 1 one reco v ers the usual de Rham differen tial and the ordinary curv ature H = dB . If ( M , g ) is oriented and equipp ed with a metric, one defines the i -th Ho dge morphism acting only on the i -th slot: ⋆ ( i ) : Ω p 1 ⊗···⊗ p i ⊗···⊗ p N ( M ) − → Ω p 1 ⊗···⊗ ( D − p i ) ⊗···⊗ p N ( M ) . (B.7) The full m ulti-form Ho dge op erator is the comp osition ⋆ := ⋆ ( N ) ◦ ⋆ ( N − 1) ◦ · · · ◦ ⋆ (1) . (B.8) An N -de Rham-lik e complex is the co c hain complex built from multi-form space with differen tial δ ( N ) and w e can define the asso ciated cohomology that turn out to b e isomorphic to de Rham cohomology . In classical Hodge theory on an orien ted Riemannian manifold ( M D , g ) , th e co differen tial d † is defined as the formal adjoin t of the de Rham differential d with resp ect to the L 2 inner pro duct induced b y g and the Riemannian volume form. The same logic extends to spaces of N -multi-forms. A natural p oin twise inner pro duct is obtained by contracting with g in eac h an tisymmetric blo ck: ⟨ A, B ⟩ x := 1 p 1 ! · · · p N ! A µ (1) 1 ··· µ (1) p 1 ··· µ ( N ) 1 ··· µ ( N ) p N B µ (1) 1 ··· µ (1) p 1 ··· µ ( N ) 1 ··· µ ( N ) p N , (B.9) and the L 2 pairing, that defines an L 2 -norm, is ⟨ A, B ⟩ := Z M ⟨ A, B ⟩ vol g . (B.10) One can think of ⟨ A, B ⟩ x v ol g as the natural analogue of α ∧ ∗ β : the pro duct is obtained b y wedging in eac h slot and then extracting the scalar density against v ol g exacly as in the Ho dge theory . W e define the co differential associated with δ ( N ) as the adjoint  δ ( N )  † suc h that ⟨ δ ( N ) A, B ⟩ = ⟨ A,  δ ( N )  † B ⟩ . (B.11) – 29 – Since δ ( N ) is a comp osition w e ha ve  δ ( N )  † =  d (1)  † ◦  d (2)  † ◦ · · · ◦  d ( N )  † . F or each slot i , the adjoin t of d ( i ) can b e written as  d ( i )  † B = ( − 1) D ( p i +1)+1 s ⋆ ( i ) d ( i ) ⋆ ( i ) B , when B has i -th degree p i . This is the direct analogue of the classical iden tit y d † = ± ∗ d ∗ for ordinary forms, applied only to the i -th factor. Because the slotwise op erators on distinct slots comm ute, one can collect all stars in to the total star and obtain  δ ( N )  † B = ( − 1) P N i =1 ( D ( p i +1)+1) s N ⋆ δ ( N ) ⋆ B . W e now restrict ourself to ( p, q ) -biforms, and relativ e mixed symmetry tensors, on a compact riemannian manifold without b oundary . W e ma y define the the laplacian ∆ δ (2) := δ (2)  δ (2)  † +  δ (2)  † δ (2) (B.12) that is of fourth order; how ever a b etter c hoice in order to hav e an elliptical op erator is to define the laplacian as e ∆ := ∆ 2 + ∆ δ (2) . (B.13) where ∆ := d (1) ( d (1) ) † + ( d (1) ) † d (1) + d (2) ( d (2) ) † + ( d (2) ) † d (2) = ∆ 1 + ∆ 2 . (B.14) Indeed ev en if ∆ δ (2) is not elliptic, the sum e ∆ is, let us see wh y in the next prop osition. Prop osition