Manifest electric-magnetic duality in linearized conformal gravity
aa r X i v : . [ h e p - t h ] J a n Manifest electric-magnetic duality in linearized conformal gravity
Hap´e Fuhri Snethlage, Sergio H¨ortner ∗ Institute for Theoretical Physics, University of Amsterdam,Science Park 904, 1090 GL Amsterdam, The Netherlands
We derive a manifestly duality-symmetric formulation of the action principle for conformal gravitylinearized around Minkowski space-time. The analysis is performed in the Hamiltonian formulation,the fourth-order character of the equations of motion requiring the formal treatment of the three-dimensional metric perturbation and the extrinsic curvature as independent dynamical variables.The constraints are solved in terms of two symmetric potentials that are interpreted as a dualthree-dimensional metric and a dual extrinsic curvature. The action principle can be written interms of these four dynamical variables, duality acting as simultaneous rotations in the respectivespaces spanned by the three-dimensional metrics and the extrinsic curvatures. A twisted self-dualityformulation of the equations of motion is also provided.
I. INTRODUCTION
Understanding dualities is a major challenge in moderntheoretical physics. Despite their widespread presence indiverse branches –including field theory, (super)gravity,string theory and condensed matter– a comprehensivetheoretical framework that explains the origin of the phe-nomenon and describes its full implications is lacking.In the case of gravitational theories, dualities are inti-mately related to the emergence of hidden symmetriesupon toroidal compactifications (supplemented by Hodgedualizations of Kaluza-Klein fields). For instance, it haslong been recognized that the reduction to three dimen-sions of the four-dimensional Einstein-Hilbert action inthe presence of a Killing vector, followed by the dual-ization of the Kaluza-Kelin vector to a scalar, exhibitsa SL (2 , R ) invariance acting on the scalar sector. Thelatter, conformed by a dilaton and axion, parametrizesa SL (2 , R ) /SO (2) coset space. This SL (2 , R ) invarianceis commonly referred to in the literature as the Ehlerssymmetry [1]. In the presence of two commuting Killingvectors, the reduction to two dimensions can be achievedin two different ways: either by a direct compactificationto two dimensions, or by a reduction to three dimensionsfollowed by a dualization of the vector field to a scalarand a final reduction to two dimensions. Each of theseroutes yields a different SL (2 , R ) /SO (2) sigma modelin the scalar sector. The first one is associated to the SL (2 , R ) Matzner-Misner group, which originates fromcoordinate transformations preserving the vector spacespanned by the Killing vectors. In the second case the SL (2 , R ) group is properly the Ehlers group. The inter-twining of the Matzner-Misner and Ehlers groups gen-erates an infinite-dimensional group realized non-locally,as described by Geroch [2]. ∗ Electronic address: [email protected]
A similar situation occurs in supergravity [3]: the (11- d ) toroidal compactification of the eleven-dimensionaltheory yields maximally supersymmetric supergravity indimension d with a symmetry structure hidden withinthe non-gravitational degrees of freedom in the reducedbosonic sector, namely p -forms and scalars. After Hodgedualization of the p -forms to their lowest possible rank,they combine in an irreducible representation of a non-compact group G acting globally, whereas the scalar sec-tor is described by the non-linear sigma model G/H , with H the maximal compact subgroup of G . In even dimen-sions, the global symmetry G is realized as an electric-magnetic duality transformation interchanging equationsof motion and Bianchi identities. Reduction to five, fourand three dimensions yields as G the exceptional Liegroups E , E and E , respectively.The study of the rich algebraic structure that underliesthe emergence of hidden symmetries in compactificationsof (super)gravity has led to conjecture the existence of aninfinite-dimensional Kac-Moody algebra acting as a fun-damental symmetry of the uncompactified theory [4]-[7],encompassing the duality symmetries that appear upondimensional reduction. A key property of these algebrasis that they involve all the bosonic fields in the theoryand their Hodge duals, including the graviton and