Julien Quéva
email: queva@univ-corse.fr

A conformally invariant gauge fixing equation and a field strength for the symmetric traceless field over d = 4 conformally flat Einstein spacetimes

This article investigates the properties of a set of conformally invariant equations on conformally flat Einstein spacetimes. These equations are shown to be gauge invariant if d = 4. We provide a conformally invariant gauge condition to that equation which generalizes in a simple manner, on those spacetimes, the Eastwood-Singer gauge condition. A byproduct of this conformally invariant gauge fixing equation is an alternate proof of Branson's factorization formula of GJMS operators on Einstein manifolds for d = 4. A field strength F is built upon the field A, its properties are worked out in details.

I. INTRODUCTION

This article aims to provide a gauge fixing equation and a field strength F to a field A solution of a restriction to conformally flat Einstein spacetimes (CFES) of a conformally invariant equation [START_REF] Wünsch | On conformally invariant differential operators[END_REF]Prop.3.2], namely: (E s (A)) µ1...µs = ( + c s R)A µ1...µs + a s s∇ (µ1 ∇ ρ A µ2...µs)ρ + b s s(s -1)g (µ1µ2 ∇ ρ ∇ σ A µ3...µs)ρσ = 0, [START_REF] Wünsch | On conformally invariant differential operators[END_REF] with A(x) ∈

• S s T x M ≡ S s • a traceless symmetric tensor of rank s, A µ1...µs = g αβ ∇ α ∇ β A µ1...µs , R the scalar curvature of the underlying lorentzian spacetime (M, g) of dimension d ≥ 3, the coefficients in [START_REF] Wünsch | On conformally invariant differential operators[END_REF] are given by:

a s = - 4 d + 2s -2 , b s = 4 (d + 2s -4)(d + 2s -2) , c s = - d 2 -2d + 4s 4d(d -1) , (2)
a 0 = b 0 = b 1 = 0 and (µ 1 . . . µ s ) is the normalized symmetrization of the enclosed indices. Equation ( 1) is explicitly shown to admit gauge solutions for tensors of rank s ≥ 1 on CFES of dimension d = 4. We demonstrate that the gauge solutions of (1) at d = 4 might be restricted, while keeping the conformal invariance, thanks to a gauge condition which generalizes (to higher ranks) the Eastwood-Singer gauge fixing equation on CFES. To be precise we show that the set:

E s (A) = 0, E 0 (φ) = 0, (d = 4) (3)
is conformally invariant and fixes the gauge, with φ = ∇ µ1 . . . ∇ µs A µ1...µs . This article is organized as follows.

In section II we will recall elementary, yet needed, facts on conformal invariance. Section III illustrates the content of the article by studying fields from rank 0 to rank 2. In such cases one sees the emergence of gauge freedom in dimension d = 4, the underlying structure between equations of different ranks when formulated on CFES. Section IV shows that these properties are indeed found for any A of rank s ≥ 2 satisfying [START_REF] Wünsch | On conformally invariant differential operators[END_REF]. That is: invariance under Weyl transformations between CFES and that there are gauge solutions for d = 4. A, generally, conformally invariant equation is recovered in (53), its restriction to CFES coincides with [START_REF] Wünsch | On conformally invariant differential operators[END_REF]. However, since we remain unclear on its gauge (or absence of) freedom we will refrain to analyze that equation. In section V we exhibit a conformally invariant gauge fixing equation. We show how it arises from the field equation in arbitrary dimensions and how it generalizes, to CFES, the Eastwood-Singer equation. These equations are precisely the set [START_REF] Branson | Sharp inequalities, the functional determinant, and the complementary series[END_REF]. We then show how by plugging a pure gauge field in the gauge fixing equation, thus inspecting the residual gauge freedom, one gets, as a byproduct, a conformally invariant equation of order 2(s+ 1) acting on scalars of weight s-1, for d = 4. This equation is recognized to be a GJMS operator [START_REF] Fefferman | The ambient metric[END_REF] fulfilling Branson's factorization formula on Einstein spaces [START_REF] Branson | Sharp inequalities, the functional determinant, and the complementary series[END_REF][START_REF] Gover | Laplacian operators and Q-curvature on conformally Einstein manifolds[END_REF]. Then our study shows a new way to derive this formula. Section VI establishes the properties of the field strength F associated to the field A. The gauge invariance of F is explicitly shown, a lagrangian giving rise to (1) is retro-engineered on those F , a set of first order equations corresponding to Maxwell's equations on the observable fields is uncovered, a decomposition of F on E-B fields is performed and its duality is shown. Appendix A comments on (the lack of) content of (1) in the case d = 2.

Those results set the ground for the quantization of fields satisfying [START_REF] Wünsch | On conformally invariant differential operators[END_REF], when d = 4, with the help of the conformally invariant gauge fixing equation [START_REF] Branson | Sharp inequalities, the functional determinant, and the complementary series[END_REF] and the scalar product (165). A task which will be addressed elsewhere.

II. A REMINDER ON CONFORMAL INVARIANCE

Let us recall the known fact that "conformal invariance" recovers two different notions, namely the invariance under Weyl rescalings and invariance of a space of solutions under the action of the conformal group. See, say, the review of Kastrup [START_REF] Kastrup | On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics[END_REF] for further references on the application of conformal invariance in theoretical physics.

A. Conformal invariance w.r.t. Weyl rescalings

On the one hand [6, App.D], starting from a manifold M equipped with a metric g a Weyl rescaling might, roughly, be defined as the application:

(M, g) → (M, g) s.t. g µν (x) = ω 2 (x)g µν (x), ω ∈ C ∞ (M ). [START_REF] Gover | Laplacian operators and Q-curvature on conformally Einstein manifolds[END_REF] Owing to this definition quantities derived from the metric, such as the curvature scalar, change according to:

Γ ρ µν = Γ ρ µν + ω -1 (δ ρ µ δ σ ν + δ ρ ν δ σ µ -g µν g ρσ )ω ,σ , (5)
R µ ρνσ = R µ ρνσ -ω -1 (δ µ [ν δ τ σ] δ ϕ ρ -g ρ[ν δ τ σ] g µϕ )ω ;τ ϕ + ω -2 (2δ µ [ν δ τ σ] δ ϕ ρ -2g ρ[ν δ τ σ] g µϕ + g ρ[ν δ µ σ] g τ ϕ )ω ,τ ω ,ϕ , (6)
R µν = R µν -ω -1 (g µν g ρσ + (d -2)δ ρ µ δ σ ν )ω ;ρσ + ω -2 (2(d -2)δ ρ µ δ σ ν -(d -3)g µν g ρσ )ω ,ρ ω ,σ , (7)
R = ω -2 R -2(d -1)ω -3 ω ;µ µ -(d -4)(d -1)ω -4 ω ,µ ω ,µ . (8)
In the above equations the semi-colon refers to the covariant derivation with respect to g, Γ is the Levi-Civita connection, and [αβ] = αβ -βα is the antisymmetric part of the enclosed indices. Notice that in practice M often will be a common subset of the spacetimes we are interested in.

Then, an equation, depending on the metric and symbolically written, E(A) = 0 is said to be Weyl invariant if and only if there exists a conformal weight h(A) ∈ R such that:

E(A) = ω h ′ E(A), A = ω h A. (9)
That is, for a given solution A of the equation E(A) = 0, the rescaled field A = ω h A is a solution of E(A) = 0 on the Weyl related spacetime (M, g). Conditions such as indices symmetry and tracelessness are obviously Weyl invariant. On the other hand, the conformal group C g of (M, g) is, by definition, the set of transformations letting the causality invariant. A convenient characterization of its infinitesimal generators is that they fulfill the conformal Killing equation:

(L X g)(x) = 2f X (x)g(x), (10)
with L X the Lie derivative along X. Notice that, according to (5), the characterization through [START_REF] Binegar | de Sitter QED[END_REF] implies that the conformal groups of (M, g) and (M, g = ω 2 g) are, at least, locally isomorphic:

C g ≃ C g ≃ C [g]
, where [g] stands for the equivalence class g ∼ ω 2 g. For the purpose of this article, since conformally flat spacetimes are (locally) Weyl related to the Minkowski spacetime, we will exploit the minkowskian conformal group. The latter is obtained by completing the Poincaré group by dilations and special conformal transformations which, in the usual rectangular coordinates, respectively read:

x µ → λx µ , λ ∈ R + * , (11)
x µ → x µ + b µ x 2 1 + 2b • x + b 2 x 2 . ( 12
)
These two transformations fulfill equation [START_REF] Binegar | de Sitter QED[END_REF] with: f D (x) = 1, f Kµ (x) = -2x µ , with D and K µ the generators of dilations and special conformal transformations respectively. It is often convenient to write the special conformal transformation [START_REF] Eastwood | A conformally invariant Maxwell gauge[END_REF] as the product I • T b • I, with I the inversion x µ → x µ /x 2 and T b the translation

x µ → x µ + b µ .
Finally, for the sake of completeness, let us recall the commutations relations of the minkowskian conformal group:

[X µν , X ρσ ] = η σ[µ X ν]ρ -η ρ[µ X ν]σ , [P ρ , D] = P ρ , [K ρ , D] = -K ρ , (13)
[P µ , K ν ] = 2(X µν -η µν D), [P ρ , X µν ] = η ρ[µ P ν] , [K ρ , X µν ] = η ρ[µ K ν] , (14)
[X µν , D] = 0, [P µ , P ν ] = 0, [K µ , K ν ] = 0, (15)
with X µν = -X νµ the generators of Lorentz transformations, P µ those of translations and