B.1. L et e ∆ := ∆ 2 + ∆ δ (2) the fourth-or der op er ator wher e ∆ and ∆ δ (2) ar e define d in ( B.14 ) and ( B.12 ) . Then e ∆ is el liptic on Ω p ⊗ q ( M ) and so is on a Y oung pr oje cte d subsp ac e. Pr o of. Let ξ ∈ T ∗ x M . The principal sym b ols of the first-order op erators are σ 1  d ( i )  ( x, ξ ) = ε i ( ξ ) := ξ ∧ ( i ) , σ 1  ( d ( i ) ) †  ( x, ξ ) = ι i ( ξ ) := ι ( i ) ξ ♯ . It follo ws that the principal sym b ol of each ∆ ( i ) is σ 2  ∆ ( i )  ( x, ξ ) = | ξ | 2 Id , ⇒ σ 2 (∆)( x, ξ ) = 2 | ξ | 2 Id , and therefore σ 4  ∆ 2  ( x, ξ ) =  2 | ξ | 2  2 Id = 4 | ξ | 4 Id . Moreo v er, σ 2  δ (2)  ( x, ξ ) = ε 2 ( ξ ) ε 1 ( ξ ) , σ 2  ( δ (2) ) †  ( x, ξ ) = ι 1 ( ξ ) ι 2 ( ξ ) , – 30 – so σ 4 (∆ δ (2) )( x, ξ ) is p ositive semidefinite but generally not in vertible since σ 4  δ (2) ( δ (2) ) †  ( x, ξ ) = ε 2 ε 1 ι 1 ι 2 = | ξ | 4 Π (1) + ( ξ ) Π (2) + ( ξ ) , σ 4  ( δ (2) ) † δ (2)  ( x, ξ ) = ι 1 ι 2 ε 2 ε 1 = | ξ | 4 Π (1) − ( ξ ) Π (2) − ( ξ ) . where Π ( i ) + ( ξ ) := 1 | ξ | 2 ε i ( ξ ) ι i ( ξ ) , Π ( i ) − ( ξ ) := 1 | ξ | 2 ι i ( ξ ) ε i ( ξ ) . Therefore σ 4  ∆ δ (2)  ( x, ξ ) = | ξ | 4  Π (1) + ( ξ )Π (2) + ( ξ ) + Π (1) − ( ξ )Π (2) − ( ξ )  , (B.15) whic h has k ernel on Ω p 1 ⊗ p 2 ( M ) . In particular, ∆ δ (2) is not elliptic b y itself. F or ξ  = 0 , the principal symbol of e ∆ is the sum σ 4 ( e ∆)( x, ξ ) = 4 | ξ | 4 Id + σ 4 (∆ δ (2) )( x, ξ ); moreo v er, • since ∆ is a second-order elliptic op erator, its symbol σ ∆ ( x, ξ ) is strictly p ositiv e definite. Consequen tly , the sym b ol of ∆ 2 is also strictly p ositiv e definite: ⟨ σ ∆ 2 ( x, ξ ) v , v ⟩ > 0 , ∀ v  = 0 , ∀ ξ  = 0 . (B.16) • the op erator ∆ δ (2) is non-elliptic, meaning its symbol σ ∆ δ (2) ( x, ξ ) may ha ve a non-trivial k ernel in certain d irections. Ho w ev er, its algebraic structure (it is defined b y the Ho dge t yp e structure) ensures it is semi-definite p ositive: ⟨ σ ∆ δ (2) ( x, ξ ) v , v ⟩ ≥ 0 , ∀ v  = 0 , ∀ ξ  = 0 . (B.17) Therefore, ⟨ σ e ∆ ( x, ξ ) v , v ⟩ = ⟨ σ ∆ 2 ( x, ξ ) v , v ⟩ + ⟨ σ ∆ δ (2) ( x, ξ ) v , v ⟩ > 0 , ∀ v  = 0 , ∀ ξ  = 0; (B.18) since σ e ∆ ( x, ξ ) is strictly positive definite, all its eigenv alues are strictly p ositiv e, ensuring it is in v ertible. This prov es that e ∆ is an elliptic op erator. Let T ∈ Ω p,q ( M ) , so ⟨ e ∆ T , T ⟩ = ⟨ ∆ 2 T , T ⟩ + ⟨ ∆ δ (2) T , T ⟩ ; (B.19) using the definitions of co differen tials and that ∆ is selfadjoin t we get  ∆ δ (2) T , T  =  δ (2) ( δ (2) ) † T , T  +  ( δ (2) ) † δ (2) T , T  = ∥ ( δ (2) ) † T ∥ 2 + ∥ δ (2) T ∥ 2 ⟨ ∆ 2 T , T ⟩ = ⟨ ∆ T , ∆ T ⟩ = ∥ ∆ T ∥ 2 . (B.20) – 31 – Therefore, T ∈ ker( e ∆) ⇔ ∆ T = 0 , δ (2) T = 0 , ( δ (2) ) † T = 0 , equiv alen tly k er( e ∆) = ker(∆) ∩ k er  δ (2)  ∩ k er  ( δ (2) ) †  = = k er  d (1)  ∩ k er  ( d (1) ) †  ∩ k er  d (2)  ∩ k er  ( d (2) ) †  ∩ k er  δ (2)  ∩ k er  ( δ (2) ) †  , (B.21) where w e used that im ( d (1) ) ⊥ im ( d (2) ) with resp ect to the pro duct ( B.9 ) and similar for co differentials. Since if T ∈ k er (∆) then T ∈ k er  δ (2)  ∩ k er  ( δ (2) ) †  , this is b ecause δ (2) and its co differential are composition of d (1) , d (2) and their co differentials, it follo ws that k er(∆) ⊂ k er(∆ δ (2) ) and so k er( e ∆) = ker(∆) ∩ k er(∆ δ (2) ) = ker(∆) . By ellipticity of e ∆ and compactness of M the kernel is finite-dimensional and im ( e ∆ ) = ( k er ( e ∆)) ⊥ . W e define the space of 2-harmonic ( p, q ) -biforms as H p ⊗ q ( M ) := { T ∈ Ω p ⊗ q ( M ) | e ∆ T = 0 } . (B.22) Theorem B.2. If M is c omp act then H p ⊗ q ( M ) ∼ = H p ⊗ H q wher e H k is the standar d Ho dge sp ac e of harmonic k -forms. Pr o of. Since M is compact H p ⊗ q ( M ) is finite-dimensional. Moreo v er, since k er ( e ∆ ) = k er(∆) ∩ k er(∆ δ (2) ) = ker(∆) w e hav e H p ⊗ q ( M ) := { T ∈ Ω p ⊗ q ( M ) | ∆ T = 0 } = k er (∆) . Since ∆ = ∆ 1 + ∆ 2 it follo ws that k er (∆) = ker (∆ 1 ) ∩ k er (∆ 2 ) and k er (∆ 1 ) = { T ∈ Ω p ⊗ q ( M ) | T ∈ H p ⊗ Ω q ( M ) } , k er (∆ 2 ) = { T ∈ Ω p ⊗ q ( M ) | T ∈ Ω p ( M ) ⊗ H q } ; so k er (∆) = ( H p ⊗ Ω q ( M )) ∩ (Ω p ( M ) ⊗ H q ) ∼ = H p ⊗ H q In the end, the space Ω p ⊗ q ( M ) can b e decomp osed as Ω p ⊗ q ( M ) ∼ = im ( e ∆) ⊕ H p ⊗ q ( M ); (B.23) explicitly if T ∈ Ω p ⊗ q w e ha ve T = T harm + d (1) A +  d (1)  † B + d (2) C +  d (2)  † D + + d (1) d (2) ϕ + d (1)  d (2)  † Ψ +  d (1)  † d (2) Ξ +  d (1)  †  d (2)  † Θ , (B.24) – 32 – where the harmonic part T harm satisfies T harm ∈ H p ⊗ q ( M ) = ker( e ∆) = ker(∆) ∼ = H p ( M ) ⊗ H q ( M ) , (B.25) and the p oten tials ha ve the follo wing bidegrees: A ∈ Ω ( p − 1) ⊗ q ( M ) , B ∈ Ω ( p +1) ⊗ q ( M ) , C ∈ Ω p ⊗ ( q − 1) ( M ) , D ∈ Ω p ⊗ ( q +1) ( M ) , ϕ ∈ Ω ( p − 1) ⊗ ( q − 1) ( M ) , Ψ ∈ Ω ( p − 1) ⊗ ( q +1) ( M ) , Ξ ∈ Ω ( p +1) ⊗ ( q − 1) ( M ) , Θ ∈ Ω ( p +1) ⊗ ( q +1) ( M ) . (B.26) This is the general decomp osition for a ( p, q ) -biform; ho wev er for a ( p, q ) mixed symmetry tensor the p oten tials has to b e in appropriate representations, meaning that the Y oung diagram asso ciated to them has to b e admissible. In particular this means that the decomp osition for a ( p, p ) mixed symmetry tensor reduces to T = T harm +  d (1)  † B + d (2) C + δ (2) ϕ +  d (1)  † d (2) Ξ +  δ (2)  † Θ ; (B.27) furthermore, some p oten tial could b e v anishing once w e require to hav e the right indices symmetries to app ear in the decomp osition. C Useful comm utation relations and asymptotic gauge fixing C.1 V anishing of the antisymmetric part of S ij Using the pair-exc hange symmetry ( k i ) ↔ ( ℓj ) , Y kj ℓi = Y ℓikj , w e compute S j i = ∇ k ∇ ℓ Y kj ℓi = ∇ k ∇ ℓ Y ℓikj = ∇ ℓ ∇ k Y kiℓj , where in the last step w e only relab eled the dumm y indices k ↔ ℓ . Hence S ij − S j i =  ∇ k ∇ ℓ − ∇ ℓ ∇ k  Y kiℓj = [ ∇ k , ∇ ℓ ] Y kiℓj . F or any (0 , 4) -tensor T kiℓj , [ ∇ a , ∇ b ] T kiℓj = − R m kab T miℓj − R m iab T kmℓj − R m ℓab T kimj − R m j ab T kiℓm . Applying this to T = Y and contracting with g ka g ℓb yields S ij − S j i = g ka g ℓb [ ∇ a , ∇ b ] Y kiℓj = T 1 + T 2 + T 3 + T 4 , – 33 – where T 1 : = − g ka g ℓb R m kab Y miℓj , T 2 : = − g ka g ℓb R m iab Y kmℓj , T 3 : = − g ka g ℓb R m ℓab Y kimj , T 4 : = − g ka g ℓb R m j ab Y kiℓm . On S 3 (a space form) the Riemann tensor is R abcd = K ( g ac g bd − g ad g bc ) . Raising the first index, R m kab = g mc R ckab = K ( δ m a g kb − δ m b g ka ) . Define the “Ricci con traction” of Y b y y ij : = Y k ikj = g kℓ Y kiℓj . Computation of T 1 . g ka g ℓb R m kab = K  g ka g ℓb δ m a g kb − g ka g ℓb δ m b g ka  = K  g km g ℓb g kb − g ℓm g ka g ka  = K  g km δ ℓ k − g ℓm δ k k  = K ( g ℓm − 3 g ℓm ) = − 2 K g ℓm . Therefore T 1 = − ( − 2 K g ℓm ) Y miℓj = 2 K g ℓm Y miℓj = 2 K y ij . Computation of T 2 . Using R m iab = K ( δ m a g ib − δ m b g ia ) , w e get g ka g ℓb R m iab Y kmℓj = K  g ka g ℓb δ m a g ib Y kmℓj − g ka g ℓb δ m b g ia Y kmℓj  = K  g km δ ℓ i Y kmℓj − g ℓm δ k i Y kmℓj  = K  g km Y kmij − g ℓm Y imℓj  . The first term v anishes since Y kmij is an tisymmetric in ( k , m ) while g km is symmetric: g km Y kmij = 0 . F or the second term, using Y imℓj = − Y miℓj , − g ℓm Y imℓj = − g ℓm ( − Y miℓj ) = g ℓm Y miℓj = y ij . Hence g ka g ℓb R m iab Y kmℓj = K y ij , T 2 = − K y ij . – 34 – Computation of T 3 . Using R m ℓab = K ( δ m a g ℓb − δ m b g ℓa ) , first con tract with g ℓb : g ℓb R m ℓab = K  g ℓb δ m a g ℓb − g ℓb δ m b g ℓa  = K  δ m a g ℓb g ℓb − g ℓm g ℓa  = K  3 δ m a − δ m a  = 2 K δ m a . Therefore g ka g ℓb R m ℓab = g ka (2 K δ m a ) = 2 K g km , and th us T 3 = − (2 K g km ) Y kimj = − 2 K y ij . Computation of T 4 . Similarly , g ka g ℓb R m j ab Y kiℓm = K  g km Y kij m − g ℓm Y j iℓm  . The second term v anishes since Y j iℓm is an tisymmetric in ( ℓ, m ) : g ℓm Y j iℓm = 0 . F or the first term, use pair exc hange Y kij m = Y j mk i and then an tisymmetry in the first pair: g km Y kij m = g km Y j mk i = − g km Y mj k i = − g mk Y kj mi = − g km Y kj mi = − y j i . Hence g ka g ℓb R m j ab Y kiℓm = − K y j i , T 4 = − ( − K y j i ) = K y j i . Summing up, S ij − S j i = (2 K y ij ) + ( − K y ij ) + ( − 2 K y ij ) + ( K y j i ) = K ( y j i − y ij ) . Using the first Bianc hi iden tity , Y kiℓj + Y iℓkj + Y ℓkij = 0 , and con tracting with g kℓ , w e obtain g kℓ Y kiℓj + g kℓ Y iℓkj + g kℓ Y ℓkij = 0 . The first term is y ij . The third term v anishes since Y ℓkij is an tisymmetric in ( ℓ, k ) : g kℓ Y ℓkij = 0 . F or the second term, use Y iℓkj = − Y ℓikj and then pair exc hange Y ℓikj = Y kj ℓi : g kℓ Y iℓkj = − g kℓ Y ℓikj = − g kℓ Y kj ℓi = − y j i . Therefore y ij − y j i = 0 = ⇒ y ij = y j i . Plugging this in to S ij − S j i = K ( y j i − y ij ) giv es S ij − S j i = 0 = ⇒ S ij = S j i . – 35 – C.2 Asymptotic gauge fixing The gauge condition equation reads 0 = ( D i H ruij ) ′ = D i H ruij + δ ( D i H ruij ) (C.1) where δ ( H ruij ) = − 2 D i ∂ j Λ ru + 2 r γ ij ( ∂ u Λ ru + ∂ r Λ ru ) , (C.2) and we ha ve used the gauge-for-gauge ( 2.5 ) . Therefore, expanding order b y order, thee equation for order D − 4 2 is D i H  D − 4 2  ruij − 2∆ ∂ j Λ  D − 4 2  ru +2 D j ∂ u Λ  D − 2 2  ru − ( D − 4) D j Λ  D − 4 2  ru − 2 D j ¯ Λ  D − 4 2  ru = 0; (C.3) that turns out to be compatible and there are enough arbitrary functions to ensure solv abilit y . C.3 Other useful commutators F or tensor H ij w e ha ve [ D a , D b ] H ij = R i mab H mj + R j mab H im . (C.4) On a 3D space form of sectional curv ature K , R abcd = K  γ ac γ bd − γ ad γ bc  , R ab = 2 K γ ab . (C.5) A direct con traction of ( C.4 ) b y γ bℓ and b y using ( C.5 ) we get [ D j , D ℓ ] H ij = K H ℓi + 2 K H iℓ − K γ iℓ H ; (C.6) using v i := D j H ij w e get, D j D ℓ H ij = D ℓ D j H ij + [ D j , D ℓ ] H ij = D ℓ v i + K H ℓi + 2 K H iℓ − K γ iℓ H . (C.7) Con tracting ( C.4 ) on the pair ( i, j ) we get [ D i , D j ] H ij = R mj H mj − R mi H im = 0; (C.8) since D i H ij = 0 w e hav e D j D i H ij = 0 , hence 0 = D j D i H ij = D i D j H ij + [ D j , D i ] H ij = D i D j H ij = D i v i . (C.9) – 36 – References [1] L. Ciambelli, F r om Asymptotic Symmetries to the Corner Pr op osal , PoS Mo da v e2022 (2023) 002 [ 2212.13644 ]. [2] A. Strominger, L e ctur es on the infr ar e d structur e of gr avity and gauge the ory , 1703.05448 . [3] F. Manzoni, Duality and asymptotic symmetries in gr avisolitons and gauge the ories , Ph.D. thesis, Univ ersità degli Studi di Roma T re, 2025. [4] C. Heissenberg, T opics in Asymptotic Symmetries and Infr ar e d Effe cts , Ph.D. thesis, Pisa, Scuola Normale Sup eriore, 2019. 1911.12203 . [5] G. Comp ère and A. Fiorucci, A dvanc e d L e ctur es on Gener al R elativity , 1801.07064 . [6] H. Bondi, M.G.J. v an der Burg and A.W.K. Metzner, Gr avitational waves in gener al r elativity. 7. W aves fr om axisymmetric isolate d systems , Pr o c. R oy. So c. L ond. A 269 (1962) 21 . [7] R. Sachs, Asymptotic symmetries in gr avitational the ory , Phys. R ev. 128 (1962) 2851 . [8] R.K. Sachs, Gr avitational waves in gener al r elativity. 