PHYSICAL REVIEW D 103, 105014 (2021) - Manifest electric-magnetic duality in linearized conformal gravity
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PHYSICAL REVIEW D 103, 105014 (2021)
Manifest electric-magnetic duality in linearized conformal gravity
*
Hapé 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
ence of a Killing vector, followed by the dualization of the infinite-dimensional Kac-Moody algebra acting as a fun-
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.
nongravitational degrees of freedom in the reduced bosonic A SOð2Þ-invariant action principle for linearized gravity
sector. After Hodge dualization of the p-forms to their has been derived [8–10],1 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
*
sergio.hortner@uva.nl
1
Interestingly, 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 SocietyHAPÉ 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 ; h̃ij Þ and
duality-invariant action in [8]. the extrinsic curvatures ðK ij ; K̃ ij Þ.
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
magnetic components of the Weyl tensor, exhibiting a 1 pffiffiffiffiffiffi
S½gμν ¼ − d4 x −gW μνρσ W μνρσ ; 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
the exchange of electric and magnetic fields. Unlike [13], W μ νρσ the Weyl tensor
duality is discussed in terms of the electric and magnetic
components of the curvature, and not at the level of the W μ νρσ ≡ Rμ νρσ − 2ðgμ ½ρ S σν − gν½ρ S σ μ Þ: ð2:2Þ
dynamical degrees of freedom of the theory.
In the present article we focus on linearized conformal Here Rμ νρσ and S μν 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
S μν ≡ Rμν − gμν R : ð2:3Þ
working in the Hamiltonian formalism, the identification of 2 6
2
Recall 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.
105014-2MANIFEST ELECTRIC-MAGNETIC DUALITY IN LINEARIZED … PHYS. REV. D 103, 105014 (2021)
The Weyl tensor W μ νρσ is invariant under diffeomor- 1
Rμνρσ ¼ − ½∂ μ ∂ ρ hνσ þ ∂ ν ∂ σ hμρ − ∂ μ ∂ σ hνρ − ∂ ν ∂ ρ hμσ
phisms 2
ð2:14Þ
δgμν ¼ ∇μ ξν þ ∇ν ξμ ; ð2:4Þ
and Sμν the linearized Schouten tensor. The action principle
and local conformal rescalings of the metric and equations of motion reduce to
Z
gμν → g0μν ¼ Ω2 ðxÞgμν : ð2:5Þ 1
S½hαβ ¼ − d4 xW μνρσ ½hW μνρσ ½h ð2:15Þ
4
These transformations also determine the symmetries of the
action principle (2.1). and
The Weyl tensor satisfies the same tensorial symmetry
properties as the Riemann tensor, ∂ μ ∂ ν W μρνσ ½h ¼ 0: ð2:16Þ
W μνρσ ¼ −W νμρσ ¼ −W μνσρ ¼ W ρσμν ; 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
W ½μνσρ ¼ 0; ð2:7Þ ∂ μ W μνρσ ½h ¼ −Cνρσ ½h: ð2:17Þ
W μ νμσ ¼ 0; ð2:8Þ This allows for a rewriting of the linearized Bach equation
in terms of the Cotton tensor:
and
∂ ρ Cνρσ ½h ¼ 0: ð2:18Þ
μ
∇μ W νρσ ¼ −Cνρσ ; ð2:9Þ
Let us now introduce the Hodge dual of the linearized
where we have introduced the Cotton tensor Weyl tensor:
Cνρσ ≡ 2∇½ρ S σν : ð2:10Þ 1
W μνρσ ½h ≡ ϵμν αβ W αβρσ ½h: ð2:19Þ
2
Equation (2.9) is a consequence of the Bianchi identity for
By construction it possesses the same symmetries as W μνρσ ,
the Riemann tensor.
namely
The fourth-order equation of motion derived from the
conformal gravity action principle (2.1) reads
W μνρσ ¼ W ρσμν ¼ −W νμρσ ¼ −W μνσρ : ð2:20Þ
ð2∇ρ ∇σ þ Rρ σ ÞW ρ μσν ¼ 0: ð2:11Þ
It also satisfies the cyclic identity
This is usually referred to as the Bach equation, the left-
W ½μνρσ 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.
