Manifest Electric-Magnetic Duality in Linearized Conformal Gravity

PHYSICAL REVIEW D 103, 105014 (2021)

Manifest electric-magnetic duality in linearized conformal gravity

Hap´e Fuhri Snethlage and Sergio Hörtner * Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1090 GL Amsterdam, Netherlands

(Received 5 March 2021; accepted 29 April 2021; published 18 May 2021)

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.

DOI: 10.1103/PhysRevD.103.105014

I. INTRODUCTION whereas the scalar sector is described by the nonlinear sigma model G=H, with H the maximal compact subgroup Understanding dualities is a major challenge in modern of G. In even dimensions, the global symmetry G is theoretical physics. Despite their widespread presence in realized as an electric-magnetic duality transformation field theory, (super)gravity and string theory, the current interchanging equations of motion and Bianchi identities. understanding of their origins and full implications is rather Reduction to five, four and three dimensions yields as G the limited. In the case of gravitational theories, dualities are exceptional Lie groups E6ð6Þ, E7ð7Þ and E8ð8Þ, respectively. intimately related to the emergence of hidden symmetries upon toroidal compactifications. For instance, it has long The study of the rich algebraic structure that underlies been recognized that the reduction to three dimensions of the emergence of hidden symmetries in compactifications the four-dimensional Einstein-Hilbert action in the pres- of (super)gravity has led to conjecture the existence of an infinite-dimensional Kac-Moody algebra acting as a fun- ence of a Killing vector, followed by the dualization of the – Kaluza-Klein vector to a scalar, exhibits a SLð2;RÞ damental symmetry of the uncompactified theory [4 7], invariance acting on the scalar sector, commonly referred encompassing the duality symmetries that appear upon to in the literature as the Ehlers symmetry, with its SOð2Þ dimensional reduction. A key property of these algebras is subgroup [1] acting as a gravitational analogue of electric- that they involve all the bosonic fields in the theory and magnetic duality. In the presence of two commuting Killing their Hodge duals, including the graviton and its dual field. vectors, the reduction to two dimensions yields the infinite- In four dimensions, the graviton and its dual are each dimensional Geroch group [2] as a hidden symmetry. described by a symmetric rank-two tensor field, and a A similar phenomenon occurs in supergravity [3]: the duality symmetry relating them is expected to emerge, (11-d) toroidal compactification of the eleven-dimensional inherited from the underlying infinite-dimensional theory yields maximally supersymmetric supergravity in algebraic structure. This has motivated the search of dimension d with a symmetry structure hidden within the duality-symmetric action principles involving gravity. ð2Þ nongravitational degrees of freedom in the reduced bosonic A SO -invariant action principle for linearized gravity – 1 sector. After Hodge dualization of the p-forms to their has been derived [8 10], making use of the Hamiltonian lowest possible rank, they combine in an irreducible formalism and therefore breaking manifest space-time representation of a non-compact group G acting globally, covariance, along the lines of [13]. Duality acts as rotations among potentials that solve the constraints in the canonical

*[email protected] 1Interestingly, gravitational duality at the linearized level and Published by the American Physical Society under the terms of electric-magnetic duality have been related via the double copy the Creative Commons Attribution 4.0 International license. [11,12]: electric-magnetic duality rotates a Coulomb charge into a Further distribution of this work must maintain attribution to dyon, and the double copy maps this process as the trans- the author(s) and the published article’s title, journal citation, formation of linearized Schwarzschild into linearized Taub-NUT and DOI. Funded by SCOAP3. produced by gravitational duality.