8. W aves in asymptotic al ly flat sp ac e-times , Pr o c. R oy. So c. L ond. A 270 (1962) 103 . [9] S. Pasterski, Asymptotic Symmetries and Ele ctr omagnetic Memory , JHEP 09 (2017) 154 [ 1505.00716 ]. [10] N. Jokela, K. Ka jantie and M. Sarkkinen, Memory effe ct in Y ang-Mil ls the ory , Phys. R ev. D 99 (2019) 116003 [ 1903.10231 ]. [11] H. Afshar, E. Esmaeili and M.M. Sheikh-Jabbari, String memory effe ct , Journal of High Ener gy Physics 2019 (2019) . [12] M. Pate, A.-M. Raclariu and A. Strominger, Color memory: A yang-mil ls analo g of gr avitational wave memory , Physic al R eview L etters 119 (2017) . [13] A. Strominger, Asymptotic symmetries of yang-mil ls the ory , Journal of High Ener gy Physics 2014 (2014) . [14] D. F rancia and C. Heissenberg, Two-F orm Asymptotic Symmetries and Sc alar Soft The or ems , Phys. R ev. D 98 (2018) 105003 [ 1810.05634 ]. [15] Y. Hamada and S. Sugishita, Soft pion the or em, asymptotic symmetry and new memory effe ct , Journal of High Ener gy Physics 2017 (2017) . [16] N. de Aguiar Alves and A.G.S. Landulfo, Sound as a gauge the ory and its infr ar e d triangle , 2512.15796 . [17] E. Esmaeili, p -form gauge fields: char ges and memories , Ph.D. thesis, IPM, T ehran, 9, 2020. 2010.13922 . [18] G. Barnich and C. T ro essaert, Bms char ge algebr a , Journal of High Ener gy Physics (2011) 105 [ 1106.0213 ]. – 37 – [19] G. Barnich and C. T ro essaert, Symmetries of asymptotic al ly flat 4 dimensional sp ac etimes at nul l infinity r evisite d , Physic al R eview L etters 105 (2010) 111103 [ 0909.2617 ]. [20] M. Campiglia and A. Laddha, Asymptotic symmetries and suble ading soft gr aviton the or em , Physic al R eview D 90 (2014) 124028 [ 1408.2228 ]. [21] M. Campiglia and A. Laddha, New symmetries for the gr avitational s-matrix , Journal of High Ener gy Physics (2015) 076 [ 1502.02318 ]. [22] D. Kap ec, V. Lysov, S. P asterski and A. Strominger, Semiclassic al vir asor o symmetry of the quantum gr avity s-matrix , Journal of High Ener gy Physics (2014) 058 [ 1406.3312 ]. [23] L. F reidel, R. Oliv eri, D. Pranzetti and S. Speziale, The weyl bms gr oup and einstein ’s e quations , Journal of High Ener gy Physics (2021) 170 [ 2104.05793 ]. [24] K. T anab e, N. T anahashi and T. Shiromizu, On asymptotic structur e at nul l infinity in five dimensions , Journal of Mathematic al Physics 51 (2010) 062502 [ 0909.0426 ]. [25] K. T anab e, S. Kinoshita and T. Shiromizu, Asymptotic flatness at nul l infinity in arbitr ary dimensions , Physic al R eview D 84 (2011) 044055 [ 1104.0303 ]. [26] D. Kap ec, V. Lysov, S. Pasterski and A. Strominger, Higher-dimensional sup ertr anslations and Weinb er g’s soft gr aviton the or em , Annals of Mathematic al Scienc es and Applic ations 2 (2017) 69 [ 1502.07644 ]. [27] S. Hollands and A. Ishibashi, Asymptotic flatness and b ondi ener gy in higher dimensional gr avity , Journal of Mathematic al Physics 46 (2005) 022503 [ gr-qc/0304054 ]. [28] O. F uen tealba, M. Henneaux, J. Matulic h and C. T ro