Aspects of Weyl Supergravity

In this paper we study the spectrum of all conformal, ${\cal N}$-extended supergravities (${\cal N}=1,2,3,4$) in four space-time dimensions. When these theories are obtained as massless limit of Einstein plus Weyl$^2$supergravity, the appropriate counting of the enhanced gauge symmetries allow us to derive the massless spectrum which consist of a dipole ghost graviton multiplet, a ${\cal N}$-fold tripole ghost gravitino, the third state belonging to a spin 3/2 multiplet and a residual vector multiplet present for non-maximal ${\cal N}<4$ theories. These theories are not expected to have a standard gravity holographic dual in five dimensions.


Introduction
There is recently a renewed interest in higher curvature theories. These are theories of the general form R + R n , where R is the standard Einstein term and R n denotes collectively n th power of the Riemann, Ricci, Weyl tensors or the curvature scalar [1][2][3][4][5][6][7][8]. A recent discussion on this class of models can be found in [9]. In particular, the R + R 2 supergravity has been studied [10], especially in connection to the inflationary dynamics [11]. In fact, the R + R 2 theory, known as the Starobinsky model [12] for inflation, propagates besides the usual massless graviton, an additional massive spin-0 state, known as the "scalaron field" or the so called "no-scale field". It is this mode that can be identified with the inflaton field and makes the theory so appealing as inflation is driven entirely by gravity itself and not by some external scalar field. Furthermore, the R + R 2 theory can also be embedded consistently in supergravity, whereas the linearized N = 1 theory has been analysed in [13] and the N = 2 in [14].
On the other hand the R + R 2 theory does not include all possible quadratic curvature theories.
Indeed, a second independent quadratic curvature invariant is the square of the Weyl tensor, whereas terms quadratic in the Riemann (or Ricci) tensor can be traded for a Weyl square term and the 4D Gauss-Bonnet topological term. However, when the Weyl square term is included in the lowenergy gravitational effective action, the spectrum changes and includes an additional massive spin-2 ghost [1]. 1 In the present paper we first discuss the massive Weyl 2 theory and its supersymmetric extensions, namely N -extended Weyl 2 supergravities, which contain not only the Weyl 2 term but also the Einstein term m 2 R. 2 The latter can therefore be seen as a mass deformation of the massless theory.
In addition, due to the relation between massive and massless Weyl supergravity, the bound N = 8 of Poincare supergravity tranfers to N = 4 in the case of Weyl supergravity [15].
Second, we are particularly interested in the massless limit m → 0 of the m 2 R + Weyl 2 theory in conformal supergravity [16][17][18]. Although these theories contain propagating ghosts, they are nevertheless very interesting, since in the massless limits, i.e. in the absence of the Einstein term, they provide unique examples of (super)conformal gravitational theories with up to four derivative terms. Namely in the limit m → 0 the spectrum gets re-organized and the symmetry gets enhanced, namely from (super) Poincare is enhanced to (super) conformal. In additional also the R-symmetries become local gauge symmetries. Furthermore in this limit, there are various primary operators, like the Weyl tensor itself. Another conformal tensor is the Bel-Robinson tensor, which is basically the square of the Weyl tensor. They have a well-defined conformal weight, i.e. transform under conformal transformations in a homogeneous way. This is true in the massive case and also in the limit of zero mass, i.e. the limit is continuous with respect to the scaling weights. Furthermore the Bel-Robinson 1 It should be noted that the problem with such ghosts states is that one cannot maintain at the same time unitarity and forward propagation in time of positive energy states. Indeed with the opposite +iǫ choice one propagates negative energies forward in time but unitarity and the optical theorem is preserved, whereas with the usual −iǫ prescription, ghosts carry positive energy but negative norm [19]. 2 A very interesting double copy construction of Weyl 2 (super) gravities was recently provided in [20].
tensor is conserved in the massive theory on Ricci-flat spaces.
The paper is organized as follows: In section two we describe the bosonic Weyl 2 gravity, its spectrum and its higher dimensional operators. In section three we discuss the spetcrum of the super-Weyl theory. Here we again provide some details of the higher dimensional operators using N = 1 superfield language. In sections four, five and six, the spectra of the N = 2, N = 3 and N = 4 super-Weyl 2 theories, respectively, are determined. We close in section seven with some discussions and expectations on the holographic duals of the (super)conformal Weyl 2 gravity.