itsdual field. The associated symmetry transformation fora given tensor field relates it to all the rest of the fields,regardless their tensor structure, and in general has ahighly non-trivial form. In four dimensions, the gravi-ton and its dual field are each described by a symmetricrank-two tensor field, and a duality symmetry relatingthem is expected to emerge, inherited from the under-lying infinite-dimensional algebraic structure. This hasmotivated the search of duality-symmetric action princi-ples involving gravity [8]-[10], along the lines of the work[11] establishing a manifestly duality symmetric formu-lation of Maxwell action.It seems natural to wonder about the possibility ofderiving duality-symmetric action principles for theoriesof gravity involving higher derivatives. Among those,conformal gravity occupies a position of particular in-terest in the literature. Being constructed out of thesquare of the Weyl tensor, the action principle is invari-ant under conformal rescalings of the metric. As opposedto Einstein gravity, it is power-counting renormalizable[13],[14], albeit it presents a Ostrogradski linear instabil-ity in the Hamiltonian –due to the fourth-order characterof the equations of motion and the non-degeneracy of theLagrangian –, which is typically assumed to translateinto the presence of ghosts –negative-norm states– uponquantization. It is well known that solutions of Einsteingravity form a subset of solutions of conformal gravity, afact that has recently been exploited in [15] to show theequivalence at the classical level of Einstein gravity witha cosmological constant and conformal gravity with suit-able boundary conditions that eliminate ghosts. Anotherinteresting aspect of conformal gravity is that it admitssupersymmetric extensions for N ≤
4, the maximallysupersymmetric theory admitting different variants (see[16] for a review and [17] for recent progress). Other the-oretical advances involving conformal gravity include itsemergence from twistor string theory [18] and its appear-ance as a counterterm in the AdS/CFT correspondence[19].A generalization of electric-magnetic duality in confor-mal gravity was studied in the early work [20], where theEuclidean action with a gauge-fixed metric was expressedin terms of quadratic forms involving the electric andmagnetic components of the Weyl tensor, exhibiting adiscrete duality symmetry upon the interchange of thesecomponents. This result can be regarded as the analog ofthe duality symmetry of Euclidean Maxwell action underthe exchange of electric and magnetic fields. Unlike [11],duality is discussed in terms of the electric and magneticcomponents of the curvature, and not at the level of thedynamical degrees of freedom of the theory.In the present article we focus on linearized conformalgravity with Lorentzian signature and show that the ac-tion principle admits a manifestly duality invariant formin terms of the dynamical variables. The derivation re-quires working in the Hamiltonian formalism, the identifi-cation of the constraints –both algebraic and differential–and the resolution of the differential ones in terms ofpotentials, that we will eventually interpret as a dualmetric and a dual extrinsic curvature. The structureof the duality-symmetric action principle is new, differ-ent from duality-invariant Maxwell theory and linearizedgravity: duality acts rotating simultaneously the three-dimensional metrics ( h ij , ˜ h ij ) and the extrinsic curva-tures ( K ij , ˜ K ij ). Recall that a higher order Lagrangian is non-degenerate when-ever the highest time derivative term can be expressed in termsof the canonical variables.
The rest of the article is organized as follows. In Sec-tion II we review general features of conformal gravityand remark that, in the linearized regime, the Hodgedual of the linearized Weyl tensor obeys an identity ofthe same functional form as the equation of motion sat-isfied by the Weyl tensor itself, in complete analogy withthe symmetric character of vacuum Maxwell equationswith respect to the exchange of the field strength andits Hodge dual. Motivated by this observation, in Sec-tion III we establish a twisted self-duality form of thelinearized equations of motion of conformal gravity. Sec-tion IV deals with the generalities of the Hamiltonianformulation, including the identification of the dynami-cal