essaert, Bondi-metzner-sachs gr oup in five sp ac etime dimensions , Physic al R eview L etters 128 (2022) 051103 [ 2111.09664 ]. [29] O. F uentealba, M. Henneaux, J. Matulich and C. T ro essaert, Asymptotic structur e of the gr avitational field in five sp ac etime dimensions: Hamiltonian analysis , Journal of High Ener gy Physics (2022) 149 [ 2206.04972 ]. [30] A. Kehagias and A. Riotto, Bms in c osmolo gy , Journal of Cosmolo gy and Astr op article Physics 2016 (2016) 059–059 . [31] A. T olish and R.M. W ald, Cosmolo gic al memory effe ct , Phys. R ev. D 94 (2016) 044009 [ 1606.04894 ]. [32] B. Bonga and K. Prabh u, BMS-like symmetries in c osmolo gy , Phys. R ev. D 102 (2020) 104043 [ 2009.01243 ]. [33] F. Manzoni, Axialgr avisolitons at infinite c orner , Class. Quant. Gr av. 41 (2024) 177001 [ 2404.04951 ]. [34] R.Z. F erreira, M. Sandora and M.S. Sloth, Asymptotic Symmetries in de Sitter and Inflationary Sp ac etimes , JCAP 04 (2017) 033 [ 1609.06318 ]. – 38 – [35] A. Camp oleoni, D. F rancia and C. Heissen b erg, Ele ctr omagnetic and c olor memory in even dimensions , Phys. R ev. D 100 (2019) 085015 [ 1907.05187 ]. [36] A. Camp oleoni, D. F rancia and C. Heissen b erg, On asymptotic symmetries in higher dimensions for any spin , JHEP 12 (2020) 129 [ 2011.04420 ]. [37] A. Camp oleoni, D. F rancia and C. Heissenberg, On higher-spin sup ertr anslations and sup err otations , JHEP 05 (2017) 120 [ 1703.01351 ]. [38] M. Campiglia and A. Laddha, Asymptotic symmetries of QED and W einb er g’s soft photon the or em , JHEP 07 (2015) 115 [ 1505.05346 ]. [39] A. Camp oleoni, D. F rancia and C. Heissen b erg, Asymptotic Char ges at Nul l Infinity in A ny Dimension , Universe 4 (2018) 47 [ 1712.09591 ]. [40] H. Afshar, E. Esmaeili and M.M. Sheikh-Jabbari, Asymptotic Symmetries in p -F orm The ories , JHEP 05 (2018) 042 [ 1801.07752 ]. [41] M. Romoli, O (r N ) two-form asymptotic symmetries and r enormalize d char ges , JHEP 12 (2024) 085 [ 2409.08131 ]. [42] P . F errero, D. F rancia, C. Heissen b erg and M. Romoli, Double-c opy sup ertr anslations , Phys. R ev. D 110 (2024) 026009 [ 2402.11595 ]. [43] M. Romoli and F. Manzoni, Higher-or der p-form Asymptotic Symmetries in D = p + 2 , Int. J. The or. Phys. 65 (2026) 56 [ 2503.22572 ]. [44] D. F rancia and F. Manzoni, Asymptotic char ges of p − forms and their dualities in any D , 2411.04926 . [45] F. Manzoni, Duality, asymptotic char ges and algebr aic top olo gy in mixe d symmetry tensor gauge the ories and applic ations , 2501.05104 . [46] F. Manzoni, Duality, asymptotic char ges and higher form symmetries in p-form gauge the ories , Eur. Phys. J. C 86 (2026) 155 [ 2411.05602 ]. [47] R. Ruzziconi, Asymptotic Symmetries in the Gauge Fixing Appr o ach and the BMS Gr oup , PoS Mo da ve2019 (2020) 003 [ 1910.08367 ]. [48] X. Bek aert and N. Boulanger, The unitary r epr esentations of the Poinc ar\’e gr oup in any sp ac etime dimension , SciPost Phys. L e ct. Notes 30 (2021) 1 [ hep-th/0611263 ]. [49] T.L. Curtright, HIGH SPIN FIELDS , AIP Conf. Pr o c. 68 (1980) 985 . [50] T. Curtright, GENERALIZED GAUGE FIELDS , Phys. L ett. B 165 (1985) 304 . [51] T.L. Curtright and P .G.O. F reund, MASSIVE DUAL FIELDS , Nucl. Phys. B 172 (1980) 413 . [52] T.L. Curtright, Massive Dual Spinless Fields R evisite d , Nucl. Phys. B 948 (2019) 114784 [ 1907.11530 ]. [53] T. Curtright, Gener alize d gauge fields , Physics L etters B 165 (1985) 304 . [54] J.M.F. Labastida and T.R. Morris, MASSLESS MIXED SYMMETR Y BOSONIC FREE FIELDS , Phys. L ett. B 180 (1986) 101 . – 39 – [55] J.M.F. Labastida, Massless Particles in A rbitr ary R epr esentations of the L or entz Gr oup , Nucl. Phys. B 322 (1989) 185 . [56] I. V uk ović, Higher spin the ory , 1809.02179 . [57] M. V asiliev, Higher spin gauge the ories in various dimensions , F ortschritte der Physik 52 (2004) 702–717 . [58] B. de Wit and D.Z. F reedman, Systematics of higher-spin gauge fields , Phys. R ev. D 21 (1980) 358. [59] A. Sagnotti, Notes on strings and higher spins , 1112.4285 . [60] J. Polc hinski, String the ory. V ol. 1: A n intr o duction to the b osonic string , Cam bridge Monographs on Mathematical Physics, Cam bridge Univ ersity Press (12, 2007), 10.1017/CBO9780511816079 . [61] J. Polc hinski, String the ory. V ol. 2: Sup erstring the ory and b eyond , Cam bridge Monographs on Mathematical Physics, Cam bridge Univ ersity Press (12, 2007), 10.1017/CBO9780511618123 . [62] M.B. Green, J.H. Sch warz and E. Witten, Sup erstring The ory: 25th A nniversary Edition , vol. 1 of Cambridge Mono gr aphs on Mathematic al Physics , Cam bridge Univ ersity Press (2012), 10.1017/CBO9781139248563 . [63] M.B. Green, J.H. Sch warz and E. Witten, Sup erstring The ory V ol. 2: 25th A nniversary Edition , Cam bridge Monographs on Mathematical Ph ysics, Cambridge Univ ersity Press (11, 2012), 10.1017/CBO9781139248570 . [64] C.M. Hull, Str ongly c ouple d gr avity and duality , Nucl. Phys. B 583 (2000) 237 [ hep-th/0004195 ]. [65] C.M. Hull, Duality in gr avity and higher spin gauge fields , JHEP 09 (2001) 027 [ hep-th/0107149 ]. [66] P .d. Medeiros and C. Hull, Exotic tensor gauge the ory and duality , Communic ations in Mathematic al Physics 235 (2003) 255–273 . [67] M. Hamermesh, Gr oup The ory and Its Applic ation to Physic al Pr oblems , Addison W esley Series in Physics, Do ver Publications (1989). [68] N. Boulanger, S. Cno ck aert and M. Henneaux, A note on spin-s duality , Journal of High Ener gy Physics 2003 (2003) 060 . [69] J. Lee and R.M. W ald, L o c al symmetries and c onstr aints , J. Math. Phys. 31 (1990) 725 . – 40 –