W μνμρ ¼ 0: ð2:22Þ
A. Remarks on the linearized regime At this point, it is crucial to observe that W μνρσ satisfies the
In the linearized regime following identity:
gμν ¼ ημν þ hμν ð2:12Þ ∂ μ ∂ ν W μρνσ½h ¼ 0: ð2:23Þ
the Weyl tensor takes the form This is directly related to the identity
W μ νρσ ½h ¼ Rμ νρσ ½h − 2ðδμ ½ρ Sσν ½h − δν½ρ Sσ μ ½hÞ; ð2:13Þ Cσ½νρ;μ ½h ¼ 0 ð2:24Þ
where Rμνρσ is the linearized Riemann tensor satisfied by the linearized Cotton tensor, for
105014-3HAPÉ FUHRI SNETHLAGE and SERGIO HÖRTNER PHYS. REV. D 103, 105014 (2021)
1 Fμν Fμν 0 1
∂ μ ∂ ν W μρνσ½h ¼ ∂ μ ∂ ν ϵμραβ W αβ νσ ½h ¼S ; S¼ : ð3:1Þ
2 Hμν Hμν −1 0
1
¼ − ∂ μ ϵμραβ Cσ αβ ½h ¼ 0: 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 Fμ ½A and Hμν ½B appear on an
linearized Bach equation (2.16) and the identity (2.23) equal footing, related to each other by Hodge dualization.
As a caveat, we notice a redundancy in the twisted self-
∂ μ ∂ ρ W μνρσ ½h ¼ 0 duality equations (3.1), for either row can be obtained from
∂ μ ∂ ρ W μνρσ½h ¼ 0 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
∂ μ Fμν ½A ¼ 0
Bi ½A E i ½A
∂ μ Fμν½A ¼ 0: ð2:27Þ ¼S ; ð3:2Þ
Bi ½B E i ½B
The set of equations (2.26) is symmetric under the
replacement of the Weyl tensor and its Hodge dual. with E i and Bi 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
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 W μνρσ ½h
Given the formal resemblance between Eqs. (2.27) and for some different metric f μν :
(2.26), it seems natural to wonder about the existence of an
underlying electric-magnetic duality structure in linearized W μνρσ ½h ¼ H μνρσ ½f; 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
selecting the purely spatial components of the latter Hμ νρσ ½f ≡ Rμ νρσ ½f − 2ðδμ ½ρ Sσν ½f − δν½ρ Sσ μ 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 f μν 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
C
μνρ ½h ¼ Dμνρ ½f; ð3:5Þ
theory. Consider the vacuum equations (2.27), where we
tacitly assume Fμν ¼ ∂ μ Aν − ∂ ν Aμ . Although (2.27) are
symmetric under the exchange of Fμν ½A and Fμν ½A, these where Dμνρ ½f has the same functional form as the Cotton
quantities do not appear exactly on an equal footing: the tensor for the dual metric f μν
equation for Fμν is an identity. In other words, Fμν has
been implicitly solved in terms of the potential Aμ . Indeed, Dμνρ ½f ¼ −∂ ρ Hρ μνρ ½f: ð3:6Þ
upon use of the Poincaré lemma, one finds Fμν ¼
ϵμναβ ∂ α Aβ for some vector potential Aμ , and the definition Exactly in parallel as what happens in electromagnetism,
of the Hodge dual yields Fμν ¼ ∂ μ Aν − ∂ ν Aμ , 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,
on equal footing, with no implicit prioritization of any of ∂ μ ∂ ν W μνρσ ½h ¼ 0 is an equation of motion and
them. This is achieved [23] by considering the field ∂ μ ∂ ν W μνρσ 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 f μν, the latter implies the equation
Fμν ½A ¼ ∂ μ Aν − ∂ ν Aμ and Fμν ≡ Hμν ½B ¼ ∂ μ Bν − ∂ ν Bμ , of motion for the dual metric ∂ μ ∂ ν Hμνρσ ½f ¼ 0,
and finally imposing a first-order, twisted self-duality whereas the former is related to the differential identity
condition that takes into account that Fμν ½A and Hμν ½B ∂ μ ∂ ν Hμνρσ½f ¼ 0.
