Bimetric, Conformal Supergravity and its Superstring Embedding

We discuss the connection between Weyl$^2$ supergravity and superstrings and further discuss holography between 4-dimensional, ${\cal N}=4$ superconformal Weyl$^2$ supergravity and ${\cal N}=8$, higher spin-four theory on $AdS_5$. The Weyl$^2$ plus Einstein supergravity theory is a special kind of a bimetric gravity theory and consists of a massless graviton multiplet plus an additional massive spin-two supermultiplet. Here, we argue that the additional spin-two field and its superpartners originate from massive excitations in the open string sector; just like the ${\cal N}=4$ super Yang-Mills gauge fields, they are localized on the world volume of D3-branes. The ghost structure of the Weyl action should be considered as an artifact of the truncation of the infinitely many higher derivative terms underlying the massive spin 2 action. In field theory, ${\cal N}=4$ Weyl$^2$ supergravity exhibits superconformal invariance in the limit of vanishing Planck mass. In string theory the additional spin-two fields become massless in the tensionless limit. Therefore low string scale scenarios with large extra dimensions provide (almost) superconformal field theories with almost massless open string spin-two fields. The full ${\cal N}=4$ scalar potential including the Yang-Mills matter multiplets is presented and the supersymmetric vacua of Einstein Supergravity are shown, as expected, to be vacua of massive Weyl supergravity. Other vacua are expected to exist which are not vacua of Einstein supergravity. Finally, we identify certain spin-four operators on the 4-dimensional boundary theory that could be the holographic duals of spin-four fields in the bulk.