η µν = diag(+1, -1, • • • , -1).
Then, if one sets:

X dd+1 = D, X µd = cK µ + 1 4c P µ , X µd+1 = cK µ - 1 4c P µ , (16)
with c ∈ R * the whole commutation algebra ( 13)-( 15) is recasted as that of o(2, d):

[X AB , X CD ] = η D[A X B]C -η C[A X B]D , (17)
with A, B,

• • • = 0, 1, • • • , d, d + 1, X AB = -X BA and η AB = diag(η µν , -1, +1).
Then [START_REF] Flato | Conformal covariance of fields equations[END_REF], on (M, g), an equation E(A) = 0 is said to be invariant under C [g] , or conformally invariant, if one can realize the Lie algebra g of C [g] such that:

[E, g](A) = ξE(A), (18)
for some function ξ. That is, a solution of E(A) = 0 is mapped to another solution of E(A) = 0 under the action of the conformal group C

[g] .
Notice that, while being different, invariance w.r.t. Weyl rescalings and invariance w.r.t. to the conformal group are not unrelated. Namely invariance in the former implies invariance in the latter, this is shown by considering the composition:

(ω h ) -1 • h : (M, g) → (M, h * g = (ω h ) 2 g) → (M, g = (ω h ) -2 h * g = g) (19)
resulting in an isometry, with h ∈ C [g] , (ω h ) 2 the scaling of g induced by h which is latter compensated by the Weyl rescaling (ω h ) -1 . The theory being invariant w.r.t. Weyl rescalings and isometries yields the claim. From the point of view of the space of solutions of the wave equation, elements of C [g] map a space of solutions on itself while Weyl rescalings embed a space of solutions into another. In that respect Weyl rescalings might be thought of as intertwining operators for the conformal group C [g] between two spaces of solutions.

III. A GUIDED TOUR FROM RANK 0 TO 2
This section studies fields from rank 0 to 2. Progressing in this manner reveals the properties of (1) one after the other and hints at what is to be expected for an arbitrary rank s ≥ 2 on CFES. Moreover, it exposits in their simplest form the way the calculations will be carried in IV and V.

A. rank 0: the scalar field A textbook [START_REF] Birrell | Quantum Fields in Curved Space[END_REF] conformally invariant equation is the scalar massless conformally coupled equation:

- 1 4 d -2 d -1 R ϕ = 0. ( 20
)
This equation might also be referred to as the conformal laplacian or the Yamabe operator. The above equation is Weyl invariant and, when realized on Minkowski spacetime, has its space of solutions left invariant under the action of SO 0 (2, d) once one has taken into account the scaling of the field:

[T g ϕ](x) = α(g, g -1 .x) h ϕ(g -1 .x), ∀g ∈ SO 0 (2, d), (21)
with h(ϕ) = 1 -d/2 the conformal weight of the scalar field. The multiplier α, appearing in [START_REF] Allen | The massless minimally coupled scalar field in de Sitter space[END_REF], is defined through

dg.s 2 = (α(g, x)) 2 ds 2 , (22)
with ds 2 the squared line element. The factor α then fulfills, by construction, the 1-cocycle equation:

α(gg ′ , x) = α(g, g ′ .x)α(g ′ , x), ∀g, g ′ ∈ SO 0 (2, d).
For g an isometry of the underlying spacetime α = 1, from [START_REF] Graham | Conformally invariant power of the laplacian, i: Existence[END_REF], and [START_REF] Allen | The massless minimally coupled scalar field in de Sitter space[END_REF] agrees with the natural action on a scalar field.

B. rank 1: the vector field

For the vector field A µ , the rank 1 field, one can find that the equation:

A µ - 4 d ∇ µ ∇ • A - 2 d -2 R µ ν A ν - 1 4 d(d -4) (d -1)(d -2) RA µ = 0 ( 23
)
is invariant under Weyl rescalings, with A µ = ω -d/2 A µ , and that its space of solutions is left invariant under the action of SO 0 (2, d). Notice that (23) might be rewritten in the following form:

A µ - 4 d ∇ ν ∇ µ A ν + 2 d d -4 d -2 R µ ν A ν - 1 4 d(d -4) (d -1)(d -2)
RA µ = 0 [START_REF] Gover | Conformally invariant powers of the laplacian -A complete nonexistence theorem[END_REF] to make it obvious that at d = 4 it reduces to the free Maxwell's equations on the vector potential A.

For d = 4 equation ( 23) admits A µ = ∇ µ ϕ as gauge solutions with the scalar ϕ unconstrained. A solution A µ of (23) will be said to be determined up to (the gradient of) a scalar ϕ:

A µ → ϕ A µ = A µ + ∇ µ ϕ. (25)
The gauge freedom [START_REF] Graham | Conformal Powers of the Laplacian via Stereographic Projection[END_REF] allowed by (1) also means that A, the vector potential, is not a physically observable field. The gauge invariant quantities, the electric and magnetic fields, are components of the field strength F given by:

(F (A)) α µ = ∇ α A µ -∇ µ A α , (26)
which is indeed gauge invariant since for A µ = ∇ µ ϕ one has:

(F (A)) α µ = ∇ α ∇ µ ϕ -∇ µ ∇ α ϕ = ∇ α , ∇ µ ϕ = 0. ( 27
)
That being said, the vector potential A remains relevant since it is the U (1) gauge field through which the interaction is introduced in the lagrangian. In addition, to simplify as much as possible scattering computations in curved spacetime, a simple expression of its two-point functions would be welcome. As it has already been shown on flat and (A)dS spacetimes [START_REF] Binegar | Conformal QED[END_REF][START_REF] Binegar | de Sitter QED[END_REF][START_REF] Faci | Conformally covariant quantization of Maxwell field in de Sitter space[END_REF] keeping the conformal covariance by using a conformally invariant gauge fixing condition while quantizing A leads to simpler, more compact, results. This is the point of view embraced in this article.

An interesting property of equation ( 23), in arbitrary dimensions, is that it provides a conformally invariant gauge fixing equation: taking its divergence and using the usual commutation relations of covariant derivatives leads to:

d -4 d ∇ • A + d d -2 ∇ µ R µν A ν - 1 4 d d -1 ∇ µ RA µ = 0. ( 28
)
In the above, for d = 4, the equation within the brackets is automatically fulfilled on the space of solutions of ( 23), then is conformally invariant on this space. For d = 4 the divergence of ( 23) vanishes and ∇ • A is unconstrained. However, one would strongly suspect

G(A) = ∇ • A + 2∇ µ R µν A ν - 2 3 ∇ µ RA µ = 0 (29)
to be conformally invariant on the space of solutions of [START_REF] Graham | Conformally invariant power of the laplacian, ii: Nonexistence[END_REF]. It is indeed the case since:

G(A) = ω -4 G(A) + 2ω -5 ( A µ -∇ µ ∇ • A -R µ ν A ν )(∇ µ ω) (30)
where the equation ( 23) appears, with d = 4, in the right hand side. The equation ( 29) is the Eastwood-Singer gauge fixing equation [START_REF] Eastwood | A conformally invariant Maxwell gauge[END_REF]. Using [START_REF] Gover | Higher symmetries of the gjms operators[END_REF] indeed restricts the gauge freedom as now the field ϕ has to fulfill the equation

2 + 2∇ µ R µν ∇ ν - 2 3 ∇ µ R∇ µ ϕ = 0. (31)
Equation ( 31) is known as the d = 4 Paneitz operator [START_REF] Paneitz | A quartic conformally covariant differential operator for arbitrary pseudo-riemannian manifolds[END_REF] (a summary might be found in [START_REF] Paneitz | A quartic conformally covariant differential operator for arbitrary pseudo-riemannian manifolds (summary)[END_REF]) which is a fourth order conformally invariant operator on scalars of null conformal weight. This operator has, also, been found by other means by Riegert [START_REF] Riegert | A Nonlocal Action for the Trace Anomaly[END_REF] and Fradkin and Tseytlin [START_REF] Fradkin | Conformal Anomaly in Weyl Theory and Anomaly Free Superconformal Theories[END_REF].

C. the rank 2 field

One can try to generalize the previous scheme to the symmetric traceless rank 2 tensor A µν . Then one can check that, say, the equation

A µν - 4 d + 2 (∇ µ ∇ ρ A νρ + ∇ ν ∇ ρ A µρ ) + 8 d(d + 2) g µν ∇ ρ ∇ σ A ρσ - (d 2 -2d + 8) 4d(d -1) RA µν (32)
+ 2 d (R µ ρ A νρ + R ν ρ A µρ ) - 4(d -1) d R µ ρσ ν A ρσ - 4 d g µν R ρσ A ρσ = 0
is Weyl invariant with A µν = A νµ , g µν A µν = 0 and A µν = ω -1-d/2 A µν . Notice that one such equation pops out from time to time for various reasons, see say [START_REF] Ben Achour | Conformally invariant wave equation for a symmetric second rank tensor (spin-2) in a d-dimensional curved background[END_REF] and references therein.

When trying to find gauge solutions to [START_REF] Rainich | The Mathematics of Relativity[END_REF] the computation does not seem to yield anything special. Also, taking the divergence, or twice the divergence, of [START_REF] Rainich | The Mathematics of Relativity[END_REF] does not shed much light. The blurriness appearing in the study of the rank two tensors marks the emergence of the invariant, under Weyl rescalings, Weyl tensor:

C µ ρνσ = R µ ρνσ - 1 d -2 δ µ [ν R σ]ρ -g ρ[ν R µ σ] + 1 (d -1)(d -2) Rδ µ [ν g σ]ρ , (33)
which is the totally traceless part of the Riemann tensor. Thanks to this tensor (32) is then far from being unique as one can add a term such as

+λC µ ρσ ν A ρσ ( 34
)
where λ is unconstrained by the requirement that the resulting equation has to be Weyl invariant.

To avoid such terms in the field equation one can restrict his study to conformally flat spacetimes. This choice, however, does not seem to make things much more simpler [18]. This is the reason why we will narrow the scope of this article to conformally flat Einstein spaces (CFES), for which the Riemann and Ricci tensors are expressed as:

R µνρσ = R d(d -1) (g µρ g νσ -g µσ g νρ ), (35)
R µν = R d g µν , (36)
R = Const. ( 37
)
Examples of CFES are Minkowski and (A)dS spacetimes. Then, using ( 35)-( 37), equation ( 32) reduces to

(E 2 (A)) µν = A µν - 4 d + 2 (∇ µ ∇ ρ A νρ + ∇ ν ∇ ρ A µρ ) + 8 d(d + 2) g µν ∇ ρ ∇ σ A ρσ - (d 2 -2d + 8) 4d(d -1) RA µν = 0. (38)
One can then follow the same steps as in the vectorial case. If one considers the field A written as the derivation of a vector field V such as

A µν = ∇ µ V ν + ∇ ν V µ - 2 d g µν ∇ • V, (39)
then, after a brief computation using ( 35)-( 37), one gets

(E 2 (A)) µν = d -2 d + 2 ∇ µ (E 1 (V )) ν + ∇ ν (E 1 (V )) µ - 2 d g µν ∇ ρ (E 1 (V )) ρ . ( 40
)
Notice the appearance of E 1 in (40). In the same manner, suppose in (39) that V µ = ∇ µ ϕ, then simplifying (40) leads to

(E 2 (A)) µν = (d -2)(d -4) d(d + 2) ∇ µ ∇ ν + ∇ ν ∇ µ - 2 d g µν E 0 (ϕ), (41)
meaning that equation [START_REF] Saa | Local electromagnetic duality and gauge invariance[END_REF] for d = 4 allows the gauge freedom

A µν → ϕ A µν = A µν + ∇ µ ∇ ν - 1 d g µν ϕ ( 42
)
and A is determined up to a scalar. For higher rank fields this conclusion will still hold. A field strength F , which is gauge independent, for the field A can be found in Sec.VI in which the field strength for arbitrary ranks are worked out. Now that the gauge freedom has been shown one can search for a gauge fixing equation similar to the Eastwood-Singer equation. Taking the divergence of (38) and using ( 35)-(37) yields:

∇ µ (E 2 (A)) µν = d -2 d + 2 (E 1 (∇ • A)) ν , (43)
where (∇ • A) ν = ∇ µ A µν . In the same manner, taking the divergence of (38) twice produces:

∇ ν ∇ µ (E 2 (A)) µν = (d -2)(d -4) d(d + 2) E 0 (φ), (44)
where φ = ∇ µ ∇ ν A µν . Those two results then suggest that the set

E 2 (A) = 0, E 0 (φ) = 0, (d = 4) (45)
is conformally invariant, while restricting the gauge freedom allowed by E 2 (A) = 0. The conformal invariance is indeed preserved, see Sec.V, and the scalar field ϕ now has to fulfill

- 1 6 R + 1 3 R ϕ = 0. ( 46
)
IV. WORKING OUT THE PROPERTIES OF (1)

In the previous section the prominent properties of the field equation ( 1) have been exposed: conformal invariance and gauge freedom up to a scalar for s ≥ 1 and d = 4. In addition, the relations (40)-( 44) suggest that, on CFES, equations of various ranks are related one to another through simple tensorial derivations.

This section exhibits that indeed all of these properties are found in the higher rank realization of (1). First, since we are mainly concerned with CFES, we derive two identities which reflect the mapping of a CFES to another CFES. Secondly, it is shown that, under such Weyl rescalings, the equation ( 1) is Weyl invariant. Finally, gauge solutions of (1) are found for d = 4.

A. The restricted Weyl transformation

This work is mostly concerned with CFES and therefore only matter Weyl rescalings mapping a CFES on another (see, say, [START_REF] Brinkmann | Einstein spaces which are mapped conformally on each other[END_REF] in a Riemannian setting). This means that a smaller class of ω's has to be considered. Let us call those restricted Weyl transformations. Without getting into the details of such transformations we can derive two identities which are fulfilled by the ω's and which will soon be needed.

Asking for (4) to map a CFES to another CFES is tantamount to require that the relations ( 35)-( 37) are preserved. First consider ( 36) on (M, g), that is:

R µν = 1 d R g µν , (47)
then plugging [START_REF] Flato | Conformal covariance of fields equations[END_REF] in the left-hand side and (8) in the right-hand side of (47) and finally, since (M, g) is also a CFES, using [START_REF] Henneaux | Duality in linearized gravity[END_REF] to further reduce the equality one obtains the first identity:

∇ µ ∇ ν - 1 d g µν 1 ω = 0. (48)
Equation ( 48) merely preserves the relations ( 35) and ( 36) while the Weyl rescaled curvature scalar R, at this point, is not necessarily constant. This is taken care of using [START_REF] Deser | Duality invariance of all free bosonic and fermionic gauge fields[END_REF] and setting ∇ µ R = 0, since R is also constant one gets:

∇ µ ω = (∇ µ ω) 3ω -1 ( ω) - R d(d -1) -(d -4)ω 3 ∇ α 1 ω ∇ µ ∇ α 1 ω , (49)
with some tweakings for later convenience. Notice that the factor ω I which arises from an inversion fulfill both (48) and (49) thus the SO 0 (2, d) invariance remains implied on flat space by this weaker Weyl invariance.

B. Restricted Weyl invariance of (1)

Performing a Weyl transformation, using ( 5) and ( 8), on (1) yields:

(E s (A)) µ1...µs = ω h-2 (E s (A)) µ1...µs + 2s d ω h-4 ω(ω ;ρ ρ ) -2ω ;ρ ω ;ρ A µ1...µs (50) -2s ω h-4 ω(ω ;(µ1 σ ) -2ω ;(µ1 ω ;σ A µ2...µs)σ + 2s(s -1) d + 2s -4 ω h-4 ω(ω ;ρσ ) -2ω ;ρ ω ;σ g (µ1µ2 A µ3...µs)ρσ ,
with the conformal weight h(A) = 1 -s -d/2. Setting ρ = ω -1 eq.( 50) might also be written as:

(E s (A)) µ1...µs = ρ 2-h (E s (A)) µ1...µs + 2s ρ 2-h ∇ (µ1 ∇ σ - 1 d δ (µ1 σ ρ A µ2...µs)σ (51) - 2s(s -1) d + 2s -4 ρ 2-h (∇ ρ ∇ σ ρ)g (µ1µ2 A µ3...µs)ρσ .
Then, from the tracelessness of A and (48) the remaining terms vanish and yields the (restricted) Weyl invariance of equation ( 1) between two CFES:

E s (A) = ω h-2 E s (A). ( 52
)
Remark that the only property used to prove (52) is that (48) is fulfilled. Latter, for the gauge fixing equation, the constant curvature of (M, g) will come into play through the use of (49).

The conformal invariance on arbitrary spacetimes

Equation ( 1) has been shown to be Weyl invariant under the restricted transformation (48). However, with (50) and the relations ( 6) and ( 7) one would find that the equation

( + c s R)A µ1...µs + a s s ∇ (µ1 ∇ σ A µ2...µs)σ + b s s(s -1) g (µ1µ2 ∇ ρ ∇ σ A µ3...µs)ρσ + d s sR (µ1 σ A µ2...µs)σ + e s s(s -1) R (µ1 ρσ µ2 A µ3...µs)ρσ + f s s(s -1) R ρσ g (µ1µ2 A µ3...µs)ρσ = 0, ( 53
)
where the coefficients a s , b s and c s are given by (2) and

d s = 2 d , e s = - 2 d d -1 s -1 , f s = - 2 d(s -1) d + s -2 d + 2s -4 , (54)
is Weyl Invariant under arbitrary Weyl rescalings. Equation ( 1) is a restriction of (53) to CFES. As noticed in Sec.III C the equation ( 53) is then far from being unique as one can add a term such as

+λs(s -1) C (µ1 ρσ µ2 A µ3...µs)ρσ , (55)
that is, changing in (53) the coefficients according to

c s → c s + λ s(s -1) (d -1)(d -2) , e s → e s + λ, d s → d s -2λ s -1 d -2 , f s → f s + λ d -2 , ( 56
)
while keeping the Weyl invariance of the resulting equation intact.

C. Gauge invariance at d = 4

In Sec.III it has been established that for d = 4 and for tensors of rank 1 and 2 that the solutions of (1) are determined up to a scalar ϕ. This subsection inspects the gauge invariance of (1) for tensors of arbitrary rank and shows explicitly that they, too, remain determined up to a scalar.

