Manifest electric-magnetic duality in linearized conformal gravity

We derive a manifestly duality-symmetric formulation of the action principle for conformal gravity linearized 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 dual three-dimensional metric and a dual extrinsic curvature. The action principle can be written in terms of these four dynamical variables, duality acting as simultaneous rotations in the respective spaces spanned by the three-dimensional metrics and the extrinsic curvatures. A twisted self-duality formulation of the equations of motion is also provided.


I. INTRODUCTION
Understanding dualities is a major challenge in modern theoretical physics. Despite their widespread presence in diverse branches -including field theory, (super)gravity, string theory and condensed matter-a comprehensive theoretical framework that explains the origin of the phenomenon and describes its full implications is lacking. In the case of gravitational theories, dualities are intimately related to the emergence of hidden symmetries upon toroidal compactifications (supplemented by Hodge dualizations of Kaluza-Klein fields). For instance, it has long been recognized that the reduction to three dimensions of the four-dimensional Einstein-Hilbert action in the presence of a Killing vector, followed by the dualization of the Kaluza-Kelin vector to a scalar, exhibits a SL(2, R) invariance acting on the scalar sector. The latter, conformed by a dilaton and axion, parametrizes a SL(2, R)/SO(2) coset space. This SL(2, R) invariance is commonly referred to in the literature as the Ehlers symmetry [1]. In the presence of two commuting Killing vectors, the reduction to two dimensions can be achieved in two different ways: either by a direct compactification to two dimensions, or by a reduction to three dimensions followed by a dualization of the vector field to a scalar and a final reduction to two dimensions. Each of these routes yields a different SL(2, R)/SO(2) sigma model in the scalar sector. The first one is associated to the SL(2, R) Matzner-Misner group, which originates from coordinate transformations preserving the vector space spanned by the Killing vectors. In the second case the SL(2, R) group is properly the Ehlers group. The intertwining of the Matzner-Misner and Ehlers groups generates an infinite-dimensional group realized non-locally, as described by Geroch [2]. * Electronic address: sergio.hortner@uva.nl A similar situation occurs in supergravity [3]: the (11d) toroidal compactification of the eleven-dimensional theory yields maximally supersymmetric supergravity in dimension d with a symmetry structure hidden within the non-gravitational degrees of freedom in the reduced bosonic sector, namely p-forms and scalars. After Hodge dualization of the p-forms to their lowest possible rank, they combine in an irreducible representation of a noncompact group G acting globally, whereas the scalar sector is described by the non-linear sigma model G/H, with H the maximal compact subgroup of G. In even dimensions, the global symmetry G is realized as an electricmagnetic duality transformation interchanging equations of motion and Bianchi identities. Reduction to five, four and three dimensions yields as G the exceptional Lie groups E 6(6) , E 7(7) and E 8 (8) , respectively.
The study of the rich algebraic structure that underlies the emergence of hidden symmetries in compactifications of (super)gravity has led to conjecture the existence of an infinite-dimensional Kac-Moody algebra acting as a fundamental symmetry of the uncompactified theory [4]- [7], encompassing the duality symmetries that appear upon dimensional reduction. A key property of these algebras is that they involve all the bosonic fields in the theory and their Hodge duals, including the graviton and its dual field. The associated symmetry transformation for a given tensor field relates it to all the rest of the fields, regardless their tensor structure, and in general has a highly non-trivial form. In four dimensions, the graviton and its dual field are each described by a symmetric rank-two tensor field, and a duality symmetry relating them is expected to emerge, inherited from the underlying infinite-dimensional algebraic structure. This has motivated the search of duality-symmetric action principles involving gravity [8]- [10], along the lines of the work [11] establishing a manifestly duality symmetric formulation of Maxwell action.
It seems natural to wonder about the possibility of deriving duality-symmetric action principles for theories of gravity involving higher derivatives. Among those, conformal gravity occupies a position of particular interest in the literature. Being constructed out of the square of the Weyl tensor, the action principle is invariant under conformal rescalings of the metric. As opposed to Einstein gravity, it is power-counting renormalizable [13], [14], albeit it presents a Ostrogradski linear instability in the Hamiltonian -due to the fourth-order character of the equations of motion and the non-degeneracy of the Lagrangian 1 -, which is typically assumed to translate into the presence of ghosts -negative-norm states-upon quantization. It is well known that solutions of Einstein gravity form a subset of solutions of conformal gravity, a fact that has recently been exploited in [15] to show the equivalence at the classical level of Einstein gravity with a cosmological constant and conformal gravity with suitable boundary conditions that eliminate ghosts. Another interesting aspect of conformal gravity is that it admits supersymmetric extensions for N ≤ 4, the maximally supersymmetric theory admitting different variants (see [16] for a review and [17] for recent progress). Other theoretical advances involving conformal gravity include its emergence from twistor string theory [18] and its appearance as a counterterm in the AdS/CFT correspondence [19].
A generalization of electric-magnetic duality in conformal gravity was studied in the early work [20], where the Euclidean action with a gauge-fixed metric was expressed in terms of quadratic forms involving the electric and magnetic components of the Weyl tensor, exhibiting a discrete duality symmetry upon the interchange of these components. This result can be regarded as the analog of the duality symmetry of Euclidean Maxwell action under the exchange of electric and magnetic fields. Unlike [11], duality is discussed in terms of the electric and magnetic components of the curvature, and not at the level of the dynamical degrees of freedom of the theory.
In the present article we focus on linearized conformal gravity with Lorentzian signature and show that the action principle admits a manifestly duality invariant form in terms of the dynamical variables. The derivation requires working in the Hamiltonian formalism, the identification of the constraints -both algebraic and differentialand the resolution of the differential ones in terms of potentials, that we will eventually interpret as a dual metric and a dual extrinsic curvature. The structure of the duality-symmetric action principle is new, different from duality-invariant Maxwell theory and linearized gravity: duality acts rotating simultaneously the threedimensional metrics (h ij ,h ij ) and the extrinsic curvatures (K ij ,K ij ).
The rest of the article is organized as follows. In Section II we review general features of conformal gravity and remark that, in the linearized regime, the Hodge dual of the linearized Weyl tensor obeys an identity of the same functional form as the equation of motion satisfied by the Weyl tensor itself, in complete analogy with the symmetric character of vacuum Maxwell equations with respect to the exchange of the field strength and its Hodge dual. Motivated by this observation, in Section III we establish a twisted self-duality form of the linearized equations of motion of conformal gravity. Section IV deals with the generalities of the Hamiltonian formulation, including the identification of the dynamical variables and the constraints of the theory. To deal with the fact that the Lagrangian contains second order time derivatives of the metric perturbation, we will formally promote the linearized extrinsic curvature to an independent dynamical variable. Section V is dedicated to the resolution of the differential constraints in terms of two potentials. These are interpreted as a dual threedimensional metric and a dual extrinsic curvature. In Section VI we present a manifestly duality invariant form of the action principle, where the two metrics and extrinsic curvatures appear on equal footing. Finally we draw our conclusions in Section VII and set out proposals for future work.

II. CONFORMAL GRAVITY
The action principle of conformal gravity is given by with g µν the metric tensor defined on a manifold M and W µ νρσ the Weyl tensor Here R µ νρσ and S µν are the Riemann and Schouten tensors, respectively. The latter is defined as We adopt the convention that indices within brackets are antisymmetrized, with an overall factor of 1/n! for the antisymmetrization of n indices.
The Weyl tensor W µ νρσ is invariant under diffeomorphisms and local conformal rescalings of the metric These transformations also determine the symmetries of the action principle (II.1).
The Weyl tensor satisfies the same tensorial symmetry properties as the Riemann tensor, as well as the identities where we have introduced the Cotton tensor Equation (II.9) is a consequence of the Bianchi identity for the Riemann tensor.
The fourth-order equation of motion derived from the conformal gravity action principle (II.1) reads This is usually referred to as the Bach equation, the left-hand side of (II.11) being dubbed the Bach tensor. Clearly, conformally flat metrics constitute a particular subset of solutions to the equations of motion (II.11). Einstein metrics constitute another subset of particular solutions.

Remarks on the linearized regime
In the linearized regime the Weyl tensor takes the form where R µνρσ is the linearized Riemann tensor (II.14) and S µν the linearized Schouten tensor. The action principle and equations of motion reduce to and The linearized Weyl tensor still obeys the symmetry properties (II.6), and the identity (II.9) takes the linearized form This allows for a rewriting of the linearized Bach equation in terms of the Cotton tensor: Let us now introduce the Hodge dual of the linearized Weyl tensor: By construction it possesses the same symmetries as It also satisfies the cyclic identity * W [µνρ]σ = 0 (II. 21) and is traceless * W µ νµρ = 0. (II.22) At this point, it is crucial to observe that * W µνρσ satisfies the following identity: This is directly related to the identity satisfied by the linearized Cotton tensor, for It is now clear that the set of equations conformed by the linearized Bach equation (II.16) and the identity (II.23) may be regarded as the analog of Maxwell equations in vacuum The set of equations (II.26) is symmetric under the replacement of the Weyl tensor and its Hodge dual.

III. TWISTED SELF-DUALITY FORM OF THE EQUATIONS OF MOTION
Given the formal resemblance between equations (II.27) and (II.26), it seems natural to wonder about the existence of an underlying electric-magnetic duality structure in linearized conformal gravity. In this section we show that the set of equations (II.26) can be cast in a covariant twisted self-duality form, and that the non-covariant subset defined by selecting the purely spatial components of the latter contains all the information of the full covariant set -which parallels the situation in electromagnetism [21] and linearized Einstein gravity [22].
In order to understand the logic underlying twisted self-duality, it is useful to briefly recall the situation in Maxwell theory. Consider the vacuum equations (II.27), where we tacitly assume , these quantities do not appear exactly on an equal footing: the equation for * F µν is an identity. In other words, * F µν has been implicitly solved in terms of the potential A µ . Indeed, upon use of Poincaré 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 formulation where F µν and * F µν appear on equal footing, with no implicit prioritization of any of them. This is achieved [21] by considering the field strength and its Hodge dual as independent variables, solving simultaneously for both in terms of potentials As a caveat, we notice a redundancy in the twisted self-duality equations (III.1), for either row can be obtained from the other by Hodge dualization. It is actually possible to identify a non-covariant subset of (III.1) that is equivalent to the original set of equations and free from redundancies [21]. This is achieved by selecting the purely spatial components of (III.1), which produces with E i and B i the usual electric and magnetic fields.
Let us now turn the discussion to linearized conformal gravity. Since the dual Weyl tensor * W µνρσ has the same algebraic and differential properties as the Weyl tensor, it can be written itself in the same functional form as can be seen as the analogue of the Bianchi identity in Maxwell theory. However, when we consider the dual theory defined by f µν , the latter implies the equation of motion for the dual metric ∂ µ ∂ ν H µνρσ [f ] = 0, whereas the former is related to the differential identity Although the set of equations (II.26) is symmetric under the formal exchange of the Weyl tensor and its Hodge dual, implicitly we have prioritized the formulation based on h µν , for f µν does not appear at all. Following the same logic as discussed above in the case of electromagnetism, it is possible though to find a set of second-order equations equivalent to (II.26) where both metrics appear on an equal footing. This is the twisted self-duality equation for linearized conformal gravity: This equation is non-redundant and contains all the information in the covariant twisted self-duality equation (III.7). It will be referred to as the non-covariant twisted self-duality equation.
From their definitions, it is straightforward to see that the electric and magnetic components are both symmetric and traceless. Moreover, their double divergence vanishes: (III.14)

IV. HAMILTONIAN FORMULATION
Having established the twisted self-duality structure underlying the equations of motion of linearized conformal gravity, the natural next step is to seek a formulation of the corresponding action principle that manifestly displays duality symmetry. In order to do so, we shall follow the same strategy as in Maxwell theory [11] and linearized gravity [8]: the Hamiltonian formulation is introduced by a 3+1 slicing of space-time, constraints are identified and solved in terms of potentials, and finally a manifest duality symmetric action is written down upon substitution in terms of potentials. This section deals with the Hamiltonian formulation of the theory and the identification of the constraints.
The action principle for linearized Weyl gravity is The squared Weyl tensor is decomposed upon a 3 + 1 slicing of space-time as follows: In order to deal with the second-order character of the Lagrangian, we shall follow the Ostrogradski method: to define a dynamical variable depending on first-order derivatives that will be formally treated as independent, and impose afterward its definition through a Lagrange multiplier. A natural choice is the linearized extrinsic curvature: 3) It will be required then to express the Weyl tensor in terms of K ij .
First, let us write the Riemann tensor and its contractions in terms of the extrinsic curvature. The components of (II.14) are: In turn, the components of the Ricci tensor read and the scalar curvature is From the definition of the Weyl tensor (II.13) one derives the relations and The Lagrangian reads The conjugate momentum associated to K ij is defined as usual: We notice that the trace of P ij vanishes identically: This shall be treated as a primary constraint.
The Hamiltonian is now introduced through the Legendre transformation Upon substitution for the generalized velocities, it can be expressed in terms of P ij and K ij : (IV.11) Now we have to take into account the fact that the definition of K ij actually depends onḣ ij by introducing in the action principle the constraint term λ ij (ḣ ij − ∂ j h 0i −∂ i h 0j −2K ij ). The factor λ ij becomes a Lagrange multiplier enforcing the definition of K ij : Clearly one can identify the Lagrange multiplier λ ij with the conjugate momentum associated to h ij , p ij ≡ λ ij . Upon integration by parts, the components h 00 and h 0j act now as Lagrange multipliers imposing the constraints and ∂ j p ij = 0. (IV.14) The latter reads exactly as in linearized gravity [8]. Ignoring total derivatives coming from the previous integration by parts, the final form of the Hamiltonian action principle is (IV.16) One notices in (IV.16) the presence of terms linear in the conjugate momenta, which points out the Ostrogradski linear instability of the theory.
There is an additional constraint that comes about by demanding the preservation of the constraint (IV.9) under time evolution. In other words, the Poisson bracket of the constraint (IV.9) with the Hamiltonian should vanish: This results in the constraint The consistency condition applied to this constraint does not produce any further ones.
Adding up the traceless constraints (IV.9) and (IV.18), the action principle reads The gauge transformations of the dynamical variables are These can be obtained directly from the definition of the dynamical variables in terms of the components of the four-dimensional metric h µν . It is straightforward to verify that the constraints (V.1), (V.2), (IV.9) and (IV.18) are first class, so the previous gauge transformations can also be derived from the Poisson bracket with the constraints. We notice that p = 0 generates the Weyl rescaling for h ij , whereas ∂ j p ij is responsible for the same three-dimensional diffeomorphism invariance of linearized Einstein gravity.

V. RESOLUTION OF THE CONSTRAINTS
We shall now focus on the resolution of the differential constraints 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: Because of the traceless condition on P ij , the antisymmetric component of ψ ij is restricted to have the form However, by the redefinition of the symmetric component of the potential ψ (ij) ≡ φ ij + ∂ i w j + ∂ j w i the solution simply takes the form with φ ij a symmetric tensor. Note that, since δP ij = 0, the ambiguities in the definition of the potential φ ij are restricted to have the form This has exactly the same form as δK ij , which already suggests 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 the Poincaré lemma and write for some symmetric potential ω ij , bearing in mind the fact that p ij is symmetric and traceless. However, the additional condition ∂ i p ij = 0 must be fulfilled, which implies that should be identically satisfied. A particular choice of ω ij that fulfills this condition is having the functional form of the linearized three-dimensional Ricci tensor for some symmetric tensor h ij -to be interpreted in the sequel as a second, dual metric. Indeed, we see that (V.12) More generally, the condition (V.10) is identically satisfied for (V. 13) with s 1 and s 2 undetermined scalar functions. The contribution to (V.9) from the last two terms in (V.13) vanishes, so they can be ignored in practice.
We shall write then where the global factor has been chosen for future convenience. This expression is invariant under transformations ofh ij having the same form as those defining the symmetries of conformal gravity, as we could have expected.

VI. MANIFEST DUALITY INVARIANCE
We can now implement equations (V.7) and (V.14) in the action principle (IV. 19). Let us first we compute the quadratic term in P ij : Remarkably, this has exactly the same form as the quadratic terms in K ij appearing in the Hamiltonian (IV. 16), and suggests an invariance under the transformation The kinetic term P ijK ij is also invariant under (VI.2) (up to total derivatives): Substituting now in the term −2p ij K ij , we find We may compare this expression with the term and see that these two are rotated into each other by the transformation (VI.2) supplemented by The kinetic termḣ ij p ij = −ḣ ij ǫ imn ∂ m R nj [h] is also invariant under (VI.6) up to total derivatives: So we conclude that, once the constraints are solved, the action principle (IV. 19) can be cast in the manifestly duality invariant form and we have dropped surface terms. One can verify that the action principle (VI.8) is actually invariant under continuous duality rotations of the dual metrics and extrinsic curvatures: (VI.10)

VII. CONCLUSIONS
We have shown that electric-magnetic duality is a hidden symmetry of linearized conformal gravity, both at the level of the equations of motion and the action principle. In order to render the symmetry manifest, i.e. to establish a formulation where the "electric" and "magnetic" degrees of freedom appear on equal footing, it seems necessary to work in a non-manifestly space-time covariant framework. The covariant equations of motion and differential identities obeyed by the Weyl tensor and its Hodge dual can be recovered from a non-covariant subset of the twisted self-duality equation, where the electric and magnetic components of the Weyl tensors for two dual metrics appear on equal footing. The action principle is cast in a manifestly duality-invariant form as well, upon resolution of the constraints in the Hamiltonian formalism. The potentials that solve these constraints are interpreted as a dual three-dimensional metric and a dual extrinsic curvature. Duality acts as simultaneous rotations in the respective spaces spanned by the two metrics and the two extrinsic curvatures.
There are several interesting directions for future work to be discussed. An important question is to determine whether a manifestly duality invariant action principle can be obtained upon linearization around more general backgrounds, in particular (anti) de Sitter space-time. The precise relation between the equations of motion obtained from the duality-symmetric action principle and the non-covariant twisted self-duality equation should be determined. Supersymmetric extensions can also be investigated, along the lines of the work [23]. Although we have not dealt with topological terms, it may be interesting to study the consequences of their presence: for instance, to investigate if they can cancel out the total derivatives produced by integration by parts in the process of rending the action principle in its manifestly duality-invariant form. Manifest space-time covariance of the action principle might be restored upon introduction of auxiliary fields, although those are expected either to enter in a non-polynomial fashion [24] or to appear in infinite number [25]. Possible obstructions to manifest duality invariance at higher perturbative orders should also be explored [26].
Electric-magnetic duality in abelian Yang-Mills theory has been discussed at the quantum level in the pathintegral formulation [27]. Here one adds to the abelian action S[A] a term i B ∧ dF featuring an additional 1-form field B, such that integrating over it produces a delta functional δ(dF ) allowing integration over not necessarily 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 original 1-form field A, but interchanging the coupling constant e 2 and the θ-parameter. Whether a similar analysis can be performed in linearized conformal gravity seems an avenue worth exploring.