Introduction
It is well known that the effective action of string theory is given in terms Einstein gravity, coupled to matter fields plus in finite series of higher derivative terms, which in particular contain an infinite series of higher curvature terms, which are suppressed by appropriate powers of the string scale M s = (α ′ ) −1 . In the so-called field theory limit of sending α ′ → 0, all higher string modes decouple and all higher derivative interactions disappear, and the effective theory is just given by the Einstein-Yang-Mills-theory. Particular string examples of those theories are brane-world models, where the Yang-Mills degrees of freedom are localized on the world-volumes of stack of D-branes, and where the gravitational fields, namely the metric field g µν and its partners, correspond to closed strings, which propagate within the entire ten-dimensional bulk space. Here will will consider the simplest case, namely a stack of N D3-branes, i.e. the open string Yang-Mills sector is confined on the 4D world-volume of the D3-branes. Now, when considering also higher curvature terms up to four derivatives [1][2][3][4][5][6][7][8], it is again well known that the R 2 action and the so-called Weyl 2 action propagate additional degrees of freedom: for R 2 there is an additional scalar mode and for Weyl 2 there exist an additional spin two field, denoted by w µν . In this paper we will discuss the physics connected to the Weyl 2 action and to spin-two field w µν and in particular the question how do they arise in string theory. Since the theory contains two spin-two metric fields, namely g µν and w µν , it is a particular example of a bimetric gravity theory [9]. As we will discuss the second spin-two mode w µν is not contained in standard closed string gravitational sector, but it corresponds to the first massive open string excitations, namely to the massive excitations of the open string Yang-Mills gauge fields. Therefore these massive fields w µν are also localized on the world volume of D3-branes, and the effective Weyl 2 is an entirely four-dimensional action on the world-volume of the D3-branes. As we will discuss, performing a particular scaling limit, the closed string gravitational modes decouple, and one is left with an effective 4D theory of only open string modes, namely massless N = 4 super Yang-Mills gauge theory plus (almost) massless N = 4 super-Weyl 2 theory, whose spectrum was recently constructed in [10]. Hence in this limit the theory becomes (almost) superconformal invariant. Note that superconformal Weyl 2 gravity [11][12][13][14], only exists for numbers of supersymmetries N ≤ 4, just like superconformal Yang-Mills gauge theories also only exist for N ≤ 4 [15]. This fact confirms our observation that Weyl 2 gravity is not originating entirely from closed strings, but is an effective open string theory, localized on D3-branes.
These theories are also of phenomenological interest, namely in the context of the low string scale scenario together with large extra dimensions, which allows for unique predictions for the production of the massive open string excitations at particle physics collider machines [16]. Namely, following the discussion of this paper, the low string scale scenario with light, open string spin-two excitations is a (almost) superconformally invariant field theory.
As we will argue in the last part of the paper, the 4D (almost) super-conformal invariant Weyl supergravity theory allows for an holographic description in terms of closed string modes in an AdS 5 bulk theory. In contrast to the standard AdS/CFT correspondence between massless open string Yang-Mills gauge theory in the 4D boundary and supergravity in the 5D bulk, the holographic description of the (almost) massless spin-two fields on the boundary is given by (almost) massless spin-four fields in the higher-dimensional bulk.
The structure of the paper is as follows: In section 2 we describe Weyl supergravity coupled to super Yang-Mills theory. In section 3, we present a string theory realization of the theory and in section 4 we present some of its holographic aspects. Finally, section 5 contains our conclusions. The most general formulation of Einstein plus curvature-square gravity is described by an action containing the standard Einstein term plus the following two terms being second order in the curvature More details can be e.g. found in [17,18]. The first term with W µνρσ being the Weyl tensor is conformally invariant, whereas the R 2 term is only scale invariant. Indeed, the conformal transformation g µν → g µν = Ω 2 g µν , (2.3) leaves the Weyl tensor inert W µ νρσ = W µ νρσ , (2.4) whereas the curvature scalar transforms as The two couplings c i in (2.1) are dimensionless. As discussed in [17], the R 2 action only propagates a scalar mode in flat four-dimensional space-time R 1,3 . Since we are in particular interested in spin-two fields and not to the additional scalar mode in the string spectrum, the R 2 action is not relevant for us, and we will set the coupling c 2 = 0. However, the action (2.1) with c 2 = 0 is not conformal invariant since the Einstein-term is not invariant under conformal transformations. Therefore the 1 There are two more linear combination of quadratic curvature terms, namely the Gauss-Bonnet and the Hirzebruch-Pontryagin action. However in four-dimensions these are total derivatives and hence we neglect them in the following. Similar considerations exist also in the supersymmetric case [19,20].
Einstein-term can be regarded as the mass term in this theory, i.e. a mass deformation, which explicitly breaks conformal invariance.
The propagator of the Einstein-Weyl 2 theory [1] described by is given by the following expression where , (2.8) and the usual transverse vector projection operator. Note that the propagator (2.7) for M 2 P = 0 (i.e., pure Weyl 2 theory) exhibits the conformal 1/k 4 behaviour. When the Einstein terms is present, we can equivalently write ∆(k) as where the massless helicity-±2 graviton is easily identified in the first term of (2.53). Moreover, we see that there is also a massive spin-2 state (the second term in (2.53)) with mass given by the pole at k 2 = g 2 W M 2 P which however has opposite residue to the usual massles graviton, and therefore describes a ghost spin-2 state. This shows that the theory contains as propagating degrees the standard, massless spin-two graviton g µν plus an additional massive spin-two field w µν .
Actually, an alternative way to see this is to write down a particular bimetric gravity theory with two spin-two fields g µν and w µν with the following two-derivative action [21]: Here G µν = R µν −1/2R g µν is the Einstein-tensor constructed from the metric g µν and the last term is a mass term for the second metric w µν . In general the action propagates also a massive scalar mode.
However setting the parameter a = 1, the scalar mode disappears and the action contains a massless spin-two field g µν plus a massive spin-two field w µν . Note that the two-derivative kinetic term for w µν is hidden in the coupling G µν (g)w µν , which can be seen by performing two partial integrations on this term. However after the partial integrations the kinetic term for w µν has the wrong sign, i.e.
w µν is a ghost-like field. Now using the equation of motion (2.13) and plugging the solution for w µν back into the action (2.12), one can show [22] that the resulting action is (classically) equivalent to the four-derivative W 2 action in eq.(2.6) by using the fact that where GB = R µνρσ R µνρσ − 4R µν R µν + R 2 is the Gauss-Bonnet term. The bimetric gravity action (2.12) for w µν can be made ghost-free by adding an infinite number of terms with a finite number of parameters to it. As shown [22], this procedure is equivalent to adding to the W 2 action an infinite number of higher derivative terms, which resemble to additional parameters of the ghost-free bimetric gravity theory. In other words, the ghost nature of the massive spin-2 excitation is an artifact of the higher derivative truncation to fourth order. En passant, let us mention theat for a = 1, the action (2.12) is (classically) equivalent to the action (2.1) with (2.15) Therefore, only for a = 1 the scalar mode associated to the R 2 term is absent.

Supersymmetric case
The above method can also be implemented in a supersymmetric setup [19]. For this, we need to recall that the graviton h µν sits in a real vector superfield Φ µ with expansion (in Wess-Zumino gauge) where A µν and A µ are the antisymmetric two-form and one-form fields of new-minimal supergravity, respectively. We can then define the real linear superfield E µ as which contains the Einstein term the field strengths of the auxiliaries A µν and A µ , respectively. We need also to define the Riemann multiplet R µν with components expansion The Weyl tensor W µνρσ is contained then in the Weyl multiplet W µν defined as as can be seen from its components expansion In terms of the real vector superfield Φ µ , the Riemann and Weyl multiplets can be written (in spinor notation), The action (2.1) (with c 2 = 0) is contained in the bosonic part of the supersymmetric Lagrangian (with M P = 1 here) The first term contains the Einstein term and the second the Weyl 2 . A supersymmetric generalization of (2.14) exists and it is written as and SBG is the supersymmetric counterpart of the usual Gauss-Bonnet term and it is such that in the real and imaginary parts of its highest θ 2 component are the Hirzebruch-Pontryagin and Gauss-Bonnet terms, respectively. We may then write (2.24) as We may linearize in E µ and W the above Lagrangian by introducing a real vector superfield V µ and a superfield H so that Then, after performing first the shift Φ µ → Φ µ − V µ and after the conformal transformation Φ µ → Φ µ + Dσ µ H + Dσ µ H, we get that the supersymmetric action (2.26) is classically equivalent to 19]. From the above Lagrangian we see that the first term in the first line describes a physical massless (2, 3 2 ) graviton multiplet (Φ µ ), whereas the second term in the first line describes a massive (2, 3 2 , 3 2 , 1) multiplet (V µ ) with mass square m 2 = 1/4c 1 [23]. The latter multiplet is not physical as its Lagrangian term opposite sign from the massless multiplet and therefore it is a ghost massive spin-2 multiplet.