First let us, for our purpose, introduce the symmetric traceless gradient (STG), defined as

(STG(f )) µ1...µs = s∇ (µ1 f µ2...µs) - s(s -1) (d + 2s -4) g (µ1µ2 ∇ σ f µ3...µs)σ , (57)
with f ∈ S s-1

•

. In addition, let us commit the abuse of language (STG(ϕ)) µ = ∇ µ ϕ. Secondly, notice that the equation (1) might then be rewritten as:

( + c s R)A + a s STG(∇ • A) = 0. ( 58
)
Now let us consider the field A = STG(f ), f ∈ S s-1

• and s ≥ 2. Since the coefficients (a s , c s ) fulfill the recurrence relations:

a s-1 = a s 1 + a s d + 2s -6 d + 2s -4 , (59)
c s-1 = 1 1 + a s d + 2s -3 d(d -1) + (s -1)(d + s -3) d(d -1) a s + c s , (60)
one gets the following identity:

E s (STG(f )) = (1 + a s ) STG(E s-1 (f )), (61)
the higher rank version of (40) where STG was given by [START_REF] Quéva | A remaining relationship between conformally invariant gauge fixing equation and GJMS operators[END_REF]. Now the identity (61) enables us to look for gauge invariance for a field A obtained from a field g of rank r, r < s, as one would obtain:

E s (STG s-r (g)) = s i=r+1 (1 + a i ) STG s-r (E r (g)).
(62)

Then, from (62), the question of the existence of gauge solutions amounts to look if there is a rank r, r < s, and a dimension d such that 1 + a r = 0. From the values of the a i 's, given in [START_REF] Fefferman | The ambient metric[END_REF], there is only one (physical) solution given by (a 1 , d = 4) corresponding to the gauge freedom up to a scalar:

A → ϕ A = A + STG s (ϕ). ( 63
)
This is this gauge freedom that we would like to restrict while keeping the conformal invariance of the whole system.

V. UNCOVERING AND DISCUSSING THE GAUGE FIXING EQUATION

In the previous section we found that for d = 4 the solutions A of (1) are determined up to a scalar ϕ (63). The purpose of this section is to exhibit a conformally invariant equation which will restrict this gauge freedom. First we will show how that equation can be obtained from (1) in arbitrary dimensions. Then, we prove that fixing the gauge in that manner is indeed conformally invariant. Finally, for this equation to be relevant one has to show that it actually constrains the gauge scalar ϕ to belong to a certain space of solutions. This is the case, as is shown in V C by inspecting the residual gauge freedom left to the scalar ϕ. Finally, in V D, it is shown that those pure gauge solutions themselves are conformally invariant. This is demonstrated by, incidentally, providing a new way to derive Branson's factorization formula of GJMS operators.

A. The derivation of the gauge fixing equation

Let us first recall that for the vector field we found, in III B, that the Eastwood-Singer gauge fixing equation [START_REF] Gover | Higher symmetries of the gjms operators[END_REF] appears when one takes the divergence of the (generally) conformally invariant equation [START_REF] Graham | Conformally invariant power of the laplacian, ii: Nonexistence[END_REF]. Then, in III C, for the rank 2 tensor the presence of gauge invariance wasn't so clear anymore. After restricting ourselves to CFES we found: on the one hand explicit gauge solutions and on the other hand that by taking the divergence of the field equation, now [START_REF] Saa | Local electromagnetic duality and gauge invariance[END_REF], one recovered the lower rank conformally invariant field equations, as seen on ( 43) and (44). Now, for an arbitrary rank s the previous section showed gauge solutions. Here we seek a gauge fixing equation which will constrain those solutions.

Taking the divergence of (1) yields the following relation:

∇ µs (E s (A)) µ1...µs = (1 + a s )(E s-1 (∇ • A)) µ1...µs-1 , (64)
through a direct computation, involving the commutation of covariant derivatives and using the fact that the Riemann tensor is given by ( 35) with R = Const. and that the coefficients (a s , b s , c s ) are solutions of the recurrence relations:

a s-1 = a s + 2b s 1 + a s , ( 65
) b s-1 = b s 1 + a s , (66)
c s-1 = 1 1 + a s d + 2s -3 d(d -1) + (s -1)(d + s -3) d(d -1) a s + c s . ( 67
)
That is, the divergence of A satisfies the equation of a rank s -1 symmetric traceless field. By induction each divergence has to fulfill the equation E i of the appropriate rank and finally:

∇ µ1 . . . ∇ µs (E s (A)) µ1...µs = s i=1 (1 + a i ) E 0 (φ). ( 68
)
Therefore, the behavior of the divergences of A is completely determined by the equation [START_REF] Wünsch | On conformally invariant differential operators[END_REF]. That is if none of the prefactor (1 + a i ), 1 ≤ i ≤ s, vanish. Similarly to IV C for d = 4 the prefactor (1 + a 1 ) vanishes. This is the confirmation of the gauge freedom shown before for which:

φ = ∇ µ1 . . . ∇ µs A µ1...µs (d = 4), (69)
is left free by the equation [START_REF] Wünsch | On conformally invariant differential operators[END_REF]. That is, up to a scalar degree of freedom. Notice that this should not come as a surprise. Indeed, if one sets b s = -a s /(d + 2s -4), to make obvious the symmetric traceless gradient in (1), as in (58), then the recurrence relations (65)-( 67) become ( 59)-(60). There is then a sort of duality, for the set [START_REF] Wünsch | On conformally invariant differential operators[END_REF], between extracting a symmetric traceless gradient, as in (61), and taking a divergence, as in (64).

To conclude, for d = 4, we have shown that the s-fold divergence φ is left free by [START_REF] Wünsch | On conformally invariant differential operators[END_REF]. To correct this we choose the following gauge fixing equation:

E 0 (φ) = - 1 6 R φ = 0, (d = 4). ( 70
)
This gauge fixing equation appears legitimate with respect to conformal invariance as on the one hand the solutions of (1) which fulfill (70) are left invariant by conformal transformations (see hereafter), and on the other hand in arbitrary dimensions d = 4 the corresponding equation is always satisfied by the solutions of (1), cf.(68). For d = 4 this pathology is rectified by enforcing (70) as a gauge fixing equation.

B. Restricted Weyl invariance of the gauge fixing equation

Similarly to the Eastwood-Singer gauge fixing equation for the vector potential (68) hints at a gauge fixing equation likely to be conformally invariant on the space of solutions of E s (A) = 0 between two CFES. In order to prove this property for an arbitrary rank s we will first consider, again, the vectorial case in order to examplify our strategy in its simplest case. Then the case s ≥ 2 will be adressed.

1. The vector field, Eastwood-Singer gauge revisited

Consider the system

A µ -∇ µ ∇ • A - 1 4 RA µ = 0, - 1 6 R ∇ • A = 0, (71)
with d = 4 and (M, g) a CFES. Performing a Weyl rescaling yields:

φ = ∇ • A = ω -2 ∇ • A + 2ω -3 (∇ µ ω)A µ . ( 72
)
which might be fed to the (rescaled) conformal laplacian:

- 1 6 R φ = ω -2 - 1 6 R + 2ω -2 (∇ α ω)∇ α + ω -3 ( ω) φ (73) = ω -4 - 1 6 R φ + 2ω -5 (∇ µ ω) A µ -∇ µ ∇ • A - 1 6 RA µ -ω -5 ( ω)∇ µ A µ + 4ω -5 (∇ α ∇ µ ω)∇ α A µ + 2ω -5 ( ∇ µ ω)A µ + 2ω -6 (∇ α ω)(∇ α ω)∇ µ A µ -8ω -6 (∇ α ω)(∇ µ ω)∇ α A µ -4ω -6 ( ω)(∇ µ ω)A µ -8ω -6 (∇ α ω)(∇ α ∇ µ ω)A µ + 12ω -7 (∇ α ω)(∇ α ω)(∇ µ ω)A µ .
As it stands the result is far from being conformally invariant on the space of solution of (71). Let us then use (48) to express, say, (∇ α ∇ µ ω) in terms of other derivatives of ω, that is:

(∇ α ∇ µ ω) = 2ω -1 (∇ α ω)(∇ µ ω) + 1 4 g αµ [( ω) -2ω -1 (∇ β ω)(∇ β ω)]. (74)
Using the above simplifies greatly (73) as one gets

- 1 6 R φ = ω -4 - 1 6 R φ + 2ω -5 (∇ µ ω) A µ -∇ µ ∇ • A - 1 6 RA µ (75) + 2ω -5 ( ∇ µ ω)A µ -6ω -6 ( ω)(∇ µ ω)A µ .
Now one can use the second identity (eq.( 49)) fulfilled by the scale factor ω of a restricted Weyl transformation, for d = 4 it reads:

( ∇ µ ω) = (∇ µ ω) 3ω -1 ( ω) - 1 12 R . (76)
With this last identity finally one gets:

- 1 6 R φ = ω -4 - 1 6 R φ + 2ω -5 (∇ µ ω) A µ -∇ µ ∇ • A - 1 4 RA µ , (77)
and thus shows the conformal invariance of (71) between two CFES. A result known to be true as this is just the restriction of (30) to our choice of spacetimes.