Massive theory
Let us first recall the bosonic Einstein plus (Weyl) 2 gravity theory in four dimensions. More details can be e.g. found in [9,21]. The action up to four orders in derivatives has the following form: is the Weyl tensor. The Weyl 2 -term in the action possesses conformal invariance as it is invariant under the conformal transformation Therefore the Einstein-term can be regarded as the mass term in this theory, i.e. a mass deformation, which explicitly breaks conformal invariance.
The equations of motion which follow from (2.1) are written as where B µν is the Bach tensor Note that the second term in B µν is needed such that the Bach tensor transforms with a uniform weight under conformal transformations. The Bach tensor is symmetric, traceless due to conformal invariance and divergence-free (due to diff. invariance) and G µν is the Einstein tensor.
Now we can recall the propagating modes corresponding to this action. For this, one analyzes the poles in the propagators generated by its quadratic part. Specifically, there are two kinds of propagating modes [1]: (i) A massless helicity-±2 graviton g µν . This mode is independent of the couplings a and κ 2 and it is the standard massless spin-two graviton.
(ii) A massive spin-two particle w µν with mass κ 2 /a. It is related to the Weyl 2 term in the action. In fact, this massive spin two particle is a ghost for a > 0, destroying unitarity, or a tachyon for a < 0, leading to an instability. We will call this part of the spectrum the non-standard sector of the theory.
Hence in summary, the Einstein plus (Weyl) 2 gravity theory contains seven propagating degrees of freedom.

Massless theory
In the following we consider the massless limit κ = 0, which is a pure Weyl 2 theory with action The pure Weyl 2 possesses conformal invariance and it propagates six degrees of freedom [23]: (i) The standard massless spin-two graviton, corresponding to a planar wave in Einstein gravity.
(ii) In the non-standard sector there is massless spin-two ghost particle, which corresponds to a non-planar wave. In addition there is a massless vector, which originates from the ±1 helicities of the massive w µν particle. However note that the helicity zero component of w µν does not correspond to a physical, propagating mode in the massless limit, since it can be gauged away by the conformal transformations (2.2).

Higher tensors
In this section we briefly discuss some higher tensors, which are also of interest in the massless Weyl 2 theory. In fact, the aim of this discussion is to construct conformal operators, which can be coupled to spin-four fields (see also the concluding section of this paper). It is known in the literature that there are no totally symmetric, quadratic in the curvature and divergence-free four-index tensors of dimension four [24] in a generic background. However, there are dimension four, divergence-free tensors which are totally symmetric just in three-indices. It is also known that there exist a unique totally symmetric, traceless and divergence-free four-index tensor, on Ricci-flat spaces, which is the This tensor is traceless T µ µνρ = 0 and satisfies Primary operators that saturate the bound satisfy the equation ∂ µ 1 J µ 1 µ 2 ···µs = 0 and therefore correspond to conserved operators. In particular, for d = 4 we find that τ = 2 for a conserved primary which is the case for the spin-two energy momentum tensor 4 . However in our case the Bel-Robsinon 3 The conformal dimension of a field in CFT is the weight under the Weyl transformation of this field with half its indices up and half down. 4 In case the energy momentum tensor is non-conserved and has dimension ∆ > 2 + s, the corresponding bulk spin-two field becomes massive [27].
operator has dimension ∆ T = 4 and spin s = 4, and therefore it has twist τ T = 0. It follows that the Bel-Robinson operator violates the unitarity bound, which is expected, since we know that the Weyl 2 contains ghosts and is therefore non-unitary. Nevertheless as the ghost states can be projected out by appropriate boundary conditions [7], we may still look for a tensor J µνρσ like the Bel-Robinson tensor that has the same symmetries with it, it is divergence-free for Bach-flat spaces and has twist τ = 2. If such a tensor exist, it could couple to a spin-4 field. In addition, the conformal dimension of J µνρσ should be ∆ J = 6, (2.18) in order to describe a spin-4 conserved operator in the CFT. It is natural to expect that, similarly to the energy momentum tenor, J µνρσ is quadratic in a tensor build out of the curvature and/or its derivatives of dimension ∆ = 3. Such a tensor with ∆ = 3 exists in conformal gravity and it is the Cotton tensor defined as is the Schouten tensor. In particular, by using the second Bianchi identity, the Cotton tensor can be written as the divergence of the Weyl tensor and therefore We can now define a conserved 4-tensor quadratic in the Cotton tensor as follows. From Eq. (2.19) we get the equation 5 whereas, using Eqs.(2.6,2.21) we find Multiplying Eq.(2.23) by C λρµ and after some algebra we arrive at and Note that under a conformal transformation with δg µν = 2ωg µν , the Cotton tensor C µνρ and the tensor J λαµν transform inhomogeneously under conformal transformations. In particular the transformation of the Cotton tensor turns out to be whereas, J λαµν transforms as This means that both C µνρ and J λαµν , although not Weyl invariant, are Weyl covariant [28] as their transformation under Weyl rescalings does not involve higher than first derivatives of the conformal factor. Therefore in order to obtain a conformal tensor one should use a Weyl-covariant derivative for the construction of an "improved" Cotton tensor, along the lines of [28].
It is easy to prove that in the linearized theory, the J λµνρ is divergence-free and has the correct If (2.32) can be extended at the full non-linear level is not known to us.