variables and the constraints of the theory. To dealwith the fact that the Lagrangian contains second ordertime derivatives of the metric perturbation, we will for-mally promote the linearized extrinsic curvature to anindependent dynamical variable. Section V is dedicatedto the resolution of the differential constraints in termsof two potentials. These are interpreted as a dual three-dimensional metric and a dual extrinsic curvature. InSection VI we present a manifestly duality invariant formof the action principle, where the two metrics and extrin-sic curvatures appear on equal footing. Finally we drawour conclusions in Section VII and set out proposals forfuture work.
II. CONFORMAL GRAVITY
The action principle of conformal gravity is given by S [ g µν ] = − Z d x √− g W µνρσ W µνρσ , (II.1)with g µν the metric tensor defined on a manifold M and W µνρσ the Weyl tensor W µνρσ ≡ R µνρσ − g µ [ ρ S σ ] ν − g ν [ ρ S µσ ] ) . (II.2)Here R µνρσ and S µν are the Riemann and Schouten ten-sors, respectively. The latter is defined as S µν ≡
12 ( R µν − g µν R ) . (II.3)We adopt the convention that indices within brackets areantisymmetrized, with an overall factor of 1 /n ! for theantisymmetrization of n indices.The Weyl tensor W µνρσ is invariant under diffeomor-phisms δg µν = ∇ µ ξ ν + ∇ ν ξ µ , (II.4)and local conformal rescalings of the metric g µν → g ′ µν = Ω ( x ) g µν . (II.5)These transformations also determine the symmetries ofthe action principle (II.1).The Weyl tensor satisfies the same tensorial symmetryproperties as the Riemann tensor, W µνρσ = −W νµρσ = −W µνσρ = W ρσµν , (II.6)as well as the identities W [ µνσ ] ρ = 0 , (II.7) W µνµσ = 0 , (II.8)and ∇ µ W µνρσ = −C νρσ , (II.9)where we have introduced the Cotton tensor C νρσ ≡ ∇ [ ρ S σ ] ν . (II.10)Equation (II.9) is a consequence of the Bianchi identityfor the Riemann tensor.The fourth-order equation of motion derived from theconformal gravity action principle (II.1) reads(2 ∇ ρ ∇ σ + R σρ ) W ρµσν = 0 . (II.11)This is usually referred to as the Bach equation, theleft-hand side of (II.11) being dubbed the Bach tensor.Clearly, conformally flat metrics constitute a particularsubset of solutions to the equations of motion (II.11).Einstein metrics constitute another subset of particularsolutions. Remarks on the linearized regime
In the linearized regime g µν = η µν + h µν (II.12)the Weyl tensor takes the form W µνρσ [ h ] = R µνρσ [ h ] − δ µ [ ρ S σ ] ν [ h ] − δ ν [ ρ S µσ ] [ h ]) , (II.13)where R µνρσ is the linearized Riemann tensor R µνρσ = −
12 [ ∂ µ ∂ ρ h νσ + ∂ ν ∂ σ h µρ − ∂ µ ∂ σ h νρ − ∂ ν ∂ ρ h µσ ](II.14)and S µν the linearized Schouten tensor. The action prin-ciple and equations of motion reduce to S [ h αβ ] = − Z d xW µνρσ [ h ] W µνρσ [ h ] (II.15)and ∂ µ ∂ ν W µρνσ [ h ] = 0 . (II.16)The linearized Weyl tensor still obeys the symmetryproperties (II.6), and the identity (II.9) takes the lin-earized form ∂ µ W µνρσ [ h ] = − C νρσ [ h ] . (II.17)This allows for a rewriting of the linearized Bach equationin terms of the Cotton tensor: ∂ ρ C νρσ [ h ] = 0 . (II.18)Let us now introduce the Hodge dual of the linearizedWeyl tensor: ∗ W µνρσ [ h ] ≡ ǫ αβµν W αβρσ [ h ] . (II.19)By construction it possesses the same symmetries as W µνρσ , namely ∗ W µνρσ = ∗ W ρσµν = − ∗ W νµρσ = − ∗ W µνσρ . (II.20)It also satisfies the cyclic identity ∗ W [ µνρ ] σ = 0 (II.21)and is traceless ∗ W µνµρ = 0 . (II.22)At this point, it is crucial to observe that ∗ W µνρσ satisfiesthe following identity: ∂ µ ∂ ν ∗ W µρνσ [ h ] = 0 . (II.23)This is directly related to the identity C σ [ νρ,µ ] [ h ] = 0 (II.24)satisfied by the linearized Cotton tensor, for ∂ µ ∂ ν ∗ W µρνσ [ h ] = ∂ µ ∂ ν ǫ µραβ W νσαβ [ h ]= − ∂ µ ǫ µραβ C σαβ [ h ] = 0 . (II.25)It is now clear that the set of equations conformed by thelinearized Bach equation (II.16) and the identity (II.23) ∂ µ ∂ ρ W µνρσ [ h ] = 0 ∂ µ ∂ ρ ∗ W µνρσ [ h ] = 0 (II.26)may be regarded as the analog of Maxwell equations invacuum ∂ µ F µν [ A ] = 0 ∂ µ ∗ F µν [ A ] = 0 . (II.27)The set of equations (II.26) is symmetric under the re-placement of the Weyl tensor and its Hodge dual. III. TWISTED SELF-DUALITY FORM OF THEEQUATIONS OF MOTION