Including Gauge Fields
Now, we will also include a four-dimensional bosonic Yang-Mills U(N) gauge theory, which is coupled to Einstein gravity. Then the action up to four orders in derivatives has the following form: F a µν is the standard Yang-Mills field strength and g 2 W and g 2 Y M are dimensionless couplings. The Yang-Mills term and the Weyl 2 -term in the action possess (classical) conformal invariance, whereas again the Einstein-term can be regarded as the mass term in this theory.
(iii) A massive spin-two particle w µν with mass It is related to the Weyl 2 term in the action. In fact as mentioned, this massive spin two particle is a ghost, destroying unitarity, but we will neglect this problem in the following and we will comment on it only in the conclusions. We will call this part of the spectrum the non-standard sector of the theory.
The Einstein plus (Weyl) 2 gravity theory contains seven propagating degrees of freedom. As already explained, this part of the theory can be considered as a bimetric theory of gravity with two spin-two fields, namely one the standard massless graviton g µν plus the non-standard massive spin-two field w µν . As we will discuss in the following, in string theory the graviton g µν originates from the closed string sector and lives in the bulk space, whereas the spin-two field w µν as well as the Yang-Mills gauge bosons A a µ come from the open string sector and will be localized on the world-volume of a stack of D3-branes.
In the following we will consider the following three limits. Later we will see how these limits are realized in string theory.
(A) Decoupling of gravity, i.e. Yang-Mills limit First we consider the infinite mass limit M P → ∞ . (2.31) In this limit gravity becomes non-dynamical and decouples from the theory. In fact, for non-zero coupling g W , both spin-two particles completely decouple, since the spin-two particle w µν becomes infinitely heavy. Alternatively one can keep M W finite, which implies that g W → 0, i.e. the spin-two Weyl modes are very weakly coupled.

(B) Massless bigravity limit
Second we consider the massless limit, namely the limit of vanishing Planck mass 2 : The propagator ∆(k) now becomes (2.33) In this limit the second spin-two field w µν will become massless and we deal with massless Weyl gravity. Therefore, for finite M P there is a Higgs effect with respect to w µν , and in the massless limit the degrees of freedom of w µν will arrange themselves into proper massless fields (see below). In this limit we deal with Yang-Mills gauge theory plus Weyl 2 theory with action This theory possesses conformal invariance and it propagates the following degrees of freedom: (i) The standard massless, closed string spin-two graviton g µν , corresponding to a planar wave in Einstein gravity.
(ii) Massless open string U(N) gauge bosons A a µ .
(iii) In the non-standard sector there is massless open string spin-two ghost particle w µν , which corresponds to a non-planar wave. In addition there is a massless open string vector w µ , 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 (2.3).
(C) Light spin-two plus massless Yang-Mills limit Now we consider the double scaling limit Therefore the coupling g W must vanish faster than M −1 P . In this limit the massless graviton decouples from theory, i.e. the standard gravitational sector gets decoupled from the massless non-standard spin-two sector. So one is left which an action that contains the massless Yang-Mills gauge fields A µ as well as the (almost) massless spin-two fields w µν . The propagator has the leading behaviour (2.33) and the dynamics is described again by the action (2.34). theory. The spectrum of the N = 4 Super-Yang-Mills plus massive N = 4 Super-Weyl theory has 2 The massless limit was also discussed in the context of bimetric theories in [22,26]. the following form [10,27]: (i) A standard massless spin-two super graviton multiplet g N =4 with n B +n F = 32 degrees of freedom and with the following helicities and SU(4) representations: together with its CPT conjugate The complex scalar corresponds to the complex complex constant τ of the N = 4 field theory, i.e.
to the massless marginal operator in the superconformal field theory.
Note that the 6N scalars Φ ij = −Φ ji , (i, j = 1, · · · , 4), of the N vector multiplets are coupled to the curvature scalar in confrormal supergravity as Therefore, the conditions where ψ j ate the gauginos, break superconformal dilatations and S-supersymmetry, lead to Poincaré supergravity and in this case the scalars parametrize the coset [28][29][30][31] SU(1, 1) In fact, the conditions (2.41) are weaker than the constraints imposed by the equations of motion of the scalars D ij kl and the fermion χ ij k , which we describe in section 2.4.3. These constraints allow to remove six vector multiplets in massless Einstein supergravity.
Notice that in rigid supersymmetry, the Yang-Mills scalar manifold is flat R 6N whereas in Poincaré supergravity the coset is SO(6, N)/SO(6) × SO(N). It looks that in massive Weyl supergravity the scalar manifold is SO(6, N)/SO(N) because 15 scalars have not been Higgsed. In other words the constraints (2.41) and (2.43) remove the 1 and 20 from 6 × 6 = 1 + 20 + 15 but do not remove the 15.
The first constraint in (2.43) coming from the D scalars which appear linearly in Einstein supergravity, is just a contribution to the scalar potential in massive Weyl supergravity because the D scalars appear now quadratically in the Lagrangian. Hence, the deformation of (2.39) to (2.42) is only true if the Weyl term is absent so that the 15 gauge fields of the superconformal multiplet are auxiliary and their equations of motion produce the deformation from R 6N to SO(6, N)/SO(6) × SO(N).
However if the Weyl action term is added, the 15 vectors are massive and propagating and the above coset is not reproduced. Poincaré supergravity is the limit M p → ∞ while Weyl supergravity is the limit M p = 0. What happen in between is a new theory we are describing. The potential of this new theory is different from Poincaré supergravity and is strictly quartic in all scalar fields before imposing the constraints as we will see below.
(iii) In the non-standard sector we have the spin-two massive Weyl multiplet of N = 4, which is irreducible with n B + n F = 2 8 = 256 states in USp (8) representations [27]: Hence in summary, the N = 4 massive super-(Weyl) 2 gravity theory contains n B + n F = 288 + 16N degrees of freedom, where N is the number of physical vector multiplets. General massive multiplets in extended supersymmetry were discussed in [32] Also note that in Einstein supergravity constraints (2.41) and (2.43) are field constraints while in massive Weyl supergravity they are VEV constraints (Higgs phase) since the six vector multiplets, which appear in the massless limit (see next section) are in this case physical degrees of freedom.
As we will now see, in massless Weyl supergravity these multiplets become unphysical gauge degrees of freedom since the massless Weyl action does not depend on compensators being superconformal invariant. So in massless Weyl supergravity coupled to Yang-Mills the moduli space is that in eq.(2.39). The massive phase is obtained when six extra singlet compensating vector multiplets are introduced.