Massive theory
Now we want to present the superfield versions of the bosonic Weyl 2 -action, first using N = 1 supersymmetry language. In conformal supergravity, one first introduces the super-Weyl tensor W αβγ , which is a chiral superfield, where α, β, γ = 1, 2 are standard SL(2, C) spinor indices. W αβγ has spin ( 3 2 , 0), i.e. its highest component is a fermionic spin-3/2 field, but W αβγ also contains the spin-two field g µν with five degrees of freedom. The Weyl superfield W αβγ satisfies and it has the following pure bosonic contributions where W δβαρ and A˙ǫ α are the spinorial equivalent or the Weyl conformal tensor and the vector auxiliary of the N = 1 supergravity, and (δ ǫ α) denotes five terms obtained by symmetrization with respect to the fermionic indices δ, ǫ and α [29].
The supersymmetric action of massive Weyl gravity is then written as where τ is the complex coupling We then find that the bosonic sector of the action is is the topological Hirzebruch-Pontryagin term. Note that in Weyl supergravity, the vector A µ is dynamical as it has a kinetic term of the form F µν F µν . Now we are ready to determine the spectrum of the massive N = 1 Super-Weyl theory. It has the following form: (i) A standard massless spin-two graviton multiplet with the following (h, q R ) helicity h components and U(1) R q R charges and with n B + n F = 4 degrees of freedom: and its CPT conjugate (ii) In the non-standard sector there is a massive spin-two supermultiplet, 6 which is the socalled massive N = 1 super-Weyl multiplet [31] with n B + n F = 16 degrees of freedom: Note that the massive states in N = 1 supergravity built representations of the group USp(2).
Hence in summary, the massive super-(Weyl) 2 gravity theory contains n B + n F = 20 degrees of freedom. 6 General massive multiplets in extended supersymmetry were discussed in [30].

Massless theory
The Weyl equation of motion eq.(2.10) now reads at linearized level [22]: This equation is equivalent to the linearized Weyl equation of motion of the bosonic Weyl tensor, as given in eq.(2.11). Then the spectrum of the massless N = 1 Super-Weyl theory has the following form: (i) A standard massless spin-two supergravity multiplet with n B + n F = 4 degrees of freedom and the following (h, q R ) helicity h components and U(1) R q R charges: together with its CPT conjugate multiplet Note that the U(1) R charges and the helicities of the fermions are correlated, i.e. the states with positive helicity have also positive U(1) R charge, and due to CPT the opposite is true for the states with negative helicities.
(ii) In the non-standard sector, we decompose the massive states into their massless helicity components. 7 Furthermore we have to decompose the USp(2) representations into the U(1) R charges of the massless states: First, we get from the massive Weyl multiplet w N =1 a massless ghost-like spin-two supermultiplet: and its CPT conjugate as in (3.11). Second we get from w N =1 a massless, physical spin-3/2 supermultiplet: together with its CPT conjugate multiplet The spin-3/2 fields together build a so-called tripole ghost, which effectively acts as a physical spin-3/2 multiplet and a dipole ghost spin-2 multiplet [22].

Higher tensors
Again, we like to consider some higher tensor in the massless N = 1 super-Weyl 2 theory. We will restrict ourselves to construct the relevant superfields in the linearized approximation of the super-Weyl 2 theory. First, the linearized super-Bel Robinson tensor has the following form: It is a tensor of spin ( 3 2 , 3 2 ) and contains a bosonic spin-four field. The bosonic part of the super-Bel Robinson tensor is T αβγδ,αβγδ = D δ DδT αβγ,αβγ | = W αβγδ Wαβ˙γδ. (3.21) Recalling that W αβγδ and Wαβ˙γδ correspond to the anti-self dual − W λ νµκ and the self dual + W λ νµκ parts of the Weyl tensor, we find the components of the Bel-Robinson tensor with Lorentz indices T µνρσ to be Therefore, we may express (3.22) as which coincides with the definition (2.12). Notice that due to eq.(3.1) the super-Bel Robinson superfield T αβγ,αβγ satisfies D α T αβγ,αβγ = DαT αβγ,αβγ = 0, (3.26) and therefore, it is conserved in Einstein supergravity.
∂ αα T αβγ,αβγ = 0. (3.27) However in Weyl supergravity, this is not the case (at least from Bach-flat but not Ricci-flat backgrounds) and therefore T αβγ,αβγ is not the correct object we are looking for, as in addition it has dimension four.
As in the bosonic case, we are considering the super-Cotton tensor, which is given as Note that Cα βγ is symmetric in the last two indices β and γ and it has spin (1, 1 2 ). Using the super-Cotton tensor the linear super-Weyl equation of motion (3.9) can be rewritten as Finally we construct the super-version of the bosonic tensor J λ αµν in eq.(2.28). It has the following form Jα βγ,αβγ = Cα βγ C αβγ . (3.30) The tensor Jα βγ,αβγ has spin ( 3 2 + 1 2 , 3 2 + 1 2 ). When writing this field in components, it contains a spin-four field, namely precisely J λµνρ , which was introduced before. Indeed, we have that Jα βγδ,αβγδ = D δ DδJα βγ,αβγ | = Cα βγδ C αβγδ = ∂ α α W αβγδ ∂α α Wαβ˙γδ. (3.31) Proceeding as above, we find that the corresponding Lorentz index tensor J νσµρ is given by which can be written in terms of the Cotton tensor as i.e., coincides with Eq.(2.28) However note that Jα βγ,αβγ is not irreducible in terms of Lorentz spins, but it also contains fields of lower spins: Jα βγ,αβγ : 3 + 3(2) + 4(1) + 2(0) . The spectrum of the massive N = 2 Super-Weyl theory has the following form: (i) A standard massless spin-two super graviton multiplet with n B + n F = 8 degrees of freedom. It contains states with the following helicities and their associated U(2) quantum numbers: In addition there are the following CPT conjugate states (ii) In the non-standard sector we have a massive spin-two supermultiplet, which is the N = 2 massive super-Weyl multiplet [14] with n B + n F = 48 degrees of freedom: w N =2 : Spin(2) + 4 × Spin(3/2) + (5 + 1) × Spin(1) + 4 × Spin(1/2) + Spin(0) .

(4.3)
Note that the massive states in N = 2 supergravity built representations of the group USp(4).
Hence in summary, the N = 2 massive super-(Weyl) 2 gravity theory contains n B + n F = 56 degrees of freedom.

Massless theory
Specifically the spectrum of the massless N = 2 Super-Weyl theory has the following form: Second we get from w N =2 two massless spin-3/2 supermultiplets with in total n B + n F = 16 degrees of freedom: As before, there are the following additional CPT conjugate states The massive Weyl multiplet w N =2 contains in addition two N = 2 vector multiplets. The first vector multiplet has the form: Its CPT conjugate multiplet is given as  It contains a neutral scalar, namely the Weyl mode, which is gauged away by the conformal transformations. Hence this vector-multiplet gets removed, it is unphysical and does not propagate.
Finally the non-standard sector also contains one CPT self-conjugate, massless N = 2 hyper multiplet: This hyper multiplet is unphysical, namely the four scalars in the 1 0 + 3 0 representations are the helicity zero components of the massive vectors inside w N =2 , which are gauged away by the local Hence in summary, the massless N = 2 super-(Weyl) 2 gravity theory contains n B + n F = 40 physical, propagating degrees of freedom.

Massive theory
The spectrum of the massive N = 3 Super-Weyl theory has the following form: (i) A standard massless spin-two super graviton multiplet g N =3 with n B +n F = 16 degrees of freedom and the following helicity and U(3) quantum numbers: In addition one obtains the CPT conjugate states: (ii) In the non-standard sector we have a massive spin-two supermultiplet, which is the N = 3 massive super-Weyl multiplet with n B + n F = 128 degrees of freedom [33]: Hence in summary, the N = 3 massive super-(Weyl) 2 gravity theory contains n B + n F = 144 degrees of freedom.

Massless theory
The spectrum of the massless N = 3 Super-Weyl theory has the following form: (i) A standard massless spin-two super graviton multiplet with n B + n F = 16 degrees of freedom, as given in eqs.(5.1) and (5.2).
(ii) In the non-standard sector, in order to obtain the massless states, we have to decompose the massive representations into massless representations. This is done via the branching rules of the massive USp(6) R-symmetry group into the R-symmetry group U(3) of the massless states. The specific decomposition of USp(6) → SU(3) × U(1) for the relevant representations is as follows: Then we get first from the massive Weyl multiplet w N =3 a spin-two supermultiplet, again with the states as given eqs.(5.1) and (5.2). They constitute a massless ghost-like spin-two supermultiplet with n B + n F = 16.
Second we get from w N =3 a helicity +3/2 supermultiplet with the U(3) charge assignements plus the CPT conjugate multiplet of the form These two multiplets together build three massless, physical spin-3/2 supermultiplets with in total n B + n F = 48. They contain the nine gauge bosons, which transform as 1 0 + 8 0 , of the local U(3) R gauge symmetry.
Finally the massive Weyl multiplet w N =3 contains in addition the following helicity states: There are also the CPT conjugate states of the form: (ii) In the non-standard sector, we get first from the massive Weyl multiplet w N =4 a massless ghostlike spin-two supermultiplet with n B + n F = 32 and with the helicites and SU(4) quantum numbers, again as given eqs.(6.1) and (6.2).
Second we get from w N =4 four massless spin-3/2 supermultiplets (in total n B + n F = 128) with the following helicities and SU(4) representations, namely We should note that the dipole ghost graviton and the tripole ghost spin-3/2 sector are accompanied by a dipole ghost complex scalar since the action is a higher-derivative action. Indeed, the equations of motion are fourth-order for the spin-2 and third order for the spin-3/2 states. This fact is also discussed in [20] at the Lagrangian level. This is not the case for the SU(4) gauge bosons which have standard Yang Mills action. The sugra higher derivative action also contains a singlet vector mode which, together with the gauge bosons, is part of the higher derivative gravitino action (which as pointed out above obeys third order equations of motion). In other words, the cubic gravitino action simultaneously describes the gravitino, the partner of the graviton, as well as the gravitini of the gravitino multiplet.
Hence in summary, the massless N = 4 super-(Weyl) 2 gravity theory contains n B + n F = 192 physical, propagating degrees of freedom. The same spectrum was also obtained in [35] using the string twistor formalism for the construction of N = 4 super-(Weyl) 2 gravity.

Conclusions and Outlook
In this paper we obtained the spectrum of all existing N -extended Weyl 2 supergravities (N = 1, 2, 3, 4) in four space-time dimensions. We are summarizing the physical spectrum, ie. the number of super-multiplets of the massless theories after subtracting the gauge multiplets, in the following Note that in the supersymmetric Weyl 2 theories one can have bosonic terms with four derivatives and also some with only two derivatives. This happens when the two-derivative terms would be auxiliary fields in Einstein supergravity. In particular this is the case in U(N ) Weyl 2 supergravity, where the Langrangian for the U(N ) vector bosons is given by the canonical F µν F µν term. 9 In addition we also discussed some operators of higher dimension, corresponding to the Bell-Robinson tensor, which is basically the square of the Weyl tensor, or the square of the Cotton tensor.
Another issue we would like to point out here is that the (super)-conformal symmetry of the (super)-Weyl 2 theory is a classical symmetry. It is true that these theories are power-counting renormalizable but the one-loop beta-functions turns out to be non-vanishing [37] and therefore they suffer from a conformal anomaly. The latter leads to serious problems in Weyl gravity as the conformal symmetry is gauged and therefore leads to inconsistencies [38][39][40]. Let us also note that one may arrive at the same conclusion by the calculation of the chiral gauge anomalies of the SU (4) R-symmetry [41] and by recalling that all anomalies are arranged in the same multiplet of the N = 4 superconformal symmetry.
However, surprisingly, the N = 4 super-Weyl 2 gravity can be made UV-finite [39,42], and thus anomaly-free [41] as well. For N = 4 Poincaré supergravity it has been conjectured in [43] a hidden superconformal symmetry. This can be achieved by appropriate coupling the Weyl 2 supergravity to four vector supermultiplets. Although this result has been shown to hold at one-loop level, it is expected to hold to all loops since the contributions to the β-function and the conformal anomaly of SYM is only an one-loop effect for N > 1 conformal supergravities [42]. For the N = 4 case in 9 An analogous effect can be seen from the Wess-Zumino Langrangian which contains three Wess-Zumino multiplets: the first term is a dipole spin 0, the second is a tripole ghost spin 1/2, whereas the last term describes two standard scalars. particular, the above statement is strengthen by the fact that the conformal anomaly is connected by supersymmetry to the SU(4) chiral anomaly which arises at one-loop also.
Let us notice that super-Weyl 2 gravity is not the only superconformal theory. As it is very well known, the N = 4 SYM theory is also invariant under the rigid superconformal supergroup SU(2, 2|4). Therefore, it can naturally source the N = 4 super-Weyl 2 gravity. In addition, the latter may affect the standard AdS/CFT holography. Indeed, allowing a boundary Weyl 2 operator still preserves the superconformal symmetry and it may correspond to a deformation of the holographic bulk theory.
Moreover, it could be, as pointed out in [44] that the pure N -extended (Weyl) 2 supergravity theory in four dimensions is the holographically dual boundary theory of an AdS 5 bulk theory, which is a higher spin theory with a spin-four multiplet of the 2N -extended supersymmetry algebra in five dimensions. These kind of theories, denoted by W-supergravities, were recently constructed [44] in flat four-dimensional space-time using a double copy and S-fold construction. Within such a duality, one would expect that a conformal operator respectively higher tensor T µνρσ on the boundary, which is quadratic in the curvature tensors and their derivatives, and which is also divergence-free on-shell, is coupled to a spin-four field in the bulk. It would be interesting to find the exact 5D holographic dual, if any, of the 4D Weyl supergravity.
Alternatively, one may consider super-Weyl 2 gravity as a bulk theory. In that case, AdS 4 is still a vacuum solution of the theory and there should exists a holographic 3D theory in the boundary, which it might not be unitary. Indeed, the massless graviton will still be coupled to a conserved spin-2 operator (energy-momentum tensor) at the boundary, the massless vector will be coupled to a conserved spin-1 operator, whereas it is expected that the massive spin-2 ghost to couple to a nonunitary spin-2 operator. Furthermore, it would be very interesting to see if there is a corresponding string construction of holographic duality, possibly using the twistor string approach of [35] for the construction of the superconformal Weyl 2 theory.