Given the formal resemblance between equations(II.27) and (II.26), it seems natural to wonder aboutthe existence of an underlying electric-magnetic dualitystructure in linearized conformal gravity. In this sec-tion we show that the set of equations (II.26) can becast in a covariant twisted self-duality form, and thatthe non-covariant subset defined by selecting the purelyspatial components of the latter contains all the informa-tion of the full covariant set –which parallels the situationin electromagnetism [21] and linearized Einstein gravity[22].In order to understand the logic underlying twistedself-duality, it is useful to briefly recall the situation inMaxwell theory. Consider the vacuum equations (II.27),where we tacitly assume F µν = ∂ µ A ν − ∂ ν A µ . Although(II.27) are symmetric under the exchange of F µν [ A ] and ∗ F µν [ A ], these quantities do not appear exactly on anequal footing: the equation for ∗ F µν is an identity. Inother words, ∗ F µν has been implicitly solved in terms ofthe potential A µ . Indeed, upon use of Poincar´e lemma,one finds ∗ F µν = ǫ µναβ ∂ α A β for some vector potential A µ , and the definition of the Hodge dual yields F µν = ∂ µ A ν − ∂ ν A µ , as expected. We seek instead a formula-tion where F µν and ∗ F µν appear on equal footing, withno implicit prioritization of any of them. This is achieved[21] by considering the field strength and its Hodgedual as independent variables, solving simultaneously forboth in terms of potentials F µν [ A ] = ∂ µ A ν − ∂ ν A µ and ∗ F µν ≡ H µν [ B ] = ∂ µ B ν − ∂ ν B µ , and finally imposing afirst-order, twisted self-duality condition that takes into account that F µν [ A ] and H µν [ B ] are not independent butactually related by Hodge dualization: ∗ (cid:18) F µν H µν (cid:19) = S (cid:18) F µν H µν (cid:19) , S = (cid:18) − (cid:19) . (III.1)Clearly, equation (III.1) implies the second-orderMaxwell equations (II.27) by taking the divergence.However, in this first-order formulation F µ [ A ] and H µν [ B ] appear on an equal footing, related to each otherby Hodge dualization.As a caveat, we notice a redundancy in the twistedself-duality equations (III.1), for either row can be ob-tained from the other by Hodge dualization. It is actu-ally possible to identify a non-covariant subset of (III.1)that is equivalent to the original set of equations and freefrom redundancies [21]. This is achieved by selecting thepurely spatial components of (III.1), which produces (cid:18) B i [ A ] B i [ B ] (cid:19) = S (cid:18) E i [ A ] E i [ B ] (cid:19) , (III.2)with E i and B i the usual electric and magnetic fields.Let us now turn the discussion to linearized conformalgravity. Since the dual Weyl tensor ∗ W µνρσ has the samealgebraic and differential properties as the Weyl tensor,it can be written itself in the same functional form as W µνρσ [ h ] for some different metric f µν : ∗ W µνρσ [ h ] = H µνρσ [ f ] , (III.3)with H µνρσ [ f ] ≡ R µνρσ [ f ] − δ µ [ ρ S σ ] ν [ f ] − δ ν [ ρ S µσ ] [ f ]) , (III.4)the relation between h µν and f µν being non-local. Sim-ilarly, it is easy to verify that the Cotton tensor and itsHodge dual have the same properties, and therefore wecan write ∗ C µνρ [ h ] = D µνρ [ f ] , (III.5)where D µνρ [ f ] has the same functional form as the Cot-ton tensor for the dual metric f µν D µνρ [ f ] = − ∂ ρ H ρµνρ [ f ] . (III.6)Exactly in parallel as what happens in electromag-netism, Hodge duality exchanges equations of motionand differential identities. In the theory defined by h µν , ∂ µ ∂ ν W µνρσ [ h ] = 0 is an equation of motion and ∂ µ ∂ ∗ ν W µνρσ [ h ] can be seen as the analogue of the Bianchiidentity in Maxwell theory. However, when we considerthe dual theory defined by f µν , the latter implies theequation of motion for the dual metric ∂ µ ∂ ν H µνρσ [ f ] = 0,whereas the former is related to the differential identity ∂ µ ∂ ν ∗ H µνρσ [ f ] = 0.Although the set of equations (II.26) is symmetric un-der the formal exchange of the Weyl tensor and its Hodgedual, implicitly we have prioritized the formulation basedon h µν , for f µν does not appear at all. Following the samelogic as discussed above in the case of electromagnetism,it is possible though to find a set of second-order equa-tions equivalent to (II.26) where both metrics appear onan equal footing. This is the twisted self-duality equationfor linearized conformal gravity: ∗ (cid:18) W µνρσ [ h ] H µνρσ [ f ] (cid:19) = S (cid:18) W µνρσ [ h ] H µνρσ [ f ] (cid:19) , S = (cid:18) − (cid:19) . (III.7)Equation (III.7) is