2470-0010=2021=103(10)=105014(10) 105014-1 Published by the American Physical Society HAPE´ FUHRI SNETHLAGE and SERGIO HÖRTNER PHYS. REV. D 103, 105014 (2021) formalism. On the other hand, a different approach based the constraints—both algebraic and differential—and the on the light-cone formalism [14] suggests that the Ehlers resolution of the differential ones in terms of potentials, that symmetry in three dimensions can be lifted to the unre- we will eventually interpret as a dual metric and a dual duced theory, appearing as a remnant of an uplift of E8ð8Þ to extrinsic curvature. The structure of the duality-symmetric four dimensions obtained by a field redefinition in d ¼ 3 action principle is new, different from duality-invariant and a subsequent “oxidation” procedure back to d ¼ 4, and Maxwell theory and linearized gravity: duality acts rotating ˜ provides some hope for a nonlinear extension of the simultaneously the three-dimensional metrics ðhij; hijÞ and ˜ duality-invariant action in [8]. the extrinsic curvatures ðKij; KijÞ. It seems natural to wonder about the possibility of The rest of the article is organized as follows. In Sec. II deriving duality-symmetric action principles for theories we review general features of conformal gravity and remark of gravity involving higher derivatives. Among those, that, in the linearized regime, the Hodge dual of the conformal gravity occupies a position of particular interest linearized Weyl tensor obeys an identity of the same in the literature. Being constructed out of the square of the functional form as the equation of motion satisfied by Weyl tensor, the action principle is invariant under con- the Weyl tensor itself, in complete analogy with the formal rescalings of the metric. As opposed to Einstein symmetric character of vacuum Maxwell equations with gravity, it is power-counting renormalizable [15,16], respect to the exchange of the field strength and its Hodge albeit it presents a Ostrogradski linear instability in dual. Motivated by this observation, in Sec. III we establish the Hamiltonian—due to the fourth-order character of a twisted self-duality form of the linearized equations of the equations of motion and the nondegeneracy of the motion of conformal gravity. Section IV deals with the Lagrangian,2 which is typically assumed to translate into generalities of the Hamiltonian formulation, including the the presence of ghosts—negative-norm states—upon quan- identification of the dynamical variables and the constraints tization. It is well known that solutions of Einstein gravity of the theory. To deal with the fact that the Lagrangian form a subset of solutions of conformal gravity, a fact that contains second order time derivatives of the metric has recently been exploited in [17] to show the equivalence perturbation, we will formally promote the linearized at the classical level of Einstein gravity with a cosmological extrinsic curvature to an independent dynamical variable. constant and conformal gravity with suitable boundary Section V is dedicated to the resolution of the differential conditions that eliminate ghosts. Another interesting aspect constraints in terms of two potentials. These are interpreted of conformal gravity is that it admits supersymmetric as a dual three-dimensional metric and a dual extrinsic extensions for N ≤ 4, the maximally supersymmetric curvature. In Section VI we present a manifestly duality theory admitting different variants (see [18] for a review invariant form of the action principle, where the two and [19] for recent progress). Other theoretical advances metrics and extrinsic curvatures appear on equal footing. involving conformal gravity include its emergence from Finally we draw our conclusions in Sec. VII and set out twistor string theory [20] and its appearance as a counter- proposals for future work. term in the AdS/CFT correspondence [21]. A generalization of electric-magnetic duality in con- II. CONFORMAL GRAVITY formal gravity was studied in the early work [22], where the Euclidean action with a gauge-fixed metric was expressed The action principle of conformal gravity is given by in terms of quadratic forms involving the electric and Z 1 magnetic components of the Weyl tensor, exhibiting a 4 pffiffiffiffiffiffi μνρσ S½gμν¼− d x −gWμνρσW ; ð2:1Þ discrete duality symmetry upon the interchange of these 4 components. This result can be regarded as the analog of the duality symmetry of Euclidean Maxwell action under with gμν the metric tensor defined on a manifold M and μ the exchange of electric and magnetic fields. Unlike [13], W νρσ the Weyl tensor duality is discussed in terms of the electric and magnetic μ μ μ μ components of the curvature, and not at the level of the W νρσ ≡ R νρσ − 2ðg ½ρSσν − gν½ρSσ Þ: ð2:2Þ dynamical degrees of freedom of the theory. μ In the present article we focus on linearized conformal Here R νρσ and Sμν are the Riemann and Schouten tensors, gravity with Lorentzian signature and show that the action respectively. The latter is defined as principle admits a manifestly duality invariant form in terms of the dynamical variables. The derivation requires 1 1 Sμν ≡ Rμν − gμνR : ð2:3Þ working in the Hamiltonian formalism, the identification of 2 6

2Recall that a higher order Lagrangian is nondegenerate We adopt the convention that indices within brackets are whenever the highest time derivative term can be expressed in antisymmetrized, with an overall factor of 1=n! for the terms of the canonical variables. antisymmetrization of n indices.

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μ The Weyl tensor W νρσ is invariant under diffeomor- 1 Rμνρσ ¼ − ½∂μ∂ρhνσ þ ∂ν∂σhμρ − ∂μ∂σhνρ − ∂ν∂ρhμσ phisms 2 ð2:14Þ δgμν ¼ ∇μξν þ ∇νξμ; ð2:4Þ and Sμν the linearized Schouten tensor. The action principle and local conformal rescalings of the metric and equations of motion reduce to

0 2 Z gμν → gμν ¼ Ω ðxÞgμν: ð2:5Þ 1 ½ ¼− 4 ½ μνρσ½ ðÞ S hαβ 4 d xWμνρσ h W h 2:15 These transformations also determine the symmetries of the action principle (2.1). and The Weyl tensor satisfies the same tensorial symmetry μρνσ properties as the Riemann tensor, ∂μ∂νW ½h¼0: ð2:16Þ

Wμνρσ ¼ −Wνμρσ ¼ −Wμνσρ ¼ Wρσμν; ð2:6Þ The linearized Weyl tensor still obeys the symmetry properties (2.6), and the identity (2.9) takes the linearized as well as the identities form

μνρσ νρσ W½μνσρ ¼ 0; ð2:7Þ ∂μW ½h¼−C ½h: ð2:17Þ

μ W νμσ ¼ 0; ð2:8Þ This allows for a rewriting of the linearized Bach equation in terms of the Cotton tensor: and νρσ ∂ρC ½h¼0: ð2:18Þ μ ∇μW νρσ ¼ −Cνρσ; ð2:9Þ Let us now introduce the Hodge dual of the linearized where we have introduced the Cotton tensor Weyl tensor: 1 Cνρσ ≡ 2∇½ρSσν: ð2:10Þ ½ ≡ ϵ αβ ½ ð Þ Wμνρσ h 2 μν Wαβρσ h : 2:19 Equation (2.9) is a consequence of the Bianchi identity for By construction it possesses the same symmetries as Wμνρσ, the Riemann tensor. namely The fourth-order equation of motion derived from the conformal gravity action principle (2.1) reads Wμνρσ ¼ Wρσμν ¼ − Wνμρσ ¼ − Wμνσρ: ð2:20Þ σ σ ρ ð2∇ρ∇ þ Rρ ÞW μσν ¼ 0: ð2:11Þ It also satisfies the cyclic identity

This is usually referred to as the Bach equation, the left- W½μνρσ ¼ 0 ð2:21Þ hand side of (2.11) being dubbed the Bach tensor. Clearly, conformally flat metrics constitute a particular subset of solutions to the equations of motion (2.11). Einstein and is traceless metrics constitute another subset of particular solutions. μ W νμρ ¼ 0: ð2:22Þ

A. Remarks on the linearized regime At this point, it is crucial to observe that Wμνρσ satisfies the In the linearized regime following identity:

μρνσ gμν ¼ ημν þ hμν ð2:12Þ ∂μ∂ν W ½h¼0: ð2:23Þ the Weyl tensor takes the form This is directly related to the identity

μ ½ ¼Rμ ½ − 2ðδμ ½ − δ μ½ Þ ð Þ W νρσ h νρσ h ½ρSσν h ν½ρSσ h ; 2:13 Cσ½νρ;μ½h¼0 ð2:24Þ where Rμνρσ is the linearized Riemann tensor satisfied by the linearized Cotton tensor, for

105014-3 HAPE´ FUHRI SNETHLAGE and SERGIO HÖRTNER PHYS. REV. D 103, 105014 (2021) 1 01 ∂ ∂ μρνσ½ ¼ ∂ ∂ ϵμραβ νσ½ Fμν Fμν μ ν W h 2 μ ν Wαβ h ¼ S ; S ¼ : ð3:1Þ Hμν Hμν −10 1 μραβ σ ¼ − ∂μϵ C αβ½h¼0: ð2:25Þ 2 Clearly, equation (3.1) implies the second-order Maxwell equations (2.27) by taking the divergence. However, in this It is now clear that the set of equations conformed by the first-order formulation Fμ½A and Hμν½B appear on an linearized Bach equation (2.16) and the identity (2.23) equal footing, related to each other by Hodge dualization. μνρσ As a caveat, we notice a redundancy in the twisted self- ∂μ∂ρW ½h¼0 duality equations (3.1), for either row can be obtained from μνρσ ∂μ∂ρ W ½h¼0 ð2:26Þ the other by Hodge dualization. It is actually possible to identify a noncovariant subset of (3.1) that is equivalent to may be regarded as the analog of Maxwell equations in the original set of equations and free from redundancies vacuum, [23]. This is achieved by selecting the purely spatial components of (3.1), which produces μν ∂μF ½A¼0 μν Bi½A Ei½A ∂μ F ½A¼0: ð2:27Þ ¼ S ; ð3:2Þ Bi½B Ei½B The set of equations (2.26) is symmetric under the E B replacement of the Weyl tensor and its Hodge dual. with i and i the usual electric and magnetic fields. Let us now turn the discussion to linearized conformal gravity. Since the dual Weyl tensor Wμνρσ has the same III. TWISTED SELF-DUALITY FORM OF THE algebraic and differential properties as the Weyl tensor, it EQUATIONS OF MOTION can be written itself in the same functional form as Wμνρσ½h Given the formal resemblance between Eqs. (2.27) and for some different metric fμν: (2.26), it seems natural to wonder about the existence of an underlying electric-magnetic duality structure in linearized Wμνρσ½h¼Hμνρσ½f; ð3:3Þ conformal gravity. In this section we show that the set of equations (2.26) can be cast in a covariant twisted self- with duality form, and that the noncovariant subset defined by μ μ μ μ selecting the purely spatial components of the latter H νρσ½f ≡ R νρσ½f − 2ðδ ½ρSσν½f − δν½ρSσ ½fÞ; ð3:4Þ contains all the information of the full covariant set— which parallels the situation in electromagnetism [23] and the relation between hμν and fμν being nonlocal. Similarly, linearized Einstein gravity [24]. it is easy to verify that the Cotton tensor and its Hodge dual In order to understand the logic underlying twisted self- have the same properties, and therefore we can write duality, it is useful to briefly recall the situation in Maxwell theory. Consider the vacuum equations (2.27), where we Cμνρ½h¼Dμνρ½f; ð3:5Þ tacitly assume Fμν ¼ ∂μAν − ∂νAμ. Although (2.27) are symmetric under the exchange of Fμν½A and Fμν½A, these where Dμνρ½f has the same functional form as the Cotton quantities do not appear exactly on an equal footing: the tensor for the dual metric fμν equation for Fμν is an identity. In other words, Fμν has μ ρ been implicitly solved in terms of the potential A . Indeed, Dμνρ½f¼−∂ρH μνρ½f: ð3:6Þ upon use of the Poincar´e lemma, one finds Fμν ¼ α β μ ϵμναβ∂ A for some vector potential A , and the definition Exactly in parallel as what happens in electromagnetism, of the Hodge dual yields Fμν ¼ ∂μAν − ∂νAμ, as expected. Hodge duality exchanges equations of motion and We seek instead a formulation where Fμν and Fμν appear differential identities. In the theory defined by hμν, ∂ ∂ μνρσ½ ¼0 on equal footing, with no implicit prioritization of any of μ νW h is an equation of motion and μνρσ them. This is achieved [23] by considering the field ∂μ∂νW ½h can be seen as the analog of the Bianchi strength and its Hodge dual as independent variables, identity in Maxwell theory. However, when we consider the solving simultaneously for both in terms of potentials dual theory defined by fμν, the latter implies the equation μνρσ Fμν½A¼∂μAν − ∂νAμ and Fμν ≡ Hμν½B¼∂μBν − ∂νBμ, of motion for the dual metric ∂μ∂νH ½f¼0, and finally imposing a first-order, twisted self-duality whereas the former is related to the differential identity μνρσ condition that takes into account that Fμν½A and Hμν½B ∂μ∂ν H ½f¼0. are not independent but actually related by Hodge Although the set of equations (2.26) is symmetric under dualization: the formal exchange of the Weyl tensor and its Hodge dual,

105014-4 MANIFEST ELECTRIC-MAGNETIC DUALITY IN LINEARIZED … PHYS. REV. D 103, 105014 (2021) implicitly we have prioritized the formulation based on hμν, and similarly for Hμνρσ for fμν does not appear at all. Following the same logic as discussed above in the case of electromagnetism, it is Eij½f ≡ H0i0j½f; possible though to find a set of second-order equations 1 B ½ ≡ − ½ ¼− ϵ mn½ ð Þ equivalent to (2.26) where both metrics appear on an equal ij f H0i0j f 2 0imnH0j f ; 3:12 footing. This is the twisted self-duality equation for linearized conformal gravity: Eq. (3.10) can be cast in the form Wμνρσ½h Wμνρσ½h 01 ¼ S ; S ¼ : ð3:7Þ Eij½h Bij½h 01 Hμνρσ½f Hμνρσ½f −10 ¼ S ; S ¼ : ð3:13Þ Eij½f Bij½f −10

Equation (3.7) is obtained from (2.26) by solving for Wμνρσ This equation is nonredundant and contains all the infor- and Hμνρσ ≡ Wμνρσ, treated as independent field strengths, and imposing the condition that they are actually related by mation in the covariant twisted self-duality equation (3.7). Hodge dualization. On the other hand, taking the double It will be referred to as the noncovariant twisted self-duality divergence on both sides, the twisted self-duality equa- equation. tion (3.7) reproduces immediately the equations of motion From their definitions, it is straightforward to see that the electric and magnetic components are both symmetric and for Wμνρσ½h and Hμνρσ½f in virtue of the differential traceless. Moreover, their double divergence vanishes: identities satisfied by their Hodge duals, exactly as what happens in the Maxwell theory. We notice (3.7) also ∂ ∂ E ½ ¼∂ ∂ E ½ ¼∂ ∂ B ½ implies the vanishing of the trace of Wμνρσ½h and i j ij h i j ij f i j ij h μνρσ½ H f , owing to the respective cyclic identities. This ¼ ∂i∂jBij½f¼0: ð3:14Þ is consistent with their definitions as the traceless part of the Riemann tensor for the corresponding metrics hμν and fμν. The set of Eqs. (3.7) is redundant, in the sense that either IV. HAMILTONIAN FORMULATION row can be obtained as the Hodge dual of the other one. So Having established the twisted self-duality structure we can keep only the subset of equations associated to the underlying the equations of motion of linearized conformal first row in (3.7): gravity, the natural next step is to seek a formulation of the corresponding action principle that manifestly displays 0 0 0 0 W i j ¼ H i j; duality symmetry. In order to do so, we shall follow the W0ijk ¼ H0ijk; same strategy as in Maxwell theory [13] and linearized gravity [8]: the Hamiltonian formulation is introduced by a ijkl ¼ ijkl ð Þ W H : 3:8 3 þ 1 slicing of space-time, constraints are identified and solved in terms of potentials, and finally a manifest duality Moreover, we see that the third equation in (3.8) can be symmetric action is written down upon substitution in obtained from the second one, for terms of potentials. This section deals with the Hamiltonian formulation of the theory and the identification of the ijkl ¼ ijkl ⇔ ϵij0m kl ¼ ijkl W H W0m H constraints. 1 The action principle for linearized Weyl gravity is ⇔ W0mkl ¼ − ϵ 0mHijkl ¼ −H0mkl; ð3:9Þ 2 ij Z 1 ¼ − 4 μνρσ ð Þ the last expression being the Hodge dual of the second S 4 d xWμνρσW : 4:1 equation in (3.8). Thus, the only independent components of the covariant twisted self-duality equations (3.7) are The squared Weyl tensor is decomposed upon a 3 þ 1 slicing of space-time as follows: W0i0j ¼ H0i0j; 0ijk 0ijk μνρσ 0i0j 0ijk ijkl W ¼ H : ð3:10Þ WμνρσW ¼ 4W0i0jW þ 4W0ijkW þ WijklW 0i0j 0ijk ¼ 8W0i0jW þ 4W0ijkW : ð4:2Þ Defining the electric component Eij and magnetic component B of the Weyl tensor Wμνρσ as ij In order to deal with the second-order character of the Lagrangian, we shall follow the Ostrogradski method to E ½h ≡ W0 0 ½h; ij i j define a dynamical variable depending on first-order 1 mn derivatives that will be formally treated as independent. B ½h ≡ − W0 0 ½h¼− ϵ0 W0 ½h; ð3:11Þ ij i j 2 imn j A natural choice is the linearized extrinsic curvature:

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1 _ The conjugate momentum associated to Kij is defined as K ¼ ðh − ∂ h0 − ∂ h0 Þ: ð4:3Þ ij 2 ij i j j i usual:

It will be required then to express the Weyl tensor in terms ∂L of K . Pij ¼ ij ∂K_ First, let us write the Riemann tensor and its contractions ij 1 in terms of the extrinsic curvature. The components of _ ij i j imj ¼ − K þ ∂ ∂ h00 − R (2.14) are 2 m 1 1 1 1 − Δ δij þ mnδij − _ δij ð Þ ¼ − ½∂ ∂ þ ∂ ∂ − ∂ ∂ − ∂ ∂ h00 Rmn K : 4:8 Rijkl 2 i khjl j lhik i lhjk j khil ; 6 3 3 1 ¼ − _ − ∂ ∂ ij R0i0j Kij 2 i jh00; We notice that the trace of P vanishes identically:

R0 ¼ −ð∂ K − ∂ K Þ: ð4:4Þ ijk j ik k ij P ¼ 0: ð4:9Þ In turn, the components of the Ricci tensor read This shall be treated as a primary constraint. 1 i _ The Hamiltonian is now introduced through the R00 ¼ R0 0 ¼ −K − Δh00 i 2 Legendre transformation k R0i ¼ ∂ Kik − ∂iK H ¼ ij _ − L ð Þ 1 P Kij : 4:10 ¼ − _ − ∂ ∂ þ k ð Þ Rij Kij 2 i jh00 Rikj ; 4:5 Upon substitution for the generalized velocities, it can be ij and the scalar curvature is expressed in terms of P and Kij:

i ij _ ij R ¼ −2R0i0 þ Rij ¼ 2K þ Δh00 þ Rij : ð4:6Þ 1 1 H ¼ − ij − ij∂ ∂ þ m ij − ijk 2 PijP 2 P i jh00 Rimj P W0ijkW0 : From the definition of the Weyl tensor (2.13) one derives the relations ð4:11Þ 1 1 ¼ ð þ mÞ − δ ð mn þ mÞ Now we have to take into account the fact that W0i0j 2 R0i0j Rimj 6 ij Rmn R0m0 _ the definition of Kij actually depends on hij by intro- 1 1 ducing in the action principle the constraint term ¼ − _ − ∂ ∂ þ m Kij i jh00 Rimj ij _ ij 2 2 λ ðh − ∂ h0 − ∂ h0 − 2K Þ. The factor λ becomes a ij j i i j ij 1 1 Lagrange multiplier enforcing the definition of Kij: − δ − _ − Δ þ mn 6 ij K 2 h00 Rmn S½h ;pij;K ;Pij;λij ijZ ij and 1 1 ¼ 4 ij _ þ ij þ ij∂ ∂ − ij m d x P Kij 2PijP 2P i jh00 P Rimj W0ijk ¼ ∂kKij − ∂jKik 1 þ 2∂ K ∂jKik þ 2∂ K∂ Kkl − ∂ K∂lK − 3∂lK ∂ Kkm þ ðδ ð∂l − ∂ Þ − δ ð∂l − ∂ ÞÞ j ik k l l kl m 2 ij Klk kK ik Kjl jK : ij _ þ λ ðhij −∂jh0i − ∂ih0j − 2KijÞ : ð4:12Þ The Lagrangian reads

1 λij L ¼ − μνρσ Clearly one can identify the Lagrange multiplier with 4WμνρσW ij ≡ λij the conjugate momentum associated to hij, p . Upon 1 integration by parts, the components h00 and h0 act now as ¼ − ð _ _ ij þ m inj j 2 KijK Rimj R n Lagrange multipliers imposing the constraints

_ i j _ ij m imj þ Kij∂ ∂ h00 − 2K Rimj − ∂i∂jh00R mÞ ij ∂i∂jP ¼ 0 ð4:13Þ 1 − ð _ 2 þ ij mn þ _ Δ − 2 _ ij − Δ mnÞ 6 K Rij Rmn K h00 KRij h00Rmn and 1 þ Δ Δ − ijk ð Þ h00 h00 W0ijkW0 : 4:7 ij 12 ∂jp ¼ 0: ð4:14Þ

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The latter reads exactly as in linearized gravity [8]. Ignoring the four-dimensional metric hμν. It is straightforward to total derivatives coming from the previous integration by verify that the constraints (5.1), (5.2), (4.9) and (4.18) are parts, the final form of the Hamiltonian action principle is first class, so the previous gauge transformations can also be derived from the Poisson bracket with the con- ij ij S½hij;p ;Kij;P ;h0i;h00 straints. We notice that p ¼ 0 generates the Weyl rescaling Z ij for hij, whereas ∂jp is responsible for the same three- ¼ 4 _ þ ij _ d x PijKij p hij dimensional diffeomorphism invariance of linearized Einstein gravity. 1 −H þ 2∂ ij − ∂ ∂ ij ð Þ 0 jp h0i 2 i jP h00 4:15 V. RESOLUTION OF THE CONSTRAINTS with We shall now focus on the resolution of the differential constraints 1 ij ij m ij j ik −H0 ¼ P P − P R − 2p K þ 2∂ K ∂ K 2 ij imj ij j ik ij ∂i∂jP ¼ 0 ð5:1Þ l l l km þ 2∂kK∂ Kkl − ∂lK∂ K − 3∂ Kkl∂mK : ð4:16Þ and One notices in (4.16) the presence of terms linear in the ij conjugate momenta, which points out the Ostrogradski ∂jp ¼ 0; ð5:2Þ linear instability of the theory. There is an additional constraint that comes about by subject to the traceless constraints demanding the preservation of the constraint (4.9) under time evolution. In other words, the Poisson bracket of the P ¼ 0 ð5:3Þ constraint (4.9) with the Hamiltonian should vanish: Z and P; d3xH ¼ 0: ð4:17Þ p ¼ 0: ð5:4Þ

This results in the constraint Let us first focus on (5.1). Taking into account that P ¼ 0, this can be solved in terms of some tensor potential ψ ij as p ¼ 0: ð4:18Þ follows [8]:

The consistency condition applied to this constraint does ij m nj m ni P ¼ ϵimn∂ ψ þ ϵjmn∂ ψ : ð5:5Þ not produce any further ones. Adding up the traceless constraints (4.9) and (4.18), the Because of the traceless condition on Pij, the antisym- action principle reads metric component of ψ ij is restricted to have the form ½ ij ij λ λ S hij;p ;Kij;P ;h0i;h00; 1; 2 ψ ¼ ∂ − ∂ ð Þ Z ½ij iwj jwi: 5:6 1 4 _ ij _ ij ij ¼ d x P K þ p h − H0 þ 2∂ p h0 − ∂ ∂ P h00 ij ij ij j i 2 i j However, by the redefinition of the symmetric component of the potential ψ ðijÞ ≡ ϕij þ ∂iwj þ ∂jwi the solution þ λ1P þ λ2p : ð4:19Þ simply takes the form

ij ¼ ϵ ∂mϕnj þ ϵ ∂mϕni ð Þ The gauge transformations of the dynamical variables are P imn jmn ; 5:7 ϕ δ ij ¼ 0 δh ¼ ∂ ξ þ ∂ ξ þ δ ξ; with ij a symmetric tensor. Note that, since P , the ij i j j i ij ϕ 1 ambiguities in the definition of the potential ij are δ ¼ ½−2∂ ∂ ξ þ δ ξ_ restricted to have the form Kij 2 i j 0 ij ; δpij ¼ 0; δϕij ¼ ∂i∂jξ þ δijθ: ð5:8Þ δPij ¼ 0: ð4:20Þ This has exactly the same form as δKij, which already ϕ These can be obtained directly from the definition of suggests that Kij and ij can be treated on equal footing, ˜ the dynamical variables in terms of the components of and justifies the renaming ϕij ≡ Kij.

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ij In order to solve the constraint (5.2), we may use the Substituting now in the term −2p Kij, we find Poincar´e lemma and write [8] ij m nj ˜ −2p Kij ¼ 2ϵimn∂ R ½hKij ij nl p ¼ ϵimnϵjkl∂m∂kω ð5:9Þ nj ˜ ¼ −2ϵimnR ½h∂mKij þ total derivatives: ð6:4Þ ωnl for some symmetric potential . The traceless condition We may compare this expression with the term p ¼ 0 (which was not present in [8]) imposes a further ij constraint on ω : ij ˜ −P Rij½h¼−2ϵimn∂mKnjRij½hð6:5Þ Δω − ∂ ∂ ωij ¼ 0 ð Þ i j : 5:10 and see that these two are rotated into each other by the transformation (6.2) supplemented by Once this is taken into account, (5.9) becomes h → h˜ ; h˜ → −h : ð6:6Þ 1 ij ij ij ij ij m nj ˜ m ni ˜ p ¼ − ðϵimn∂ R ½hþϵjmn∂ R ½hÞ; ð5:11Þ 2 _ ij _ ˜ The kinetic term hijp ¼ −hijϵimn∂mRnj½h is also invari- ˜ ant under (6.6) up to total derivatives: with Rij½h having the functional form of the linearized three-dimensional Ricci tensor for some symmetric tensor _ ˜ 1 _ ˜ ˜ ˜ — −h ϵ ∂ R ½h¼ h ϵ ∂ ðΔh − ∂ ∂ h Þ hij to be interpreted in the sequel as a second, dual ij imn m nj 2 ij imn m nj j k kn — metric , and the global factor has been chosen for future 1 _ ˜ → − h˜ ϵ ∂ ðΔh − ∂ ∂ h Þ convenience. Clearly we can choose the dual metric hij to 2 ij imn m nj j k kn coincide with the metric f introduced in Sec. III,uptoa 1 ij ¼ _ ϵ ∂ ðΔ˜ − ∂ ∂ ˜ Þ gauge transformation. The expression (5.11) is invariant 2 hij imn m hnj j khkn ˜ under transformations of hij having the same form as those þ total derivatives: ð6:7Þ defining the symmetries of conformal gravity, So we conclude that, once the constraints are solved, the δ˜ ¼ ∂ χ þ ∂ χ þ δ χ ð Þ hij i j j i ij ; 5:12 action principle (4.19) can be cast in the manifestly duality invariant form as we could have expected. Z ½ ˜ ˜ ¼ 4 ½2ϵ ∂ ˜ _ VI. MANIFEST DUALITY INVARIANCE S hij; hij;Kij; Kij; d x imn mKjnKij _ ˜ We can now implement Eqs. (5.7) and (5.11) in the − hijϵimn∂mRnj½h − H; ð6:8Þ action principle (4.19). Let us first compute the quadratic term in Pij: where

1 − H ¼ 2ϵ ∂m ˜ ½ − 2ϵ ∂m ½˜ ij ¼ 2∂ ˜ ∂j ˜ ik þ 2∂k ˜ ∂l ˜ imn KnjRij h imn KijRnj h 2 P Pij jKik K K Kkl 2∂ ∂j ik þ 2∂k ∂l − ∂ ∂l − 3∂l ∂ km ˜ l ˜ l ˜ ˜ km jKik K K Kkl lK K Kkl mK − ∂lK∂ K − 3∂ Kkl∂mK : ð6:1Þ ˜ j ˜ ik k ˜ l ˜ ˜ l ˜ þ 2∂jKik∂ K þ 2∂ K∂ Kkl − ∂lK∂ K l ˜ ˜ km Remarkably, this has exactly the same form as the quadratic − 3∂ Kkl∂mK ð6:9Þ terms in Kij appearing in the Hamiltonian (4.16), and suggests an invariance under the transformation and we have dropped surface terms. One can verify that the action principle (6.8) is actually invariant under ˜ ˜ Kij → Kij; Kij → −Kij: ð6:2Þ continuous duality rotations of the dual metrics and extrinsic curvatures: ij _ The kinetic term P Kij is also invariant under (6.2) (up to hij cos θ sin θ hij total derivatives): ¼ ˜ − θ θ ˜ hij sin cos hij ij _ ˜ _ ij ˜_ ij P Kij ¼ 2ϵimn∂mKjnK → −2ϵimn∂mKjnK Kij cos θ sin θ Kij ¼ : ð6:10Þ ˜ _ ij ˜ − θ θ ˜ ¼ 2ϵimn∂mKjnK þ total derivatives: ð6:3Þ Kij sin cos Kij

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VII. CONCLUSIONS consequences of their presence: for instance, to investigate if they can cancel out the total derivatives produced by We have shown that electric-magnetic duality is a hidden integration by parts in the process of rending the action symmetry of linearized conformal gravity, both at the level principle in its manifestly duality-invariant form. Manifest of the equations of motion and the action principle. In order space-time covariance of the action principle might be to render the symmetry manifest, i.e., to establish a formulation where the “electric” and “magnetic” degrees of restored upon introduction of auxiliary fields, although freedom appear on equal footing, it seems necessary to those are expected either to enter in a nonpolynomial fashion work in a nonmanifestly space-time covariant framework. [26] or to appear in infinite number [27] (see [28], however, for a construction of actions for self-dual 2n þ 1-forms with The covariant equations of motion and differential identities 4 þ 2 obeyed by the Weyl tensor and its Hodge dual can be auxiliary fields in n dimensions that do not display recovered from a noncovariant subset of the twisted self- these drawbacks). It should be determined whether obstruc- duality equation, where the electric and magnetic compo- tions to manifest duality invariance at higher perturbative nents of the Weyl tensors for two dual metrics appear on orders are present [29]. Connections with the double copy equal footing. The action principle is cast in a manifestly [11,12] at the linearized level constitute another interesting duality-invariant form as well, upon resolution of the aspect to be investigated. constraints in the Hamiltonian formalism. The potentials Electric-magnetic duality in Abelian Yang-Mills theory that solve these constraints are interpreted as a dual has been discussed at the quantum level in the path-integral ½ three-dimensional metric and a dual extrinsic curvature. formulationR [30]. Here one adds to the Abelian action S A Duality acts as simultaneous rotations in the respective a term i B ∧ dF featuring an additional 1-form field B, spaces spanned by the two metrics and the two extrinsic such that integrating over it produces a delta functional curvatures. δðdFÞ allowing integration over not necessarily closed There are several interesting directions for future work to 2-forms F. If we instead integrate over F, the partition be discussed. An important question is to determine whether function written as an integral over B takes the same form a manifestly duality invariant action principle can be as expressed in terms of the original 1-form field A,but obtained upon linearization around more general back- interchanging the coupling constant e2 and the θ-parameter. grounds, in particular (anti) de Sitter space-time. The precise Whether a similar analysis can be performed in linearized relation between the equations of motion obtained from the conformal gravity seems an avenue worth exploring. duality-symmetric action principle and the noncovariant twisted self-duality equation should be determined. ACKNOWLEDGMENTS Supersymmetric extensions can also be investigated, along the lines of the work [25]. Although we have not dealt The work of S. H. is supported by a fellowship from with topological terms, it may be interesting to study the Ramon Areces Foundation (Spain).

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