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,
105014-4MANIFEST ELECTRIC-MAGNETIC DUALITY IN LINEARIZED … PHYS. REV. D 103, 105014 (2021)
implicitly we have prioritized the formulation based on hμν , and similarly for H μνρσ
for f μν does not appear at all. Following the same logic as
discussed above in the case of electromagnetism, it is E ij ½f ≡ H0i0j ½f;
possible though to find a set of second-order equations 1
equivalent to (2.26) where both metrics appear on an equal Bij ½f ≡ −H0i0j ½f ¼ − ϵ0imn H0j mn ½f; ð3:12Þ
footing. This is the twisted self-duality equation for 2
linearized conformal gravity: Eq. (3.10) can be cast in the form
μνρσ μνρσ
W ½h W ½h 0 1
¼ S ; S¼ : ð3:7Þ E ij ½h Bij ½h 0 1
μνρσ
H ½f μνρσ
H ½f −1 0 ¼S ; S¼ : ð3:13Þ
E ij ½f Bij ½f −1 0
Equation (3.7) is obtained from (2.26) by solving for W μνρσ
and Hμνρσ ≡ W μνρσ , treated as independent field strengths, This equation is nonredundant and contains all the infor-
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
for W μνρσ ½h and Hμνρσ ½f in virtue of the differential electric and magnetic components are both symmetric and
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
implies the vanishing of the trace of W μνρσ ½h and ∂ i ∂ j E ij ½h ¼ ∂ i ∂ j E ij ½f ¼ ∂ i ∂ j Bij ½h
Hμνρσ ½f, owing to the respective cyclic identities. This ¼ ∂ i ∂ j Bij ½f ¼ 0: ð3:14Þ
is consistent with their definitions as the traceless part of the
Riemann tensor for the corresponding metrics hμν and f μν .
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
W 0i0j ¼ H 0i0j ; duality symmetry. In order to do so, we shall follow the
W 0ijk ¼ H 0ijk ; same strategy as in Maxwell theory [13] and linearized
gravity [8]: the Hamiltonian formulation is introduced by a
W ijkl ¼ H ijkl : ð3:8Þ 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
W ijkl ¼ Hijkl ⇔ ϵij0m W 0m kl ¼ Hijkl constraints.
1 The action principle for linearized Weyl gravity is
⇔ W 0mkl ¼ − ϵij 0m Hijkl ¼ −H0mkl; ð3:9Þ
2 Z
1
the last expression being the Hodge dual of the second S¼− d4 xW μνρσ W μνρσ : ð4:1Þ
4
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:
W 0i0j ¼ H 0i0j ;
W 0ijk ¼ H 0ijk : ð3:10Þ W μνρσ W μνρσ ¼ 4W 0i0j W 0i0j þ 4W 0ijk W 0ijk þ W ijkl W ijkl
¼ 8W 0i0j W 0i0j þ 4W 0ijk W 0ijk : ð4:2Þ
Defining the electric component E ij and magnetic compo-
nent Bij of the Weyl tensor W μνρσ as
In order to deal with the second-order character of the
Lagrangian, we shall follow the Ostrogradski method to
E ij ½h ≡ W 0i0j ½h;
define a dynamical variable depending on first-order
1 derivatives that will be formally treated as independent.
Bij ½h ≡ −W 0i0j ½h ¼ − ϵ0imn W 0j mn ½h; ð3:11Þ
2 A natural choice is the linearized extrinsic curvature:
105014-5HAPÉ FUHRI SNETHLAGE and SERGIO HÖRTNER PHYS. REV. D 103, 105014 (2021)
1 The conjugate momentum associated to K ij is defined as
K ij ¼ ðh_ ij − ∂ i h0j − ∂ j h0i Þ: ð4:3Þ
2 usual:
It will be required then to express the Weyl tensor in terms ∂L
of K ij . Pij ¼
∂ K_ ij
First, let us write the Riemann tensor and its contractions
in terms of the extrinsic curvature. The components of 1
¼ − K_ ij þ ∂ i ∂ j h00 − Rimj m
(2.14) are 2
1 1 1 1 _ ij
Rijkl ¼ − ½∂ i ∂ k hjl þ ∂ j ∂ l hik − ∂ i ∂ l hjk − ∂ j ∂ k hil ; − Δh00 δ þ Rmn δ − Kδ :
ij mn ij ð4:8Þ
2 6 3 3
1
R0i0j ¼ −K_ ij − ∂ i ∂ j h00 ; We notice that the trace of Pij vanishes identically:
2
R0ijk ¼ −ð∂ j K ik − ∂ k K ij Þ: ð4:4Þ
P ¼ 0: ð4:9Þ
In turn, the components of the Ricci tensor read
This shall be treated as a primary constraint.