obtained from (II.26) by solving for W µνρσ and H µνρσ ≡ ∗ W µνρσ , treated as independent fieldstrengths, and imposing the condition that they are ac-tually related by Hodge dualization. On the other hand,taking the double divergence on both sides, the twistedself-duality equation (III.7) reproduces immediately theequations of motion for W µνρσ [ h ] and H µνρσ [ f ] in virtueof the differential identities satisfied by their Hodge du-als, exactly as what happens in the Maxwell theory. Wenotice (III.7) also implies the vanishing of the trace of W µνρσ [ h ] and H µνρσ [ f ], owing to the respective cyclicidentities. This is in consistency with their definitions asthe traceless part of the Riemann tensor for the corre-sponding metrics h µν and f µν .The set of equations (III.7) is redundant, in the sensethat either row can be obtained as the Hodge dual of theother one. So we can keep only the subset of equationsassociated to the first row in (III.7): ∗ W i j = H i j ∗ W ijk = H ijk ∗ W ijkl = H ijkl . (III.8)Moreover, we see that the third equation in (III.8) canbe obtained from the second one, for ∗ W ijkl = H ijkl ⇔ ǫ ij m W kl m = H ijkl ⇔ W mkl = − ǫ mij H ijkl = − ∗ H mkl , (III.9)the last expression being the Hodge dual of the secondequation in (III.8). Thus, the only independent com-ponents of the covariant twisted self-duality equations(III.7) are ∗ W i j = H i j ∗ W ijk = H ijk . (III.10) Defining the electric component E ij and magnetic com-ponent B ij of the Weyl tensor W µνρσ as E ij [ h ] ≡ W i j [ h ] B ij [ h ] ≡ − ∗ W i j [ h ] = − ǫ imn W mn j [ h ] , (III.11)and similarly for H µνρσ E ij [ f ] ≡ H i j [ f ] , B ij [ f ] ≡ − ∗ H i j [ f ] = − ǫ imn H mn j [ f ] , (III.12)equation (III.10) can be cast in the form (cid:18) E ij [ h ] E ij [ f ] (cid:19) = S (cid:18) B ij [ h ] B ij [ f ] (cid:19) , S = (cid:18) − (cid:19) . (III.13)This equation is non-redundant and contains all the in-formation in the covariant twisted self-duality equation(III.7). It will be referred to as the non-covariant twistedself-duality equation.From their definitions, it is straightforward to see thatthe electric and magnetic components are both symmet-ric and traceless. Moreover, their double divergence van-ishes: ∂ i ∂ j E ij [ h ] = ∂ i ∂ j E ij [ f ] = ∂ i ∂ j B ij [ h ] = ∂ i ∂ j B ij [ f ] = 0 . (III.14) IV. HAMILTONIAN FORMULATION
Having established the twisted self-duality structureunderlying the equations of motion of linearized confor-mal gravity, the natural next step is to seek a formula-tion of the corresponding action principle that manifestlydisplays duality symmetry. In order to do so, we shallfollow the same strategy as in Maxwell theory [11] andlinearized gravity [8]: the Hamiltonian formulation is in-troduced by a 3+1 slicing of space-time, constraints areidentified and solved in terms of potentials, and finally amanifest duality symmetric action is written down uponsubstitution in terms of potentials. This section dealswith the Hamiltonian formulation of the theory and theidentification of the constraints.The action principle for linearized Weyl gravity is S = − Z d xW µνρσ W µνρσ . (IV.1)The squared Weyl tensor is decomposed upon a 3 + 1slicing of space-time as follows: W µνρσ W µνρσ = 4 W i j W i j + 4 W ijk W ijk + W ijkl W ijkl = 8 W i j W i j + 4 W ijk W ijk . (IV.2)In order to deal with the second-order character of theLagrangian, we shall follow the Ostrogradski method:to define a dynamical variable depending on first-orderderivatives that will be formally treated as independent,and impose afterward its definition through a Lagrangemultiplier. A natural choice is the linearized extrinsiccurvature: K ij = 12 ( ˙ h ij − ∂ i h j − ∂ j h i ) . (IV.3)It will be required then to express the Weyl tensor interms of K ij .First, let us write the Riemann tensor and its contrac-tions in terms of the extrinsic curvature. The componentsof (II.14) are: R ijkl = −
12 [ ∂ i ∂ k h jl + ∂ j ∂ l h ik − ∂ i ∂ l h jk − ∂ j ∂ k h il ] R i j = − ˙ K ij − ∂ i ∂ j h R ijk = − ( ∂ j K ik − ∂ k K ij ) . (IV.4)In turn, the components of the Ricci tensor read R = R i i = − ˙ K −
12 ∆ h R i = ∂ k K ik − ∂ i KR ij = − ˙ K ij − ∂ i ∂ j h + R kikj , (IV.5)and the scalar curvature is R = − R i i + R ijij = 2 ˙ K + ∆ h + R ijij . (IV.6)From the definition of the Weyl tensor (II.13) one derivesthe relations W i j = 12 ( R i j + R mimj ) − δ ij ( R mnmn + R m m )= 12 ( − ˙ K ij − ∂ i ∂ j h + R mimj ) − δ ij ( − ˙ K −