The general case at s ≥ 2

To prove the invariance of (3) under Weyl rescalings between two conformally flat Einstein spaces one needs the above ingredients (74) and ( 76) and the properties arising from the symmetry and tracelessness of Each formula appearing in the vectorial case has to be generalized. Let us begin with the generalization of (72):

φ = ∇ µ1 . . . ∇ µs ω h A µ1...µs , (78)
in which h(A) = 1 -s -d/2. Let us rewrite eq.( 48) as

∇ µ ∇ ν - 1 d g µν ρ = -ω -2 ∇ µ ∇ ν - 1 d g µν ω = 0, ρ = 1 ω . (79)
Thanks to the tracelessness of A one then realizes that there cannot be derivatives of ω of degree greater or equal to 2 contracted with A since

(∇ µi ∇ µj ω)A µ1...µs = ∇ µi ∇ µj - 1 d g µiµj ω A µ1...µs = 0, (80)
in which one uses the tracelessness of A and then that (79) is fulfilled by ω. This simplifies greatly the expansion of φ as one then gets:

φ = s i=0 Γ(h + 1) Γ(h + 1 -i) s i ω h-i (∇ω) i ∇ s-i A = s i=0 Γ(h + 1) Γ(h + 1 -i) s i ω h-i (∇ω) i ∇ s-i A, (81)
using the notation in which an index contracted with A is not written. For instance a generic term in (81) reads as:

(∇ω) i ∇ s-i A = (∇ µ1 ω) . . . (∇ µi ω)(∇ µi+1 . . . ∇ µs A µ1...µs ). (82)
Indeed, any term in the expansion of φ might be brought into the form (82) since A is fully contracted and symmetric. Now one can express ∇ n A in terms of ∇ and ω. By induction on n one would get

∇ n A = n i=0 Γ(d + 2s -n -1 + i) Γ(d + 2s -n -1) n i (-1) i ρ -i (∇ρ) i ∇ n-i A = n i=0 Γ(d + 2s -n -1 + i) Γ(d + 2s -n -1) n i ω -i (∇ω) i ∇ n-i A, (83)
using the same argument about the derivatives of ρ with (79) and finally ω -1 (∇ω) = -ρ -1 (∇ρ) to recast the result in terms of ω solely. Then plugging (83) in (81) and inverting the order of summation yields:

φ = s i=0 i j=0 i j Γ(h + 1) Γ(h + 1 -j) Γ(d + s -1 + i) Γ(d + s -1 + j) s i ω h-j (∇ω) i ∇ s-i A (84) = s i=0 Γ(d + s -1 + i) Γ(d + s -1) 2 F 1 (-i, -h; d + s -1; 1) s i ω h-j (∇ω) i ∇ s-i A (85) = s i=0 s! s -i! (i + 1) ω -1-s-i (∇ω) i ∇ s-i A. (86)
In which, from (84) to (86), h has been set to its value and d = 4. Replacing φ by (86) and applying the conformal laplacian to it yields:

- 1 6 R φ = s i=0 s! s -i! (i + 1) ω -s-i-3 (∇ω) i - 1 6 R + i(∇ω) i-1 ( ∇ω) (87) + i(i -1)(∇ω) i-2 (∇ α ∇ω)(∇ α ∇ω) + 2i(∇ω) i-1 (∇ α ∇ω)∇ α -(s + i)ω -s-i-4 (∇ω) i ( ω) + 2(∇ω) i (∇ α ω)∇ α + 2i(∇ω) i-1 (∇ α ω)(∇ α ∇ω) + (s + i)(s + i + 1)ω -s-i-5 (∇ω) i (∇ α ω)(∇ α ω) ∇ s-i A.
Using (74) simplifies the above equation to:

- 1 6 R φ = s i=0 s! s -i! (i + 1)ω -s-i-3 (∇ω) i - 1 6 R ∇ s-i A (88) - s i=1 s! s -i! 2i ω -s-i-3 (∇ω) i-1 (∇ α ω)∇∇ α ∇ s-i A + s i=2 s! s -i! (i -1) ω -s-i-3 (∇ω) i-2 (∇ α ω)(∇ α ω)∇ 2 ∇ s-i A - s-1 i=1 s! s -i! i(i + 1) ω -s-i-4 ( ω)(∇ω) i ∇ s-i A -3s s!ω -2s-4 ( ω)(∇ω) s A + s i=1 s! s -i! i(i + 1) ω -s-i-3 (∇ω) i-1 ( ∇ω)∇ s-i A.
In the latter one can recognize, abusing a bit the notation, the contraction of (∇ω) i with STG ∇ ∇ s-i A in the second and third line. Substituting (76) in the sixth line cancels the fourth and fifth line and changes the coupling to the curvature to: 1/6 + i/12 = (i + 2)/12. Hence, one gets:

- 1 6 R φ = s i=0 s! s -i! (i + 1)ω -s-i-3 (∇ω) i - 2 i + 1 STG ∇ - 2 + i 12 R ∇ s-i A = s i=0 s! s -i! (i + 1)ω -s-i-3 (∇ω) i E i (∇ s-i A), (d = 4). ( 89
)
On the space of solutions of E s (A) = 0, according to (62), each term with 1 ≤ i ≤ s vanishes thus leaving:

- 1 6 R φ = ω -s-3 - 1 6 R φ. (90)
The identity (90) concludes the proof of the invariance of the gauge fixed set (3) with respect to Weyl rescalings between two CFES.

C. The residual gauge freedom

Equation (70) provides a conformally invariant gauge fixing equation of [START_REF] Wünsch | On conformally invariant differential operators[END_REF]. Here it is shown that ϕ in (63) is no longer arbitrary and its remaining gauge freedom allowed by (3) is found.

To begin let us consider a pure gauge field:

A = STG s (ϕ). (91)
Then, plugging (91) into (70) yields:

- 1 6 R ∇ s STG s (ϕ) = - 1 6 R ∇ s-1 (∇ • STG)(STG s-1 (ϕ)) (92) = - 1 6 R ∇ s-1 U s-1 (STG s-1 (ϕ)) (93) = α s - 1 6 R U s-1′ (∇ s-1 STG s-1 (ϕ)) (94) with U s (A) = A + d + 2s -4 d + 2s -2 STG(∇ • A) + s(d + s -2) d(d -1) RA (95)
for A ∈ S s • and for φ a scalar field:

U s′ (φ) = + s(d -1 + s) d(d -1) R φ, (96)
and α s a non-vanishing numerical factor (namely

α s = (1 + s)(d + s -2)(d + 2s -2) -1
). Going from (93) to (94), that is obtaining (96) from (95), is performed in the same vein as the computation already carried in V A, for which the commutation relations between divergences and a second order equation akin to (95) were obtained. Then, carrying this scheme in (92) till the end with (96) and minding that d = 4 one gets that the scalar gauge field ϕ fulfills:

- 1 6 R + 1 3 R × • • • × + (s -1)(s + 2) 12 R ϕ = 0. ( 97
)
Therefore in the gauge transformation (63) the scalar field ϕ is no longer arbitrary since it has now to fulfill the above equation.

Remark on the residual gauge freedom on de Sitter

Notice that, if the underlying spacetime is the de Sitter spacetime, for which R = -d(d-1)H 2 with H 2 the constant Hubble radius, in the scalar representation of the de Sitter group SO 0 (1, d) ⊂ SO 0 (2, d) the first Casimir operator reads as C 1 (SO 0 (1, d)) = -H -2 then equation ( 96) is recast as

[C 1 (SO 0 (1, d)) + j(d -1 + j)]φ = 0, j ∈ N. ( 98
)
This is precisely the equation fulfilled by a scalar field in the discrete series of SO 0 (1, d). As a consequence, if the gauge scalar ϕ transforms covariantly under the de Sitter group it decomposes as

ϕ = ϕ cc + s-1 j=0 ϕ ds(j) (99)
in which ϕ cc stands for the massless conformally coupled field, which lies in the complementary series of SO 0 (1, d), and ϕ ds(j) is the j'th term in the discrete series of SO 0 (1, d). The 0'th term of the discrete series is the, so-called, massless minimally coupled field. Those fields are known to be problematic [START_REF] Allen | Vacuum States in de Sitter Space[END_REF][START_REF] Allen | The massless minimally coupled scalar field in de Sitter space[END_REF] however their difficulties are not conveyed to the field A as they are filtered by the symmetric traceless gradient.