1 The Hamiltonian is now introduced through the
R00 ¼ R0i0 ¼ −K_ − Δh00
i
2 Legendre transformation
R0i ¼ ∂ k K ik − ∂ i K
1 H ¼ Pij K_ ij − L: ð4:10Þ
Rij ¼ −K_ ij − ∂ i ∂ j h00 þ Rikj k ; ð4:5Þ
2
Upon substitution for the generalized velocities, it can be
and the scalar curvature is expressed in terms of Pij and K ij :
R ¼ −2R0i0 i þ Rij ij ¼ 2K_ þ Δh00 þ Rij ij : ð4:6Þ 1 1
H ¼ − Pij Pij − Pij ∂ i ∂ j h00 þ Rimj m Pij − W 0ijk W 0 ijk :
From the definition of the Weyl tensor (2.13) one derives 2 2
the relations ð4:11Þ
1 1 Now we have to take into account the fact that
W 0i0j ¼ ðR0i0j þ Rimj m Þ − δij ðRmn mn þ R0m0 m Þ
2
6
the definition of K ij actually depends on h_ ij by intro-
1 _ 1 ducing in the action principle the constraint term
¼ −K ij − ∂ i ∂ j h00 þ Rimj m
2 2 λij ðh_ ij − ∂ j h0i − ∂ i h0j − 2K ij Þ. The factor λij becomes a
Lagrange multiplier enforcing the definition of K ij :
1 _ 1
− δij −K − Δh00 þ Rmn mn
6 2
S½hij ; pij ; K ij ;Pij ; λij
and Z
1 1
¼ d x Pij K_ ij þ Pij Pij þ Pij ∂ i ∂ j h00 − Pij Rimj m
4
W 0ijk ¼ ∂ k K ij − ∂ j K ik 2 2
1 þ 2∂ j K ik ∂ j K ik þ 2∂ k K∂ l K kl − ∂ l K∂ l K − 3∂ l K kl ∂ m K km
þ ðδij ð∂ l K lk − ∂ k KÞ − δik ð∂ l K jl − ∂ j KÞÞ:
2 ij _
þ λ ðhij − ∂ j h0i − ∂ i h0j − 2K ij Þ : ð4:12Þ
The Lagrangian reads
1 Clearly one can identify the Lagrange multiplier λij with
L ¼ − W μνρσ W μνρσ
4 the conjugate momentum associated to hij , pij ≡ λij . Upon
1 integration by parts, the components h00 and h0j act now as
¼ − ðK_ ij K_ ij þ Rimj m Rinj n Lagrange multipliers imposing the constraints
2
þ K_ ij ∂ i ∂ j h00 − 2K_ ij Rimj m − ∂ i ∂ j h00 Rimj m Þ
∂ i ∂ j Pij ¼ 0 ð4:13Þ
1
− ðK_ 2 þ Rij ij Rmn mn þ KΔh
_ 00 − 2KR
_ ij ij − Δh00 Rmn mn Þ
6 and
1
þ Δh00 Δh00 − W 0ijk W 0 : ijk
ð4:7Þ
12 ∂ j pij ¼ 0: ð4:14Þ
105014-6MANIFEST ELECTRIC-MAGNETIC DUALITY IN LINEARIZED … PHYS. REV. D 103, 105014 (2021)
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-
S½hij ; pij ; K ij ; Pij ; h0i ; h00 straints. We notice that p ¼ 0 generates the Weyl rescaling
Z for hij, whereas ∂ j pij is responsible for the same three-
¼ d4 x Pij K_ ij þ pij h_ ij dimensional diffeomorphism invariance of linearized
Einstein gravity.