12 ∆ h + R mnmn )and W ijk = ∂ k K ij − ∂ j K ik + 12 ( δ ij ( ∂ l K lk − ∂ k K ) − δ ik ( ∂ l K jl − ∂ j K )) . The Lagrangian reads L = − W µνρσ W µνρσ = − (cid:20)
12 ( ˙ K ij ˙ K ij + R mimj R injn + ˙ K ij ∂ i ∂ j h − K ij R mimj − ∂ i ∂ j h R imjm ) −
16 ( ˙ K + R ijij R mnmn + ˙ K ∆ h − KR ijij − ∆ h R mnmn )+ 112 ∆ h ∆ h − W ijk W ijk (cid:21) . (IV.7)The conjugate momentum associated to K ij is defined asusual: P ij = ∂L∂ ˙ K ij = − (cid:20) ˙ K ij + 12 ∂ i ∂ j h − R imjm −
16 ∆ h δ ij + 13 R mnmn δ ij −
13 ˙ Kδ ij (cid:21) . (IV.8)We notice that the trace of P ij vanishes identically: P = 0 . (IV.9)This shall be treated as a primary constraint.The Hamiltonian is now introduced through the Leg-endre transformation H = P ij ˙ K ij − L . (IV.10)Upon substitution for the generalized velocities, it can beexpressed in terms of P ij and K ij : H = − P ij P ij − P ij ∂ i ∂ j h + R mimj P ij − W ijk W ijk . (IV.11)Now we have to take into account the fact that thedefinition of K ij actually depends on ˙ h ij by introduc-ing in the action principle the constraint term λ ij ( ˙ h ij − ∂ j h i − ∂ i h j − K ij ). The factor λ ij becomes a Lagrangemultiplier enforcing the definition of K ij : S [ h ij , p ij , K ij , P ij ] = Z d x [ P ij ˙ K ij + 12 P i ij P i ij + 12 P ij ∂ i ∂ j h − P ij R mimj + 2 ∂ j K ik ∂ j K ik + 2 ∂ k K∂ l K kl − ∂ l K∂ l K − ∂ l K kl ∂ m K km + λ ij ( ˙ h ij − ∂ j h i − ∂ i h j − K ij )] . (IV.12)Clearly one can identify the Lagrange multiplier λ ij withthe conjugate momentum associated to h ij , p ij ≡ λ ij .Upon integration by parts, the components h and h j act now as Lagrange multipliers imposing the constraints ∂ i ∂ j P ij = 0 (IV.13)and ∂ j p ij = 0 . (IV.14)The latter reads exactly as in linearized gravity [8]. Ig-noring total derivatives coming from the previous integra-tion by parts, the final form of the Hamiltonian actionprinciple is S [ h ij , p ij , K ij , P ij , h i , h ] = Z d x h P ij ˙ K ij + p ij ˙ h ij −H + 2 ∂ j p ij h i − ∂ i ∂ j P ij h (cid:21) (IV.15)with − H = 12 P ij P ij − P ij R mimj − p ij K ij + 2 ∂ j K ik ∂ j K ik + 2 ∂ k K∂ l K kl − ∂ l K∂ l K − ∂ l K kl ∂ m K km . (IV.16)One notices in (IV.16) the presence of terms linear in theconjugate momenta, which points out the Ostrogradskilinear instability of the theory.There is an additional constraint that comes about bydemanding the preservation of the constraint (IV.9) un-der time evolution. In other words, the Poisson bracketof the constraint (IV.9) with the Hamiltonian should van-ish: { P, Z d x H} = 0 . (IV.17)This results in the constraint p = 0 . (IV.18)The consistency condition applied to this constraint doesnot produce any further ones.Adding up the traceless constraints (IV.9) and (IV.18),the action principle reads S [ h ij , p ij , K ij , P ij , h i , h , λ , λ ]= Z d x (cid:20) P ij ˙ K ij + p ij ˙ h ij − H + 2 ∂ j p ij h i − ∂ i ∂ j P ij h + λ P + λ p ] . (IV.19)The gauge transformations of the dynamical variables are δh ij = ∂ i ξ j + ∂ j ξ i + δ ij ξδK ij = 12 [ − ∂ i ∂ j ξ + δ ij ˙ ξ ] δp ij = 0 δP ij = 0 . (IV.20)These can be obtained directly from the definition ofthe dynamical variables in terms of the components ofthe four-dimensional metric h µν . It is straightforwardto verify that the constraints (V.1), (V.2), (IV.9) and(IV.18) are first class, so the previous gauge transfor-mations can also be derived from the Poisson bracketwith the constraints. We notice that p = 0 generates theWeyl rescaling for h ij , whereas ∂ j p ij is responsible forthe same three-dimensional diffeomorphism invariance oflinearized Einstein gravity. V. RESOLUTION OF THE CONSTRAINTS
We shall now focus on the resolution of the differentialconstraints ∂ i ∂ j P ij = 0 (V.1)and ∂ j p ij = 0 , (V.2)subject to the traceless constraints P = 0 (V.3)and p = 0 . (V.4)Let us first focus on (V.1). Taking into account that P =0, this can be solved in terms of some tensor potential ψ ij as follows: P ij = ǫ imn ∂ m ψ nj + ǫ jmn ∂ m ψ ni . (V.5)Because of the traceless condition on P ij , the antisym-metric component of ψ ij is restricted to have the form ψ [ ij ] = ∂ i w j − ∂ j w i . (V.6)However, by the redefinition of the symmetric componentof the potential ψ ( ij ) ≡ φ ij + ∂ i w j + ∂ j w i the solutionsimply takes the form P ij = ǫ imn ∂ m φ nj + ǫ jmn ∂ m φ ni , (V.7)with φ ij a symmetric tensor. Note that, since δP ij = 0,the ambiguities in the definition of the potential φ ij arerestricted to have the form δφ ij = ∂ i ∂ j ξ + δ ij θ. (V.8)This has exactly the same form as δK ij , which alreadysuggests that K ij and φ ij can be treated on equal footing,and justifies the renaming φ ij ≡ ˜ K ij .In order to solve the constraint (V.2), we may use thePoincar´e lemma and write p ij = ǫ imn ∂ m ω nj + ǫ jmn ∂ m ω ni (V.9)for some symmetric potential ω ij , bearing in mind thefact that p ij is symmetric and traceless. However, theadditional condition ∂ i p ij = 0 must be fulfilled, whichimplies that ∂ i ǫ jmn ∂ m ω ni = 0 (V.10)should be identically satisfied. A particular choice of ω ij that fulfills this condition is ω ij = R ij [˜ h ] , (V.11)with R ij [˜ h ] having the functional form of the linearizedthree-dimensional Ricci tensor for some symmetric tensor˜ h ij –to be interpreted in the sequel as a second, dualmetric. Indeed, we see that ∂ i ǫ jmn ∂ m R ni [˜ h ] = ǫ jmn ∂ m (∆ ∂ i ˜ h ik − ∂ l ∆˜ h kl ) = 0 . (V.12)More generally, the condition (V.10) is identically satis-fied for ω ij = R ij [˜ h ] + δ ij s + ∂ i ∂ j s . (V.13)with s and s undetermined scalar functions. The con-tribution to (V.9) from the last two terms in (V.13) van-ishes, so they can be ignored in practice.We shall write then p ij = − (cid:16) ǫ imn ∂ m R nj [˜ h ] + ǫ jmn ∂ m R ni [˜ h ] (cid:17) , (V.14)where the global factor has been chosen for future con-venience. This expression is invariant under transforma-tions of ˜ h ij having the same form as those defining thesymmetries of conformal gravity, δ ˜ h ij = ∂ i χ j + ∂ j χ i + δ ij χ, (V.15)as we could have expected. VI. MANIFEST DUALITY INVARIANCE
We can now implement equations (V.7) and (V.14) inthe action principle (IV.19). Let us first we compute thequadratic term in P ij :12 P ij P ij = 2 ∂ j ˜ K ik ∂ j ˜ K ik + 2 ∂ k ˜ K∂ l ˜ K kl − ∂ l ˜ K∂ l ˜ K − ∂ l ˜ K kl ∂ m ˜ K km . (VI.1)Remarkably, this has exactly the same form as thequadratic terms in K ij appearing in the Hamiltonian(IV.16), and suggests an invariance under the transfor-mation K ij → ˜ K ij , ˜ K ij → − K ij . (VI.2)The kinetic term P ij ˙ K ij is also invariant under (VI.2)(up to total derivatives): P ij ˙ K ij = 2 ǫ imn ∂ m ˜ K jn ˙ K ij → − ǫ imn ∂ m K jn ˙˜ K ij = 2 ǫ imn ∂ m ˜ K jn ˙ K ij + total derivatives . (VI.3)Substituting now in the term − p ij K ij , we find − p ij K ij = 2 ǫ imn ∂ m R nj [˜ h ] K ij = − ǫ imn R nj [˜ h ] ∂ m K ij + total derivatives . (VI.4)We may compare this expression with the term − P ij R ij [ h ] = − ǫ imn ∂ m ˜ K nj R ij [ h ] (VI.5)and see that these two are rotated into each other by thetransformation (VI.2) supplemented by h ij → ˜ h ij ˜ h ij → − h ij . (VI.6)The kinetic term ˙ h ij p ij = − ˙ h ij ǫ imn ∂ m R nj [˜ h ] is also in-variant under (VI.6) up to total derivatives: − ˙ h ij ǫ imn ∂ m R nj [˜ h ] = 12 ˙ h ij ǫ imn ∂ m (∆˜ h nj − ∂ j ∂ k ˜ h kn ) → −
12 ˙˜ h ij ǫ imn ∂ m (∆ h nj − ∂ j ∂ k h kn )= 12 ˙ h ij ǫ imn ∂ m (∆˜ h nj − ∂ j ∂ k ˜ h kn ) + total derivatives . (VI.7)So we conclude that, once the constraints are solved, theaction principle (IV.19) can be cast in the manifestlyduality invariant form S [ h ij , ˜ h ij , K ij , ˜ K ij , ] = Z d x [2 ǫ imn ∂ m ˜ K jn ˙ K ij − ˙ h ij ǫ imn ∂ m R nj [˜ h ] − H ] , (VI.8)where − H = 2 ǫ imn ∂ m ˜ K nj R ij [ h ] − ǫ imn ∂ m K ij R nj [˜ h ]2 ∂ j K ik ∂ j K ik + 2 ∂ k K∂ l K kl − ∂ l K∂ l K − ∂ l K kl ∂ m K km + 2 ∂ j ˜ K ik ∂ j ˜ K ik + 2 ∂ k ˜ K∂ l ˜ K kl − ∂ l ˜ K∂ l ˜ K − ∂ l ˜ K kl ∂ m ˜ K km (VI.9)and we have dropped surface terms. One can verify thatthe action principle (VI.8) is actually invariant undercontinuous duality rotations of the dual metrics and ex-trinsic curvatures: (cid:18) h ij ˜ h ij (cid:19) = (cid:18) cos θ sin θ − sin θ cos θ (cid:19) (cid:18) h ij ˜ h ij (cid:19)(cid:18) K ij ˜ K ij (cid:19) = (cid:18) cos θ sin θ − sin θ cos θ (cid:19) (cid:18) K ij ˜ K ij (cid:19) . (VI.10) VII. CONCLUSIONS