D. The residual gauge freedom and its relation with GJMS operators

Let us write the equation fulfilled by ϕ, once the gauge fixing equation has been imposed, as:

P 2n ϕ = n ℓ=1 + (ℓ + 1)(ℓ -2) 12 R ϕ = 0, n = p + 1, d = 4. ( 100
)
This equation is known as Branson's factorization formula of GJMS operators and is conformally invariant. Let us recall that Graham-Jenne-Mason-Sparling (GJMS) [START_REF] Graham | Conformally invariant power of the laplacian, i: Existence[END_REF][START_REF] Graham | Conformally invariant power of the laplacian, ii: Nonexistence[END_REF][START_REF] Gover | Conformally invariant powers of the laplacian -A complete nonexistence theorem[END_REF] results come from the question whether or not there is a curved analog of the conformally invariant (SO 0 (2, d)) flat operator n , n ∈ N. That is, does there exists an operator P 2n from densities of weight n -d/2 to densities of weight-n -d/2 on a conformal manifold of dimension d ≥ 3 whose leading symbol is n ? Their result is the following. For d odd there exists one such operator. For d even P 2n exists provided that the bounds 1 ≤ n ≤ d/2 are satisfied. In that respect for d = 4 the Paneitz operator ( 31) is the critical GJMS operator. Later Branson [START_REF] Branson | Sharp inequalities, the functional determinant, and the complementary series[END_REF][START_REF] Graham | Conformal Powers of the Laplacian via Stereographic Projection[END_REF], through an harmonic analysis argument, proved that on S d equipped with its standard metric that P 2n reduces to:

P 2n ϕ = n ℓ=1 + (2ℓ -2 + d)(2ℓ -d) 4d(d -1) R ϕ. ( 101
)
Notice that on conformally flat Einstein spaces, since the obstruction tensor O µν vanishes and the ambient metric can be recovered at arbitrary order [START_REF] Graham | AdS/CFT correspondence: Einstein metrics and their conformal boundaries[END_REF], P 2n exists to arbitrary order (however, for n > d/2, d even, these operators are no longer natural conformally invariant differential operators [START_REF] Gover | Laplacian operators and Q-curvature on conformally Einstein manifolds[END_REF]). Branson's factorization formula has been extended to Einstein metrics in [START_REF] Fefferman | The ambient metric[END_REF][START_REF] Gover | Laplacian operators and Q-curvature on conformally Einstein manifolds[END_REF]. For a study of the (higher) symmetries of such operators see [START_REF] Eastwood | Higher symmetries of the Laplacian[END_REF][START_REF] Eastwood | Higher symmetries of the square of the Laplacian[END_REF][START_REF] Gover | Higher symmetries of the gjms operators[END_REF].

In this subsection, we supply a proof of the conformal invariance of (100) by mimicking the calculus of the conformal invariance of [START_REF] Erdmenger | Conformally covariant differential operators: Symmetric tensor fields[END_REF] through that of [START_REF] Gover | Higher symmetries of the gjms operators[END_REF]. That is, this computation relies completely on the fact that for d = 4 both conformal invariance and gauge invariance are present. A useful identity, for our analysis, is produced in the process.

To exemplify our scheme let us, again, consider the vectorial case. The whole idea is that the gauge fixing condition ( 29) is conformally invariant on the space of solutions of ( 24) (d = 4). Then, if one plugs in (29) a pure gauge solution the "on the space of solutions of (24)" is already taken care of and one is left to look if there is a Weyl rescaling of the scalar field resulting only in the appropriate rescaling of the pure gauge vector. That is, does there exist w ∈ R such that the equation

A µ = ∇ µ ω w ϕ = ω 0 ∇ µ ϕ = A µ , (102)
is fulfilled? The answer is, obviously, positive with w = 0. Extended to our case the question now is, does there exist

w ∈ R such that STG s (ω w ϕ) = ω h STG s (ϕ) (103) with h(A) = 1 -s -d/2?
First let us study the case with s = 2. With no assumption on ω one obtains:

(STG 2 (ϕ)) µν = ω w-4 (STG 2 (ϕ)) µν + 2(w -1) ω w-5 (∇ µ ω)∇ ν + (∇ ν ω)∇ µ - 2 d g µν (∇ α ω)∇ α ϕ (104) + 2w(w -1) ω w-6 (∇ µ ω)(∇ ν ω) - 1 d g µν (∇ α ω)(∇ α ω) ϕ -2w ω w-3 ∇ µ ∇ ν - 1 d 1 ω ϕ.
If one sets w = 1 the right hand side of ( 104) is greatly simplified but, still, does not produce the desired result. Then, one can notice that under the restricted Weyl transformation (48) that the remaining term vanishes. This is the point of view that we will adopt here, but it also hints at the gauge invariance:

A µν → ϕ A µν = A µν + ∇ µ ∇ ν - 1 d g µν + 1 d -2 R µν - 1 d g µν R ϕ (105)
of ( 53) on (more) generic spacetimes. Setting ρ = ω -1 , v = -w and noticing the identity:

STG(f ) = ρ 2 STG(f ), ∀f ∈ S s • , (106)
simplifies the right hand side of (103) which might then be expressed as:

STG s (ω w ϕ) = STG s (ρ v ϕ) = ρ 2 STG(ρ 2 STG(ρ 2 STG(. . . (ρ v ϕ) . . . ))) (s times). (107)
To get a tractable formula let us also use the following notation:

× : S s1 • × S s2 • → S s1+s2 • (108) f, g → (s 1 + s 2 )! s 1 !s 2 ! (f • g) ST (109) in which | ST is the projector (| ST | ST =| ST ) onto S s1+s2
• . On scalars it reduces to the pointwise product of two functions. Then, thanks to this definition, accommodated to that of the symmetric traceless gradient (STG) given in (57), one has a Leibniz identity:

STG(f × g) = STG(f ) × g + f × STG(g). (110)
Then, beginning with the identity ρ v ϕ = ρ v × ϕ, using (48) to discard any term such as STG i (ρ) with i ≥ 2 and an induction on the degree yields:

STG s (ρ v ϕ) = s i=0 Γ(v + s) Γ(v + s -i) s i ρ v+2s-i (STG(ρ)) × i × STG s-i (ϕ). (111)
Finally, setting w = s -1, that is v = 1 -s, to cancel the terms with i ≥ 1, and restoring the ω's provides the identity:

STG s (ω s-1 ϕ) = ω -s-1 STG s (ϕ). (112)
However, for the equation (100) to be conformally invariant the scaling factor of the resulting field in (112) has to agree with that of A, namely:

h(A) = 1 -s - d 2 = -s -1, (113)
which is achieved only for d = 4. The SO 0 (2, d) invariance of (100) is inherited from that of n on Minkowski spacetime, a well known result [START_REF] Jakobsen | Wave and Dirac operators, and representations of the conformal group[END_REF].

Notice that the identity (112) has an important physical interpretation: a pure gauge solution on (M, g) is mapped on a pure gauge solution on (M, g) and similarly under SO [START_REF] Fefferman | The ambient metric[END_REF][START_REF] Gover | Laplacian operators and Q-curvature on conformally Einstein manifolds[END_REF]. This gives an invariant subspace of solutions of (1).

VI. THE FIELD STRENGTH F

It has been shown in Sec.IV C that the equation (1) admits gauge solutions, through the gauge transformation (63). Now one has to provide gauge invariant observables similar to the Faraday tensor ( 26) for the 4-potential leading to the electromagnetic fields.

To tackle this problem we proceed in complete analogy with the electromagnetic case (s = 1) and one notes that indeed almost everything carries through the s ≥ 2 case. First the equivalent of Maxwell's equations, written as a divergence, provides a candidate F for the field strength, its symmetries are also registered. It is then showed that F vanishes on gauge solutions, on CFES, hence is a field strength. The equations on A are then translated into a system of two first-order conformally invariant equations on F , one of which vanishes for F derived from a potential and (M, g) being conformally flat. Then, as the Faraday tensor decomposes in electric and magnetic fields E-B, it is explicitly shown that F decomposes in a set of 3-dimensional traceless symmetric tensors. We then show, on flat space, how the conformally invariant system of equations is written on those E-B fields and prove their duality. Finally a scalar product on the space of solutions of (1) is given.

A. Reconstructing the field strength F

To produce a field strength let us consider, instead of eq.( 1), the equation:

A - (d + 2s -4) s(d + s -3) STG(∇ • A) - (d + s -2) d(d -1) RA = 0, (114)
which, in arbitrary dimension, has the gauge invariance (63). For s = 1 (114) is the restriction of Maxwell equation ∇ µ (∇ µ A ν -∇ ν A µ ) = 0 to a CFES. For d = 4 (1) and (114) are the same. Under the assumption that ( 35)- [START_REF] Deser | Duality invariance of all free bosonic and fermionic gauge fields[END_REF] are fulfilled (114) can be recast as

∇ α (F (A)) α µ1...µs = 0, ( 115
)
where F is built upon A by: (F (A)) α µ1...µs = (DA) α µ1...µs (116)

= ∇ α A µ1...µs -∇ (µ1 A µ2...µs)α - (s -1) (d + s -3) g α(µ1 ∇ σ A µ2...µs)σ -g (µ1µ2 ∇ σ A µ3...µs)ασ , (117)
found already in [START_REF] Erdmenger | Conformally covariant differential operators: Symmetric tensor fields[END_REF] and from which we borrow the notation D. For s = 1 one has F α µ = ∇ α A µ -∇ µ A α . We will show, on CFES, that F is a field strength as it is independent of the gauge scalar field ϕ.

Notice that on a generic spacetime, without the assumption that ( 35)-( 37) are fulfilled, the equation resulting from (115) is still for d = 4 a conformally invariant equation as one, schematically, gets:

∇ α (F (A)) α µ1...µs = (53) + (55), λ = s + 2 s(s -1) , d = 4. (118)
From (116) one can workout the identities fulfilled by the field strength, namely:

F (α µ1...µs) = 0, g µiµj F α µ1...µs = 0, g αµi F α µ1...µs = 0, (119)
thanks to which one can find that F possesses

dof(F ) = (d + s -4)! (d -3)!(s + 1)! s(d + 2s -2)(d + s -2) (120)
independent components. Such as the Faraday tensor those relations could serve as the defining properties of F and under the appropriate assumptions eventually there would exist a potential A such that F is given by (116). Later, in VI C, it is shown that the equation ∇ α F α µ1...µs = 0 can be completed by another equation of motion which vanishes as soon as there exists a potential A (similar to Maxwell's equations without charges) and g is conformally flat. Finally one can notice that the relations (119) are not independent as one clearly gets the third from the other two.