1
−H0 þ 2∂ j pij h0i − ∂ i ∂ j Pij h00 ð4:15Þ
2
V. RESOLUTION OF THE CONSTRAINTS
with We shall now focus on the resolution of the differential
constraints
1
−H0 ¼ Pij Pij − Pij Rimj m − 2pij K ij þ 2∂ j K ik ∂ j K ik
2 ∂ i ∂ j Pij ¼ 0 ð5:1Þ
þ 2∂ k K∂ l K kl − ∂ l K∂ l K − 3∂ l K kl ∂ m K km : ð4:16Þ
and
One notices in (4.16) the presence of terms linear in the
conjugate momenta, which points out the Ostrogradski ∂ j pij 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 P¼0 ð5:3Þ
constraint (4.9) with the Hamiltonian should vanish:
Z and
3
P; d xH 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 Pij ¼ ϵimn ∂ m ψ nj þ ϵjmn ∂ m ψ ni : 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
S½hij ;pij ; K ij ;Pij ; h0i ;h00 ;λ1 ;λ2 ψ ½ij ¼ ∂ i wj − ∂ j wi : ð5:6Þ
Z
1
¼ d x Pij K_ ij þ pij h_ ij − H0 þ 2∂ j pij h0i − ∂ i ∂ j Pij h00
4
2 However, by the redefinition of the symmetric component
of the potential ψ ðijÞ ≡ ϕij þ ∂ i wj þ ∂ j wi the solution
þ λ1 P þ λ2 p : ð4:19Þ simply takes the form
The gauge transformations of the dynamical variables are Pij ¼ ϵimn ∂ m ϕnj þ ϵjmn ∂ m ϕni ; ð5:7Þ
δhij ¼ ∂ i ξj þ ∂ j ξi þ δij ξ; with ϕij a symmetric tensor. Note that, since δPij ¼ 0, the
ambiguities in the definition of the potential ϕij are
1 _
δK ij ¼ ½−2∂ i ∂ j ξ0 þ δij ξ; restricted to have the form
2
δpij ¼ 0; δϕij ¼ ∂ i ∂ j ξ þ δij θ: ð5:8Þ
δPij ¼ 0: ð4:20Þ
This has exactly the same form as δK ij , which already
These can be obtained directly from the definition of suggests that K ij and ϕij can be treated on equal footing,
the dynamical variables in terms of the components of and justifies the renaming ϕij ≡ K̃ ij .
105014-7HAPÉ FUHRI SNETHLAGE and SERGIO HÖRTNER PHYS. REV. D 103, 105014 (2021)
In order to solve the constraint (5.2), we may use the Substituting now in the term −2pij K ij , we find
Poincaré lemma and write [8]
−2pij K ij ¼ 2ϵimn ∂ m Rnj ½h̃K ij
p ¼ ϵimn ϵjkl ∂ m ∂ k ω
ij nl
ð5:9Þ ¼ −2ϵimn Rnj ½h̃∂ m K ij þ total derivatives: ð6:4Þ
for some symmetric potential ωnl . The traceless condition We may compare this expression with the term
p ¼ 0 (which was not present in [8]) imposes a further
constraint on ωij : −Pij Rij ½h ¼ −2ϵimn ∂ m K̃ nj Rij ½h ð6:5Þ
Δω − ∂ i ∂ j ωij ¼ 0: ð5:10Þ 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
hij → h̃ij ; h̃ij → −hij : ð6:6Þ
1
p ¼ − ðϵimn ∂ m Rnj ½h̃ þ ϵjmn ∂ m Rni ½h̃Þ;
ij
ð5:11Þ
2 The kinetic term h_ ij pij ¼ −h_ ij ϵimn ∂ m Rnj ½h̃ is also invari-
ant under (6.6) up to total derivatives:
with Rij ½h̃ having the functional form of the linearized
three-dimensional Ricci tensor for some symmetric tensor 1
h̃ij —to be interpreted in the sequel as a second, dual −h_ ij ϵimn ∂ m Rnj ½h̃ ¼ h_ ij ϵimn ∂ m ðΔh̃nj − ∂ j ∂ k h̃kn Þ
2
metric—, and the global factor has been chosen for future 1
convenience. Clearly we can choose the dual metric h̃ij to → − h̃_ ij ϵimn ∂ m ðΔhnj − ∂ j ∂ k hkn Þ
2
coincide with the metric f ij introduced in Sec. III, up to a 1_
gauge transformation. The expression (5.11) is invariant ¼ hij ϵimn ∂ m ðΔh̃nj − ∂ j ∂ k h̃kn Þ
2
under transformations of h̃ij 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
δh̃ij ¼ ∂ i χ j þ ∂ j χ i þ δij χ; ð5:12Þ action principle (4.19) can be cast in the manifestly duality
invariant form
as we could have expected. Z
VI. MANIFEST DUALITY INVARIANCE S½hij ; h̃ij ; K ij ; K̃ ij ; ¼ d4 x½2ϵimn ∂ m K̃ jn K_ ij
We can now implement Eqs. (5.7) and (5.11) in the − h_ ij ϵimn ∂ m Rnj ½h̃ − H; ð6:8Þ
action principle (4.19). Let us first compute the quadratic
term in Pij : where
1 ij − H ¼ 2ϵimn ∂ m K̃ nj Rij ½h − 2ϵimn ∂ m K ij Rnj ½h̃
P Pij ¼ 2∂ j K̃ ik ∂ j K̃ ik þ 2∂ k K̃∂ l K̃ kl
2
2∂ j K ik ∂ j K ik þ 2∂ k K∂ l K kl − ∂ l K∂ l K − 3∂ l K kl ∂ m K km
− ∂ l K̃∂ l K̃ − 3∂ l K̃ kl ∂ m K̃ km : ð6:1Þ
þ 2∂ j K̃ ik ∂ j K̃ ik þ 2∂ k K̃∂ l K̃ kl − ∂ l K̃∂ l K̃
Remarkably, this has exactly the same form as the quadratic − 3∂ l K̃ kl ∂ m K̃ km ð6:9Þ
terms in K ij 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
K ij → K̃ ij ; K̃ ij → −K ij : ð6:2Þ continuous duality rotations of the dual metrics and
extrinsic curvatures:
The kinetic term Pij K_ ij is also invariant under (6.2) (up to
total derivatives):
hij cos θ sin θ hij
¼
h̃ij − sin θ cos θ h̃ij
P K_ ij ¼ 2ϵimn ∂ m K̃ jn K_ → −2ϵimn ∂ m K jn K̃_
ij ij
ij K ij cos θ sin θ K ij
¼ : ð6:10Þ
¼ 2ϵimn ∂ m K̃ jn K_ ij þ total derivatives: ð6:3Þ K̃ ij − sin θ cos θ K̃ ij
105014-8MANIFEST ELECTRIC-MAGNETIC DUALITY IN LINEARIZED … PHYS. REV. D 103, 105014 (2021)
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 formu-
lation 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,
The covariant equations of motion and differential identities for a construction of actions for self-dual 2n þ 1-forms with
obeyed by the Weyl tensor and its Hodge dual can be auxiliary fields in 4n þ 2 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
three-dimensional metric and a dual extrinsic curvature. formulation
R [30]. Here one adds to the Abelian action S½A
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).
[1] J. Ehlers, Transformation of static exterior solutions of Proceedings, NATO Advanced Study Institute, Cargese,
Einstein’s gravitational field equations into different solu- France, 1997, edited by L. Baulieu, P. Di Francesco, M.
tions by means of conformal mappings, in Les Theories Douglas, V. Kazakov, M. Picco, and P. Windey, NATO Sci.
relativistes de la gravitation, Colloques Internationaux du Ser. C Vol. 520 (Dordrecht, Netherlands, 1999), pp. 121–139.
CNRS 91, 275 (1962). [5] T. Damour, M. Henneaux, and H. Nicolai, E10 and a Small
[2] R. P. Geroch, A method for generating solutions of Tension Expansion of M Theory, Phys. Rev. Lett. 89,
Einstein’s equations, J. Math. Phys. (N.Y.) 12, 918 (1971); 221601 (2002).
A method for generating new solutions of Einstein’s [6] P. C. West, E(11) and M theory, Classical Quantum Gravity
equations. 2, J. Math. Phys. (N.Y.) 13, 394 (1972). 18, 4443 (2001).
[3] E. Cremmer and B. Julia, The SO(8) supergravity, Nucl. [7] N. D. Lambert and P. C. West, Coset symmetries in dimen-
Phys. B159, 141 (1979). sionally reduced bosonic string theory, Nucl. Phys. B615,
[4] B. Julia, Group disintegrations, in Superspace and Super- 117 (2001).
gravity, Nuffield Gravity Workshop, Cambridge, England, [8] M. Henneaux and C. Teitelboim, Duality in linearized
1980, edited by S. W. Hawking and M. Roček (Cambridge gravity, Phys. Rev. D 71, 024018 (2005).