We have shown that electric-magnetic duality is a hid-den symmetry of linearized conformal gravity, both at thelevel of the equations of motion and the action principle.In order to render the symmetry manifest, i.e. to estab-lish a formulation where the “electric” and “magnetic”degrees of freedom appear on equal footing, it seems nec-essary to work in a non-manifestly space-time covariantframework. The covariant equations of motion and differ-ential identities obeyed by the Weyl tensor and its Hodgedual can be recovered from a non-covariant subset of thetwisted self-duality equation, where the electric and mag-netic components of the Weyl tensors for two dual metricsappear on equal footing. The action principle is cast in a manifestly duality-invariant form as well, upon resolutionof the constraints in the Hamiltonian formalism. The po-tentials that solve these constraints are interpreted as adual three-dimensional metric and a dual extrinsic cur-vature. Duality acts as simultaneous rotations in the re-spective spaces spanned by the two metrics and the twoextrinsic curvatures.There are several interesting directions for future workto be discussed. An important question is to determinewhether a manifestly duality invariant action principlecan be obtained upon linearization around more generalbackgrounds, in particular (anti) de Sitter space-time.The precise relation between the equations of motion ob-tained from the duality-symmetric action principle andthe non-covariant twisted self-duality equation should bedetermined. Supersymmetric extensions can also be in-vestigated, along the lines of the work [23]. Although wehave not dealt with topological terms, it may be inter-esting to study the consequences of their presence: forinstance, to investigate if they can cancel out the to-tal derivatives produced by integration by parts in theprocess of rending the action principle in its manifestlyduality-invariant form. Manifest space-time covarianceof the action principle might be restored upon introduc-tion of auxiliary fields, although those are expected eitherto enter in a non-polynomial fashion [24] or to appear ininfinite number [25]. Possible obstructions to manifestduality invariance at higher perturbative orders shouldalso be explored [26].Electric-magnetic duality in abelian Yang-Mills theoryhas been discussed at the quantum level in the path-integral formulation [27]. Here one adds to the abelianaction S [ A ] a term i R B ∧ dF featuring an additional1-form field B , such that integrating over it producesa delta functional δ ( dF ) allowing integration over notnecessarily closed 2-forms F . If we instead integrate over F , the partition function written as an integral over B takes the same form as expressed in terms of the original1-form field A , but interchanging the coupling constant e and the θ -parameter. Whether a similar analysis canbe performed in linearized conformal gravity seems anavenue worth exploring. [1] J. Ehlers, “Transformation of static exterior solutions ofEinstein’s gravitational field equations into different solu-tions by means of conformal mappings,” in “Les Theoriesrelativistes de la gravitation”, Colloques Internationauxdu CNRS 91, 275 (1962).[2] R. P. Geroch, “A Method for generating solutions ofEinstein’s equations,” J. Math. Phys. , 918 (1971);R. P. Geroch, “A Method for generating new solutions ofEinstein’s equations. 2,” J. Math. Phys. , 394 (1972).[3] E. Cremmer and B. Julia, “The SO(8) Supergravity,” Nucl. Phys. B , 141 (1979).[4] B. Julia, “Group disintegrations,” in “Superspace andSupergravity”, Hawking, S.W., and Ro˘cek, M., eds.,Nuffield Gravity Workshop, Cambridge, England, June22 - July 12, 1980 (Cambridge University Press, Cam-bridge, U.K.; New York, U.S.A., 1981) p. 331-350;B. L. Julia, “Dualities in the classical supergravity lim-its: Dualizations, dualities and a detour via (4k+2)-dimensions,” in *Strings, branes and dualities. Proceed-ings, NATO Advanced Study Institute, Cargese, France, May 26-June 14, 1997* Ed. L. Baulieu, P. Di Francesco,M. Douglas, V. Kazakov, M. Picco, P. Windey, NATOSci.Ser.C 520 (1999) Dordrecht, Netherlands, p. 121-139.[5] T. Damour, M. Henneaux and H. Nicolai, “E10 and a“small tension expansion” of M Theory”, Phys. Rev.Lett. 89 221601 (2002).[6] P. C. West, “E(11) and M theory,” Class. Quant. Grav. , 4443 (2001).[7] N. D. Lambert, P. C. West, “Coset Symmetries in Dimen-sionally Reduced Bosonic String Theory,” Nucl. Phys. B , 117 (2001).[8] M. Henneaux and C. Teitelboim, “Duality in linearizedgravity,” Phys. Rev. D , 024018 (2005).[9] B. Julia, J. Levie and S. Ray, “Gravitational duality nearde Sitter space,” JHEP0511