B. The field strength F is a field strength on CFES Now we will prove that F is indeed a field strength: a gauge invariant quantity. Let (M, g) be a CFES and consider the field A = STG(f ) with f a symmetric traceless field of rank s -1. Let us also make the extra assumption that the field f satisfies the equation (114) written at the rank s -1. Then, after some algebra in which the strategy is that as soon as a laplacian hits f to replace it by the remaining of (114), one gets the following identity:

(F (STG(f ))) α µ1...µs = (s -1) ∇ (µ1 (F (f )) |α| µ2...µs) -s -

1 d + 2s -4 g (µ1µ2 ∇ σ (F (f )) |α| µ3...µs) (121) = s -1 s (STG(F (f ))) |α| µ1...µs , (122)
in which the |α| means that this indice is not involved in the STG process. Suppressing the indices one then has

F (STG(f )) = s -1 s STG(F (f )). ( 123
)
This scheme can be pushed further. Let g ∈ S r • , r < s, and in addition let g be a solution of (114) written at the rank r. Then, STG n (g) is a solution of (114) written at the rank r + n, using (62) adapted to (114). Then, from (123) one gets:

F (STG s-r (g)) = r s STG s-r (F (g)). (124)
Finally, one can consider a scalar field ϕ and a pure gauge field obtained from that scalar A = STG s (ϕ). Since (114) has solutions determined up to a scalar STG n (ϕ) is a solution whatever the rank is and the identity (124) reads as

F (STG s (ϕ)) = 1 s STG s-1 (F (STG(ϕ)) = 0, (125)
in which the last equality is nothing but [START_REF] Eastwood | Higher symmetries of the Laplacian[END_REF]. We thus showed that F is indeed a field strength as it vanishes on gauge solutions. The gauge invariance of DA was already noted in flat space in [START_REF] Erdmenger | Conformally covariant differential operators: Symmetric tensor fields[END_REF], for a generalization to more generic (w.r.t. CFES) curved background one would have to have a better understanding of the gauges solutions.

C. The equations on F and their conformal invariance

Presently we have found that, at least on CFES, one can find a field strength F and that the equation on the potential A might be rewritten as a first order equation on F (see (115)). From Maxwell's equations point of view we do know that then there lack a second of equation (the sourceless one) to get a well posed set of PDEs on F . Here we show that one can find one such set of equations and that all the properties found for Maxwell's equations remain as soon as one restricts oneself to conformally flat spacetimes.

Let us assume that F fulfills (119), with no further assumptions, and let us define

(DF ) αβ µ1...µs = ∇ [α F β] µ1...µs - s s + 1 ∇ (µ1 F |[β α]|µ2...µs) - s (d + s -4)(s + 1) s g (µ1|[α ∇ σ F β]| µ2...µs)σ (126) -(s -1)g (µ1µ2 ∇ σ F |[β α]|µ3...µs)σ + g (µ1|[α ∇ σ F |σ| β]|µ2...µs) ,
in which the notation D seems natural if one compares the coefficients in (126) and those of (116). Notice that for s = 1 one gets

(DF ) αβ µ = ∇ α F β µ -∇ β F α µ - 1 2 ∇ µ F [β α] = ∇ α F β µ + ∇ β F µ α + ∇ µ F α β , (127)
Then, if F happens to derive from a potential A as in (116) one would get (DF ) αβ µ1...µs = (D 2 A) αβ µ1...µs (128)

= (1 -s)[C αβ σ (µ1 A µ2...µs)σ + C σ (µ1µ2|[α A β]|µ3...µs)σ ] + (1 -s)(2 -s) d + s -4 C ρ ν1ν2 σ δ [α ρ g β](µ1 δ µ2 σ δ µ3 ν3 + δ [α ν3 g β](µ1 δ µ2 ρ δ µ3 σ + δ [α ν3 δ β] σ δ (µ1 ρ g µ2µ3 A µ4...µs)ν1ν2ν3
and the right hand side of (128) vanishes when either s = 1 (thanks to Bianchi identity, already used here) or if (M, g) is conformally flat for which one gets:

D 2 A = 0, (M, g) ≡ conformally flat. ( 129
)
One then is led to consider the system (DF ) αβ µ1...µs = 0, ∇ α F α µ1...µs = 0, (M, g) ≡ conformally flat.

In addition one shows, taking into account the symmetries (119), that if, and only if, d = 4 the system (130) is conformally invariant with then F = ω -3-s F . If F derives from a potential A this conformal invariance agrees with that of F with (F (A)) α µ1...µs = ω -3-s (F (A)) α µ1...µs , A µ1...µs = ω -1-s A µ1...µs , d = 4.

Finally, still for d = 4, instead of (130) one would rather consider the system ( * DF ) µ1...µs = 0, ∇ α F α µ1...µs = 0, (M, g) ≡ conformally flat, d = 4, (132) in which we set ( * DF ) µ1...µs = s ε (µ1 αβσ (DF ) |αβ| µ2...µs)σ , with ε the totally antisymmetric tensor.

To obtain a wave operator on the field strength F one could consider the system ∇ β (DF ) αβ µ1...µs = 0, ∇ α F α µ1...µs = 0.

Let us simply remark that this system is no longer conformally invariant.

D. An E-B decomposition of F Notice, from (120), that for d = 4 the following formula occurs:

dof(F ) d=4 = 2s(s + 2) = 2 s j=1 (2j + 1). (134)
That is, F possesses the appropriate number of independent components to be written as the sum over 2s symmetric traceless tensors with respect to SO(3). This subsection investigates one such decomposition and its consequences. First we propose a definition of such fields and immediately show that, indeed, F is entirely expressible in terms of those fields. Then we write how the set (132) is translated on those fields. A task greatly simplified once one has shown that (132) have a duality property. Similarly we register how the minkowskian conformal group SO(2, 4) acts upon those fields. Every computation is performed on flat space from now on. The results which are obtained here can be lifted to conformally flat spacetimes through (131).

Definition of the E-B fields

On flat space let us define

E i1...ij j = M i1...ij j -traces, B i1...ij j = 1 s + 1 N i1...ij j -traces, (135)
with 1 ≤ j ≤ s, the traces are with respect to the 3 dimensional metric δ ij = δ ij = -η ij and we defined

M i1...ij j = F 0 0...0i1...ij , N i1...ij j = jε (i1|bc F b c0...0|i2...ij ) . (136)
One can decompose F in the (analogues of) E and B if an arbitrary component can, unambiguously, be written in terms of such 3-dimensional fields, that is if one can invert (135)-(136).

The inversion of (135)-(136)

a. The M j -N j 's in terms of the E j -B j 's. From the properties (119) fulfilled by F and the definition (136) one finds

δ iai b M i1...ij j = M i1... ia... i b ...ij j-2 , δ iai b N i1...ij j = N i1... ia... i b ...ij j-2 , (137)
in which a hat over an index means that said index is omitted. From (135) and (137) one can invert the M j 's in terms of the E j 's since

E i1...ij j = ⌊j/2⌋ k=0 α jk δ (i1i2 . . . δ i 2k-1 i 2k M i 2k+1 ...ij ) j-2k , (138)
in which the coefficients α jk are determined by asking that (138) is traceless, which is well posed thanks to (137) once one has chosen an α j0 (here α j0 = 1 for E and M and α j0 = 1/s + 1 for B and N ). Then, (138) might, rightly, be viewed as an upper triangular transformation between the E j 's and the M j 's and as such can be inverted with:

M i1...ij j = E i1...ij j -α j1 δ (i1i2 E i3...ij ) j-2 -(α j2 -α j1 α j-2 1 )δ (i1i2 δ i3i4 E i5...ij ) j-4 -. . . (139) = ⌊j/2⌋ k=0 β jk δ (i1i2 . . . δ i 2k-1 i 2k E i 2k+1 ...ij ) j-2k . (140)
The same arguments apply to invert the B j 's in terms of the N j 's. With a bit of work one could find that:

α jk = (-1) k j! j -2k! 2j -1 -2k!! 2k -1!! 2j -1!! = (-1) k j! j! 2j -2k! 2k! 2j! j -k! j -2k! k! , (141)
β jk = j! j -2k! 2j -4k + 1!! 2k -1!! 2j -2k + 1!! = j! 2j -4k + 2! 2k! j -k + 1! k! j -2k! j -2k + 1! 2j -2k + 2! , (142)
for k ≥ 1 and α j0 = β j0 = 1 otherwise. In the above we used the identity on bifactorial of odd numbers: 2n -1!! = 2n!/2 n n!. b. F in terms of the M j -N j 's. We are now concerned with the second step of the inversion, that is to write an arbitrary component of F in terms of the M j 's and the N j 's. From the definition (136) an arbitrary F 0 0...0i1...ij might already be written in terms of the M j 's. What is left to show is that F l 0...0i1...ij might also be written in terms of the M j 's and the N j 's.

From (119) one can write the following (sub-)identity:

F l i1...ij 0...0 = j j + 1 F [l (i1]i2...ij )0...0 - s -j j + 1 F 0 i1...ij l0...0 , (143)
in which the second term of the r.h.s., from the above remark, is already inverted in terms of the M j 's. Then, it suffices to show that one can express the first term of the r.h.s. of (143) unambiguously in terms of the M j 's and the N j 's.