University Press, Cambridge, England, New York, U.S.A., [9] B. Julia, J. Levie, and S. Ray, Gravitational duality near
1981), pp. 331–350; B. L. Julia, Dualities in the classical de Sitter space, J. High Energy Phys. 11 (2005) 025.
supergravity limits: Dualizations, dualities and a detour [10] S. Hörtner, Manifest gravitational duality near anti de Sitter
via (4k þ 2)-dimensions, in Strings, branes and dualities. space-time, Front. Phys. 7, 188 (2019).
105014-9HAPÉ FUHRI SNETHLAGE and SERGIO HÖRTNER PHYS. REV. D 103, 105014 (2021)
[11] W. T. Emond, Y. Huang, U. Kol, N. Moynihan, and D. [22] S. Deser and R. I. Nepomechie, Electric-magnetic dua-
O’Connell, Amplitudes from Coulomb to Kerr-Taub-NUT, lity of conformal gravitation, Phys. Lett. 97A, 329
arXiv:2010.07861. (1983).
[12] A. Banerjee, E. O. Colg’ain, J. A. Rosabal, and H. Yavartanoo, [23] C. Bunster and M. Henneaux, The action for twisted self-
Ehlers as EM duality in the double copy, Phys. Rev. D 102, duality, Phys. Rev. D 83, 125015 (2011).
126017 (2020). [24] C. Bunster, M. Henneaux, and S. Hörtner, Gravitational
[13] S. Deser and C. Teitelboim, Duality transformations of electric-magnetic duality, gauge invariance and twisted self-
Abelian and non-Abelian gauge fields, Phys. Rev. D 13, duality, J. Phys. A 46, 214016 (2013).
1592 (1976). [25] C. Bunster and M. Henneaux, Supersymmetric electric-
[14] S. Ananth, L. Brink, and S. Majumdar, A hidden symmetry magnetic duality as a manifest symmetry of the action for
of quantum gravity, J. High Energy Phys. 11 (2018) 078; S. super-Maxwell theory and linearized supergravity, Phys.
Majumdar, Ehlers symmetry in four dimensions, Phys. Rev. D 86, 065018 (2012).
Rev. D 101, 024052 (2020). [26] P. Pasti, D. Sorokin, and M. Tonin, Lorentz-invariant
[15] K. Stelle, Renormalization of higher-derivative quantum actions for chiral p-forms, Phys. Rev. D 55, 6292 (1997);
gravity, Phys. Rev. D 16, 953 (1977). P. Pasti, D. Sorokin, and M. Tonin, Phys. Rev. D 52,
[16] S. L. Adler, Einstein gravity as a symmetry-breaking effect R4277 (1995).
in quantum field theory, Rev. Mod. Phys. 54, 729 (1982). [27] B. McClain, Y. S. Wu, and F. Yu, Nucl. Phys. B343, 689
[17] J. Maldacena, Einstein Gravity from Conformal Gravity, (1990); C. Wotzasek, Phys. Rev. Lett. 66, 129 (1991); F. P.
arXiv:1105.5632. Devecchi and M. Henneaux, Phys. Rev. D 54, 1606 (1996);
[18] E. Fradkin and A. A. Tseytlin, Conformal supergravity, I. Martin and A. Restuccia, Phys. Lett. B 323, 311 (1994);
Phys. Rep. 119, 233 (1985). N. Berkovits, Phys. Lett. B 388, 743 (1996).
[19] D. Butter, F. Ciceri, B. de Wit, and B. Sahoo, Construction [28] A. Sen, Self-dual forms: Action, Hamiltonian and compac-
of all N ¼ 4 Conformal Supergravities, Phys. Rev. Lett. tification, J. Phys. A 53, 084002 (2020).
118, 081602 (2017). [29] S. Deser and D. Seminara, Free spin 2 duality invariance
[20] N. Berkovits and E. Witten, Conformal supergravity in cannot be extended to GR, Phys. Rev. D 71, 081502
twistor-string theory, J. High Energy Phys. 08 (2004) 009. (2005).
[21] H. Liu and A. A. Tseytlin, Nucl. Phys. B533, 88 (1998); V. [30] E. Witten, On S-duality in Abelian gauge theory, Sel. Math.
Balasubramanian, E. G. Gimon, D. Minic, and J. Rahmfeld, 1, 383 (1995).
Phys. Rev. D 63, 104009 (2001).
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