Massless theory
Now we can consider the N = 4 supersymmetric version of the Higgs effect for the spin-two Weyl superfield w N =4 . In the limit M P → 0 the bosonic and fermionic degrees of freedom of w N =4 will arrange themselves into proper massless supermultiplets, when taking into account the additional local superconformal and gauge symmetries, which arise in the massless limit. In order to perform the massless limit we need the branching rules of the massive USp(8) R-symmetry group into the R-symmetry group SU(4) of the massless states. The specific decomposition of USp(8) → SU(4) for the relevant representations is as follows: Then for M P = 0, the spectrum of the massless N = 4 Super-Weyl theory has the following form [10]: (i) A standard massless spin-two supergravity multiplet with n B + n F = 32 degrees of freedom as given in eqs.(2.36) and (2.37).
(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.(2.36) and (2.37).
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, namelȳ 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 [33] 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, 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 [34] using the string twistor formalism for the construction of N = 4 super-(Weyl) 2 gravity. The spin 1/2 have three sources, from the spin 3/2 cubic gravitino kinetic term, the spin 1/2 cubic kinetic term and the spin 1/2 standard Majorana kinetic term.
At the end of this section, we can summarize the spectrum of Weyl supergravity in the following way. In pure Weyl supergravity without any additional massless Yang-Mills multiplets, the six vector multiplets with 36=1+15+20 helicity zero components play the role of super-goldstone bosons.

Scalar potential
In this section we will consider some couplings between the Yang-Mills sector and the Weyl sector of the theory. In particular we will discuss the potential of the scalar fields that appear in the N = 4 Yang  (4) rep. w c of the spin-two dipole ghost multiplet, E ij is symmetric, D ij kl is pseudoreal and Φ ij is antisymmetric, and in the adjoint representation of the gauge group G. They satisfy the relations whereas their complex conjugate fields are Notice that in Eq.(2.44) we have seen that the spin-two massive Weyl multiplet of N = 4 in the non-standard sector has n B + n F = 2 8 = 256 states which are arranged in USp (8)  The most general Lagrangian for the N = 4 conformal supergravity has been constructed in [31].
It turns out that it is completely specified by a single holomorphic and homogeneous of zeroth degree function H(φ α ) of the coset variables φ α . The structure of the scalar potential for N = 4 super Yang-Mills is coupled to N = 4 conformal supergravity can be read off from refs [29][30][31] and it turns out to be (in the notation of [31]) where D is the operator In rigid supersymmetry, only the last term of the potential (2.53) exists. All the other terms arise from the Weyl multiplet (terms proportional to H and its derivatives) and the gauge-matter coupling.
The scalars D ij kl are auxiliaries and can be integrated out leading to Note that a non-constant H function gives extra terms to the scalar potential (2.55). This will be the case in twistor string theory where H is an exponential in the holomorphic variable [34]. For constant H, the terms proportional to DH and D 2 H in the potential drop and it is easy to see that E = 0 and Φ in the Cartan subalgebra of the gauge group is an extremum of the potential. This is the breaking of superconformal to Poincare supergravity if 6 auxiliary vector multiplets are added with wrong sign so that a correct Einstein term and the solution D = 0 is possible. Indeed, let us recall that the fermions of the theory are the gravitini ψ i µ (in the 4 of SU(4)) associated with Q-supersymmetry, the composite φ µi (in the 4) associated with S-supesymmetry and the two spinor fields Λ i and χ ij k in the 4 and 20 of SU(4), respectively. The fermionic shifts of the spinors fields contain among others, the terms [27] δΛ i = · · · + E ij ǫ j , We note that pure massive Weyl supergravity is obtained by adding to the Weyl multiplet 6 vector multiplets of wrong sign. In this case the spectrum is the standard massless N = 4 Poincaré supergravity coupled to a massive N = 4 spin-2 ghost multiplet. The massive scalars are then 20 from the six compensatos, the 10 + 10 E scalars and 1 + 1 from the dilation dipole massive ghost.
All together they make the 42 (of USp (8)) as it should. Indeed, the constraint (2.43) is needed in Poincare supergravity because the D scalars appear linearly in the action [29,30]. However, this is not true in Weyl massive supergravity where they appear quadratically [31] so that they lead to a new potential term after integrate them out rather than to a constraint and the 20 D scalars becomes dynamical. In the higgsed phase, the 15 scalars go away and this explains 42 = 1 + 1 + 20 + 10 + 10.

String realization
Now we want to discuss how to obtain Weyl 2 gravity plus Yang-Mills gauge theory from IIB superstring theory. As already mentioned, in string theory the graviton g µν originates from the closed string sector and lives in the bulk space, whereas the spin-two field w µν as well as the Yang-Mills gauge bosons A a µ come from the open string sector and will be localized on the world-volume of a stack of D3-branes. In the following we will first discuss the closed and open string spectrum and then, how the various limits can be realized in string theory.
Here we will discuss the case of maximal supersymmetry. This means that in four-dimensional the closed string bulk theory possesses N = 8 supersymmetry (i.e. 32 supercharges), whereas the open string sector localized on the D-brane worldvolume will preserve N = 4 supersymmetry (i.e. 16 supercharges). Specifically, we will consider the type IIB superstring on R 1,3 × T 6 , with an additional stack of N D3-branes with world-volumes on R 1,3 . Possible other D-branes and/or orientifold planes do not play an important role for the discussion, and we also do not address the question of tadpole cancellation. In fact, when taking the decoupling limit of infinite T 6 volume later on, i.e. considering a non-compact six-dimensional extra space, we just deal with N D3-branes in flat ten-dimensional space-time.
The spectrum of this string theory is now as follows:

Massless open string Yang-Mills sector
Now we come to the massless open string spectrum of the D3-branes on the background R 1,3 × T 6 .
For maximally supersymmetric, toroidal compactifications of D = 10 superstring, its excitations form supermultiplets of N = 4 supersymmetry. Before discussing the first excited level, we recall the vertices of massless particles, which arise from the zero modes and include, in the NS sector, the gauge bosons A a and six real scalars φ I , I = 1, . . . , 6. In the R sector, we have four gauginos λ A , I = A, . . . , 4. All in all, these zero mode form one N = 4 gauge supermultiplet. The NS sector vertices, in the (−1)-ghost picture, read: (3.1) Here, X, ψ, Z, Ψ are the fields of N = 1 worldsheet SCFT, with the Greek indices associated to D = 4 spacetime fields X µ , ψ ν and the Latin upper case labeling internal D = 6 (e.g. Z I , Ψ I ). φ is the scalar bosonizing the superghost system.
The R sector vertices, in the (−1/2)-ghost picture, read: Here, S andS are the left and right-handed SU(2) spin fields, respectively, while Σ A and Σ A are the internal Ramond spin fields. The couplings are where g Y M is the gauge coupling. In the above definitions, T a are the Chan-Paton factors accounting for the gauge degrees of freedom of the two open string ends, meaning that all these massless states are in the adjoint representation of the U(N) gauge group.
We can also write these states in terms of the fermionic oscillators in the transversal space-time directions, denoted by b i r (i = 1, 2), and the internal oscillators b I r (I = 1, . . . , 6). Then the eight bosons bosonic states in he adjoint representation look like For the eight fermions in the adjoint representation one simply has λ a,A ∼ T a |α, A , where H m is an auxiliary vector field. Note that again the open string gauge coupling g A = (2α ′ ) 1/2 g Y M appears in this vertex operator. At this level, the on-shell condition is k 2 = − 1 α ′ . The constraints due to the requirement of BRS invariance are:

7)
B m m + k m H m = 0 .
In D = 10 all 128 bosonic degrees of freedom can be accounted for by setting H = 0, i.e. with a traceless, transverse B and transverse E.
Also for the fermions, we begin with the first massive level in D = 10. In the R sector, the fermion vertex operator [in its canonical (−1/2)-ghost picture] is parametrized by two vectors, Majorana-Weyl spinors v A m andρ nḂ of opposite chirality [35]: Here, A denotes a left-handed spinor index whileḂ is its right handed counterpart. Γ n are 16 × 16 Weyl blocks of the D = 10 gamma matrices and Θ A are the conformal weight h = 5 8 chiral spin fields. Requiring BRST invariance imposes two on-shell constraints on v A m andρ ṁ B which determineρ in terms of v and leave 144 independent components in the latter. Furthermore, a set of 16 spurious states exists which allows to takeρ and v as transverse and Γ-traceless: These 128 = 144 − 16 physical degrees of freedom match the counting for bosons.
As for the massless states, we can also write the massive states, that are created by these vertex operators, in terms of the bosonic and fermionic oscillators α n and b r . Now we split the indices into uncompactified and internal indices. Furthermore we will omit the gauge index, i.e. we drop the Chan-Paton factor T a , which means that we consider the neutral, excited states of the Abelian  In ten dimensions, the 128 massive fermions are given by the following string states: In terms of four-dimensional massive spinors this leads to: (4 +4) × Spin(3/2) + (4 +4 + 20 +20) × Spin(1/2) ,

Closed string sector
In the following we will also provide the closed string spectrum of the theory, both in the bulk and are non-linearly realized on the D3-branes. Therefore the massless closed string spectrum on the D3-branes is precisely the one of N = 4 supergravity. The corresponding massless states precisely build the standard massless spin-two super graviton multiplet g N =4 , which is displayed in eqs. (2.36) and (2.37).

Massive closed string spin-four sector
As discussed in [37], the first excited closed string states are obtained by performing the tensor product of two super-Weyl supermultiplets. This leads to a massive supermultiplet with a highest spin-four tensor field Φ 4 in the closed string sector, whereas the massive spin-two sector, i.e. the massive Weyl supermultiplets, correspond to open string excitations.
For the case under consideration with background space R 1,3 × T 6 , the bulk spectrum is then given in terms of massive spin-four N = 8 supermultiplet Φ 4 N =8 :

Effective field theory and limits
Now we will discuss the four-dimensional effective field theory on the stack N D3 branes. From the closed strings we will restrict ourselves to the massless gravitational sector, and the closed string spin-four in the bulk will be mentioned later in the next section on holography. For the open strings, we will on the massless spin-one Yang-Mills sector as well as on the massive spin-two Weyl sector.
Since both types of fields belong to open string with ends lying on the D3-branes, the Yang-Mills field as well as the Weyl fields are confined to the world-volumes of the D3-branes.

Ten-dimensional picture, non-compact space
Here we consider a stack of N D3-branes in a non-compact space R 1,9 . The ten-dimensional action can be schematically written as 19) where S bulk is the effective action of the massless gravitons and their superpartners from the closed strings, S brane is the four-dimensional effective action of the massless Yang-Mills fields and the massive spin-two field w µν on the D3-branes, (3.20) and S int describes the interactions between the open and closed string modes.
Let us now determine the effective couplings in terms of the basic string parameters, which are (i) g s = e φ , the string coupling constant, which is determined by the vev of the dilaton and (ii) M s = 1/ √ α ′ , namely the string scale.
In the string frame, the effective ten-dimensional Planck mass is given as In order to go to the Einstein frame, one has to perform a Weyl rescaling of the metric, which in D dimensions takes the form Therefore the ten-dimensional Einstein-Hilbert term transforms from the string frame into the Einstein frame as |g| 10 e −2φ R → |g| 10 R (3. 24) and in the Einstein frame the Planck mass is therefore independent of g s : Second, the gauge kinetic term of a Dp-brane transforms from the string frame into the Einstein frame as Hence, for D3-branes (p = 3) the effective gauge coupling in the Einstein frame is given as Finally for the fundamental string tension one obtains that Therefore the masses of the excited strings in the Einstein frame scale as 29) and hence the ratio between these masses and the 10d Planck scale remains invariant. In D dimensions, a mass, when measured in the Einstein metric, is related to g s as i.e. we arrive at a theory of free gravitons and its supersymmetry partners. This decoupling limit is also referred to the Maldacena limit: free type IIB supergravity in the bulk and four-dimensional SYM theory with 16 supercharges on the world-volume of the branes. To see the more precise form of the decoupling limit, which zooms into the near horizon region of the D3-brane SUGRA solution, we recall that it is defined as follows: On the gauge theory side this limit corresponds to the limit of infinite 't Hooft coupling In this the near-horizon limit the type IIB background of the N D3 branes is given by the well-known AdS 5 × S 5 geometry. Note that this limit can be obtained by sending L to infinity while keeping M s fixed, which means that the near horizon limit can be obtained for finite masses of the string excitations.

Four-dimensional picture, compact internal space
Now we switch to four dimensions and consider the theory compactified on R 1,3 × T 6 . As we have discussed in section 3.1.2 the massive open string excitations precisely agree with the N = 4 spintwo Weyl supermultiplet. The question is now, which is the correct effective action for these massive states. Since these states appear at the first mass level, the corresponding effective action must contain four derivatives. Hence a priori, it could be either the R 2 -action or the W 2 -action. Since the R 2 propagates a scalar degree of freedom, whereas the W 2 -action propagates precisely the spintwo degrees of freedom of the Weyl-supermultiplet, we can safely conclude that the W 2 -action is the correct effective action for the massive open string fields. Therefore the four-dimensional string effective active action for closed string gravity plus open string Yang-Mills plus open string massive spin-two fields has the following form: In addition to the ten-dimensional string parameters g s and M s we now gain a third parameter, namely: (iii) R: the radius of the internal space, i.e. the volume of the T 6 is given by R 6 . In units of the string length L s the size of the internal scape is given by the dimensionless parameter r = R/L s = RM s .
The three string parameters g s , M s and r are identified with the three four-dimensional coupling constants of the effective theory in the following way: (i) The four-dimensional gravitational closed string coupling M P in the Einstein frame: The open string Yang-Mills coupling g Y M for the gauge fields on the D3-branes: iii) The bimetric Weyl coupling g W : The effective 4D coupling g W can be determined by the requirement that the mass of the massive open string spin-two fields w µν is given in the Einstein frame as (see eq.3.30)) It then follows from eq.(2.30) that Observe that in the four-dimensional Einstein frame, the Weyl coupling g W is scaling with respect to g s as the gravitational coupling, because it corresponds to a coupling between closed and open strings. Moreover is proportional to the inverse of the internal volume.
Now we can consider the following four decoupling limits in the four-dimensional effective string theory, which we already mentioned before in section two: (A) Decoupling of gravity The decoupling of the closed string modes namely the decoupling of standard gravity is achieved sending the Planck mass to infinity:

(B) Massless bigravity limit
Second, we consider the massless limit, namely the limit of vanishing Planck mass: For a rank s symmetric traceless tensor, there is the following relation between the corresponding mass of the field in the (d + 1)-dimensional bulk and the scaling dimension ∆ and the spin s of the operator in the conformal field theory on the d-dimensional boundary: This formula is consistent with the unitarity bound, which is given as According to the standard holographic dictionary, the most relevant operator is the conserved boundary energy momentum tensor T ν µ , which has conformal dimension ∆ = 4 and spin s = 2 and hence it saturates the unitarity bound in four dimensions. T ν µ is is coupled to a symmetric tensor g µν , which becomes the massless spin-two graviton field in the higher-dimensional bulk theory. 4 In our concrete case of four-dimensional super Yang-Mills theory plus Weyl 2 gravity given in eq.(2.34), we can derive the energy momentum tensor from the Yang-Mills action plus the linearized gravity However, let us mention that although the original (2.34) theory is invariant under conformal transformations, it seems that (4.3) fails as the Einstein tensor transforms non-homogeneously.
Therefore, in order to restore conformal invariance of (4.3), we have to assign a non-homogeneous transformation for the field w µν . In fact, it can be verified that under an infinitesimal conformal transformation δg µν = 2λ(x)g µν , (4.4) the Einstein tensor transforms as Then, it can be verified [38] that the action (4.3) is invariant if w µν transforms as In other words, under a conformal transformation, the field w µν transforms as it would transform under a diffeomorphism generated by the gradient of the conformal factor.
It is straightforward to calculate the energy-momentum tensor for the theory (4.3) which turns out to be where The equation of motion for w µν is
where S µν is the Schouten tensor Rg µν , (4.12) and it turns out that T µν F on-shell is is the Bach tensor. The latter is symmetric, traceless and divergence-free and therefore, T µν F is also traceless (due to conformal invariance) and divergence-free (due to diff invariance). In addition B µν transforms under a conformal transformation g µν → Ω 2 g µν as and therefore it has dimension ∆ B = 4 (as the energy-momentu tensor).
Next we proceed to the massive spin-four operators on the boundary in dimension d = 4, which are coupled to massive spin-four, closed string fields in the bulk. In order to be massive their scaling dimension ∆ should be larger than 6. These fields will become massless in the limit α ′ → ∞, i.e.
M s → 0. In our concrete case, the relevant spin-four operator J µνρσ could be for example  (4.18) or products of the Weyl tensor, as for example where is the Bel-Robinson tensor [40]. The dimension of the latter is ∆ T = 4 as under conformal transformations, it transforms as T µν ρσ → T µν ρσ = Ω −4 T µν ρσ . (4.21) The operators J µσνρ above have spin s = 4, they transform under conformal transformations as J µσ νρ → Ω −8 J µσ νρ , (4.22) and their dimension is therefore ∆ J = 8. Hence these operators are then holographically coupled to massive spin-four fields in the bulk, such that we are dealing with a higher spin-four theory in the bulk. In string theory, J µνρσ can be viewed as massive composite field with mass square m 2 = 20/α ′ , corresponding to the product of two closed string graviton vertex operators. Note that this mass is the mass on AdS 5 , which is not the same as the mass of the corresponding string state on a flat Minkowski background. In the supersymmetric case, the field content and the supermultiplet structure is precisely as the one given in section 3.2.2, which is obtained by the tensor product of two N = 4 super-Weyl multiplets. Since in the decoupling limit, the spin-two fields w µν are free fields on the 4D boundary, also the spin-four field in the AdS 5 bulk space should be a free field, with the following free field equation: where R is the scalar curvature of the AdS 5 space. 5 So in summary, the Yang-Mills energy momentum tensor T µν couples to a spin-two field in the bulk, the standard graviton on AdS 5 × S 5 , whereas J µνρσ couples to a spin-four field in the bulk.
It means in particular when considering just the N -extended (Weyl) 2 supergravity theory in four dimensions without the Yang-Mills part that this theory 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 [37] in flat four-dimensional space-time using a double copy construction.
Therefore, the (almost) massless spin-two fields w µν are conjectured to be dual to N = 8 spin-four fields on AdS 5 × S 5 . To support this conjecture it would be important to compute some correlation functions of J µνρσ on the boundary and compare them with the corresponding spin-four correlation functions in the bulk.
. gauge fields, the massive spin-two graviton supermultiplet originates from open string excitations on D3-branes and hence is localized in four space-time dimensions, whereas the standard massless spintwo graviton supermultiplet is coming from the closed string sector. We then argued that effective action of this bimetric theory is given by the four-derivative, N = 4 supersymmetric Weyl 2 action, whose Weyl-supermultiplet precisely embraces the same number of degrees of freedom as the first massive open string excitations on the D3-branes. In the massless limit, where the mass of the open string "gravitons" and their superpartners go to zero, the theory becomes N = 4 superconformal.
We discussed that the holographic description of this quadratic spin-two superconformal gravity on the four-dimensional boundary is given in terms of a higher N = 8 spin-four theory in the AdS 5 bulk space. We have constructed the corresponding N = 8 spin-four supermultiplet in terms of massive closed string excitations in four space-time dimensions, which then can be lifted to the five-dimensional AdS 5 space. In addition we have identified certain spin-four operators on the fourdimensional boundary space, which, following the holographic dictionary, can couple to the spin-four fields in the five-dimensional bulk.
At the end of the paper, we like to close with the following additional remarks: • It is clear from string theory that the massive open string spin-two state cannot be a ghost state. So eventually one has to write down an effective action for this spin-two state, which is ghost-free. But here we are truncating the spectrum to the first excited level and neglecting all the higher open string excitations. In the same way we are restricting the effective action to be just with four derivatives, but we neglect all the higher derivative interactions [43]. It is now still a conjecture that the full effective action action of this open string spin-two state can be written as an infinite power series expansion of the Weyl-tensor. In fact it was recently argued in [22] that adding an infinite series of curvature tensors should provide an action which propagated a ghost-free open string spin-two particle. Truncating this series to Weyl 2 , the spin-two particle becomes a ghost.
• In case we are dealing with a stack of N D3-branes, the massive N = 4 Weyl supermultiplet is colored, just like the U(N) gauge fields. Therefore one is dealing with N 2 copies of interacting spin-two Weyl supermuliplets. In this paper we have considered the simpler case of just one single, neutral Weyl supermuliplet, which belongs to the U(1) part of the U(N) symmetry group, or simply is the relevant open string excitation for the case N = 1.
• It would be interesting to compute the string scattering amplitudes between the massless and massive string fields using techniques already applied in [35] in order to confirm the effective Weyl 2 action and the couplings between the Yang-Mills and the Weyl sectors, proposed in this paper.
• The massive closed string spin-four field can be viewed as a kind of a bound state of two massive open string spin-two states, in analogy to the massless closed string graviton, which can be regarded as the bound state of two massless open string gauge bosons. This observation relies in the structure of the string vertex operators and is also the basis of the double copy constructions, which was recently also worked out four the spin-four case [37].
• It should be possible to perform a socalled S-fold projection, getting completely get rid off the massless Yang-Mills sector. In this case one would entirely deal with strongly coupled, massive Weyl 2 supergravity on the boundary and with massive spin-four supergravity in the bulk, a theory denoted by W-supergravity, recently constructed in [37]. In the massless, superconformal limit, the spin-four W-supergravity on AdS 5 also becomes massless.
• The scalar potential should capture also the solutions which are not the one of Einstein supergravity. In the bosonic case these are the solutions where the Bach tensor vanishes but not the Ricci tensor. While the first break conformal to Poincare supergravity, the others may also break supersymmetry even partially, which still has to be discovered yet. It is likely that any conformally flat space is a solution with vanishing Bach tensor so it is conceivable that AdS or even dS space are solutions of massless Weyl supergravity, as it is true in the simplest bosonic case.
• Finally, we would like to stress that the superconformal symmetry of the supersymmetric Weyl 2 theory is a classical symmetry 6 . Althought such theories are power-counting renormalizable, their one-loop beta-functions are be non-vanishing [45] and therefore they suffer from a conformal anomaly. The latter leads to serious problems since conformal symmetry is gauged in Weyl gravity and therefore leads to inconsistencies [46][47][48]. The same conclusion can be drawn by considering the chiral gauge anomalies of the SU(4) R-symmetry [49] and recalling that all anomalies are accommodated in the same multiplet of the N = 4 superconformal symmetry.