Out of the definition (136) of the N j 's, of the properties fulfilled by the antisymmetric tensor ε ijk and those of F given in (119) one can establish the following formula:

ε kl(i1 N i2...ij )k j = jF [l (i1]i2...ij )0...0 -(s + 1)δ l(i1 F |0| i2...ij )0...0 (144) -(j -2)δ (i1i2 F [|l| i3]i4...ij )0...0 + (s + 1)δ (i1i2 F |0| i3...ij )l0...0
put in a more suitable form, for our purpose, as:

jF [l (i1]i2...ij )0...0 = (j -2)δ (i1i2 F [|l| i3]i4...ij )0...0 + ε kl(i1 N i2...ij )k j + (s + 1)(δ l(i1 M i2...ij ) j-1 -δ (i1i2 M i3...ij )l j-1
).

(145)

By constantly using the above formula one can invert F [l (i1]... in terms of the M j 's and the N j 's terminating with either

F [l i1]0...0 = ε kli1 N k 1 (146)
or

F [l i1]i20...0 + F [l i2]i10...0 = 1 2 (ε kli1 N i2k 2 + ε kli2 N i1k 2 ) + 1 2 (s + 1)(δ li1 M i2 1 + δ li2 M i1 1 -2δ i1i2 M l 1 ), (147)
depending on the parity of j. This means that the first term in (143) can unambiguously be written in terms of the N j 's and the M j 's. From (136) it is also the case for the second term in (143). Hence F l 0...0i1...ij has a well posed decomposition over the M j 's and the N j 's.

c. F in terms of the E j -B j 's. Finally, F can be written in terms of the M j 's and the N j 's, which themselves can be written in terms of the E j 's and the B j 's. Hence, an arbitrary component of F might be written uniquely in terms of E j and B j that is:

F = s j=1 (E j ⊕ B j ).
(148)

The field equation written on Ej-Bj

We now would like to see how the system (133) reads on the E j -B j fields. First, instead of (133) let us write the set under consideration as ∂ α F α µ1...µs = 0, ∂ α F α µ1...µs = 0, on (R 4 , η),

in which we set * DF ∝ ∂ F . Writing the system in this manner simplifies our study as in the definition of F very few terms survive, namely: F α µ1...µs = s s + 1 ε α(µ1 βσ F |β| µ2...µs)σ .

A careful inspection of F shows that it fulfills all the properties of (119) and as such might be decomposed over fields E j and B j according to (135)-(136). Performing one such decomposition, while knowing the E j 's and B j 's of F , one recognize that:

E j = -B j , B j = E j , (151)
and then F = -F , showing that eq.( 149) remains invariant under the electric-magnetic duality (E j , B j ) → (-B j , E j ).

(152)

This duality can be extended to infinitesimal rotations as:

δE j = -θB j , δB j = θE j , |θ| ≪ 1, (153)
or integrated to the finite rotation:

θ E j = cos θE j -sin θB j , θ B j = sin θE j + cos θB j ,

as a symmetry of the system (149) and generalize to the known duality rotation of electromagnetism [START_REF] Rainich | The Mathematics of Relativity[END_REF][START_REF] Misner | Classical physics as geometry: Gravitation, electromagnetism, unquantized charge, and mass as properties of curved empty space[END_REF]. An interesting question would be to find whether this applies also for the (on shell) action as it the case for electromagnetism [START_REF] Deser | Duality Transformations of Abelian and Nonabelian Gauge Fields[END_REF]? Notice however as showed recently [START_REF] Deser | No local Maxwell duality invariance[END_REF], following [START_REF] Henneaux | Duality in linearized gravity[END_REF][START_REF] Deser | Duality invariance of all free bosonic and fermionic gauge fields[END_REF], that one such symmetry cannot be made local, at least not without sacrificing the gauge invariance [START_REF] Saa | Local electromagnetic duality and gauge invariance[END_REF]. It might be worth to stress that the duality (152), and then rotation (154), has to be applied to all fields (E j , B j ) for all j ≤ s.

Then, if one introduces the following notations:

DIV 3 E j ≡ ∂ ij E i1...ij j , STG 3 E j-1 ≡ jδ k(i1 ∂ k E i2...ij ) j-1 -traces, STC 3 E j ≡ ε kl(i1 ∂ k E i2...ij )l j , (155)
writing H j = E j + iB j , with i 2 = -1, and thanks to the duality (152) the system (149) now reads as

∂ t H j -i s + 1 j + 1 STC 3 H j - s -j j + 1 DIV 3 H j+1 + 1 j 1 + 2(s -j) (j + 1)(j + 2) STG 3 H j-1 = 0, (156)
with H j = 0 for j ≤ 0 and for j > s. The complex conjugated equation holds on the complex conjugated fields H j .

4. The action of SO(2, 4) on the Ej-Bj fields

Let us deduce the action of SO [START_REF] Fefferman | The ambient metric[END_REF][START_REF] Gover | Laplacian operators and Q-curvature on conformally Einstein manifolds[END_REF] on the E j -B j fields from that on the field strength F by: (gE j )(F ) = E j (gF ), g ∈ o(2, 4).

(157)

Then one gets:

(P µ H j ) i1...ij = ∂ µ H i1...ij j , (158)
(DH j ) i1...ij = (x • ∂ + 2)H i1...ij j , (159)
Notice that the full conformal group acts independently on the fields H j and H j .

E. The Lagrangian and scalar product out of F Let us conclude this section with various remarks. First, notice that the equation of motion (115) can be derived from the action: S(g, A) = s 2(s + 1) (F (A)) α µ1...µs (F (A)) α µ1...µs dVol g = s 2(s + 1) (F (A)) 2 dVol g .

Notice also, in view of quantization, using (119) that the elements on the "diagonal" F 0 0...0 , F 1 1...1 , ..., are null, then having chosen a system such that the 0'th coordinate is the one by which the spacetime is foliated by Cauchy hypersurfaces one would get, in a conventional canonical quantization, that the 0'th momenta identically vanishes. This hints at a gauge freedom which is likely to persist on a broader class of spacetimes (than CFES).

Finally, again still in view of quantization, thanks to the operator D and (164), one can equip the space of solutions to (1) with a symplectic form A, A ′ = A µ1...µs (DA ′ ) α µ1...µs -A ′ µ1...µs (DA) α µ1...µs dΣ α ,

in which the integral is carried over the Cauchy hypersurface Σ. One such product, for d = 4, is conformally invariant and vanishes on pure gauge solutions.

  (X 0m H j ) i1...ij = x [0 ∂ m] H (X mn H j ) i1...ij = x [m ∂ n] H i1...ij j + (Σ mn H j ) i1...ij = x [m ∂ n] H

			i1...ij j	+ (Σ 0m H j ) i1...ij				(160)
	= x [0 ∂ m] H	i1...ij j	+ i	s + 1 j + 1	ε km(i1 H j i2...ij )k	-	j(s -j) j + 1	H	i1...ij m j+1
	-	[(s + 3)j(j + 1) + 2(s + 1)] (j + 1)(j + 2)	δ m(i1 H j-1 i2...ij )	-	j -1 2j -1	δ (i1i2 H	i3...ij )m j-1	,
								i1...ij j	+ jδ k [m δ	(i1 n] H	i2...ij )k j	,	(161)
	(K

ρ H j ) i1...ij = (x 2 ∂ ρ -2x ρ (x • ∂ + 2))H i1...ij j + x σ (Σ σρ H j ) i1...ij ,

(162)

= x σ (X σρ H j ) i1...ij -x ρ (DH j ) i1...ij -2x ρ H i1...ij j

.

the (s -1)-divergence of the field vanishes anyway. Therefore, the equation E 1 (V ) = 0 is no constraint on the space of solutions of (A3).

Appendix A: The special case of d = 2

The dimension d = 2 is far too particular with respect to conformal invariance and cannot be examined on the same footing as the others. This section describes the (lack of) content of (1) for d = 2.

One could consider the system [START_REF] Wünsch | On conformally invariant differential operators[END_REF] and wonder what happens for d = 2. First, the conformal invariance under restricted Weyl transformations remains, the same applies to SO(2, 2). However, following the same steps has in the main text, one would find that the equation admits solutions determined up to a vector ((1 + a 2 )| d=2 = 0). Similarly, one could get the impression that the set:

where V µ1 = ∇ µ2 . . . ∇ µs A µ1...µs , is conformally invariant while restricting the (newfound) gauge freedom. While the former happens to be true, the latter is not as E 1 (V ) = 0 is found to be automatically fulfilled on the space of solutions of E s (A) = 0. In order to show this, notice first that A possesses, whatever the rank s, only 2 independent components. In the usual cartesian coordinates let us choose those as: Φ ± = A 00...00 ± A 00...01 . (A2)

Then, turning to the chiral coordinates x ± = x 0 ± x 1 the equation ( 1) is rewritten as the system:

The system is conformally invariant and the rank from which Φ ± emanates is read off the way the field transforms. For d = 2 the fields admits the gauge transformation:

with a an arbitrary vector field. On the independent components Φ ± the above reads as

Now, for a given field A ≃ (Φ + , Φ -) one can find (a + , a -such that the gauge transformed field (A5) is null. This means that any field A, s ≥ 2, is in the (gauge) equivalence class of the trivial solution. Hence that field carries no physical content. This can also be seen on the gauge fixing equation since: