Two-vierbein gravity action from the gauge theory of the

PHYSICAL REVIEW D 100, 084012 (2019)

Two-vierbein gravity action from the gauge theory of the conformal group
Iva Lovrekovic *
The Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom
(Received 2 April 2019; published 7 October 2019)
We study the gravity action built from two gauge fields corresponding to the generators of the conformal group. Starting with the action from which one can obtain Einstein gravity and conformal gravity upon imposing suitable constraints, we keep two independent gauge fields and integrate out the field corresponding to the generator of Lorentz transformations. We identify the two gauge fields with two vierbeins and perturb them around anti–de Sitter space. This gives the linearized equations that differ from both Einstein gravity and conformal gravity linearized equations. We also study the linearized equations for one gauge field perturbed around the flat space and one around zero, and the case in which the gauge fields are proportional to each other.
DOI: 10.1103/PhysRevD.100.084012

I. INTRODUCTION
Conformal gravity was interpreted as a gauge theory of conformal group O(4,2) by Kaku et al. [1] in 1977. The motivation to study it was the fact that Einstein gravity has been viewed as a gauge theory of the de Sitter group O(3,2) [2], which upon contraction reduces to the Poincare´ group. Squaring the curvatures of the de Sitter group, one obtains Einstein gravity [2], while the Poincare´ group and the de Sitter group are subgroups of the conformal group O(4,2). It is natural to look at the square of the curvature of O(4,2). To achieve the invariance of a constructed action under proper conformal gauge transformations, the authors had to require that the gauge generator of the translations vanishes. The resulting action is invariant under conformal transformations, and it is a gauge theory of the conformal group. It is built out of three independent gauge fields. Upon integrating out the gauge fields, we are left with the remaining two. This situation where one encounters two different fields appears in bimetric gravity models, which contain two dynamical metrics. These models [3–5] originated from the de Rham– Gabadadze–Tolley (dRGT) massive gravity model [6–9]. It has been shown that other higher derivative theories, one of them being conformal gravity, can be rewritten and obtained from bimetric and partially massless bimetric theory [10]. This has further motivated a study of bimetric gravity [3], whose action takes the form [3]
*[email protected]
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Z pffiffiffiffiffiffiffiffiffiffiffiffiffi

Z pffiffiffiffiffiffiffiffiffiffiffiffiffi

S ¼ M2g d4x − det gRðgÞ þ M2f d4x − det fRðfÞ

Z pffiffiffiffiffiffiffiffiffiffiffiffiffi X4 qffiffiffiffiffiffiffiffiffi þ 2m2M2eff d4x − det g βnen g−1f : ð1Þ
n¼0

RðgÞ and RðfÞ are Ricci scalars with respect to metrics gμν and fμν, Mg and Mf are two different Planck masses, and Meff is an effective Planck mass. The en pareffiffiffieffiffilffieffiffimffi entary symmetric polynomials in eigenvalues of g−1f, and βn are four combinations of the mass of the graviton, the cosmological constant, and the free parameters. The graviton mass and cosmological constants for gμν and fμν are among five free parameters of the theory. Four-dimensional spin-2 theories have recently been studied within the different dimensional reduction schemes coming from five-dimensional ChernSimons gauge theories. The resulting actions were fourdimensional generalizations of Einstein-Cartan theory, conformal gravity, and bimetric gravity [11].
Here, we study linearized gravity, perturbed around maximally symmetric space, as a gauge theory of the conformal group while keeping two dynamical gauge fields. We find that perturbing the equations around anti–de Sitter (AdS) space gives degeneracy in the fields. The reason for this comes from the symmetric appearance of the gauge fields in the initial action and perturbation around maximally symmetric space. The linearized theory is different from the sum of linearized Einstein gravities for the two metrics since the equations of motion do not come from the corresponding Einstein actions, where the linearized MacDowell-Mansouri action has been studied in Ref. [12]. It also differs from linearized conformal gravity since we do not require invariance under the proper

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IVA LOVREKOVIC
conformal gauge transformations, and a vanishing of the generator of translations which has in Ref. [1] been imposed “by hand.”
Comparison with linearized Einstein gravity (EG) and conformal gravity (CG) further shows that the original action should consist out of the two Ricci scalars, one for each metric, and an additional potential. Just like CG, the action has one dimensionless parameter α, but two dynamical gauge fields, as one would expect from gauge theory for bimetric gravity. We also compare the linearized equations to the linearized equations of bimetric gravity. One could remove the degeneracy between the fields by introducing a parameter multiplying one of the gauge fields; however, the fields would still be linearly dependent. In order for them not to be linearly dependent, one would need to have the kinetic part modified. Another possibility for removing the degeneracy would be to perturb the fields around different backgrounds; for example, one of the fields could be perturbed around the AdS background, and another around a black hole. For now, we focus on the perturbations of both of the fields around AdS space, perturbation of one field around AdS space and the other around flat space, and in the nonperturbative case where gauge fields depend linearly on each other. The content of the article is as follows. Section II describes the action and corresponding equations of motion, while Sec. III analyzes them as a perturbation around the maximally symmetric spaces. In Sec. IV we obtain the linearized equations of motion for the two gauge field fluctuations, perturbed around the AdS space. In Sec. V we show an example of linearization around Minkowski space, while in Sec. VI we consider the case in which the gauge fields are proportional to each other. In Sec. VII we discuss the results and possible future prospects.

II. ACTION

The most general parity conserving quadratic action that can be constructed using the curvatures of a conformal group with no dimensional constants is [1]

Z

I ¼ α d4xϵμνρσϵabcdRμνabðJÞRρσcdðJÞ

ð2Þ

8

for α as a dimensionless constant,

RμνabðJÞ ¼ Rμνab − 2ðeaμfbν − ebμfaνÞ

þ 2ðeaνfbμ − ebνfaμÞ;

ð3Þ

and

Rμνab ¼ −∂μωνab þ ∂νωμab þ ωcμaωνcb − ωcνaωμcb: ð4Þ

It consists of the gauge fields eaμ and faμ, which appear symmetrically in the action, and spin connection ωμab.

PHYS. REV. D 100, 084012 (2019)

If we rewrite the action using Eq. (3) and omit the topologically invariant, Gauss-Bonnet term ðRμνabðωÞÞ2, the action becomes

αZ I¼
8Z ¼α
8

d4xϵμνρσϵabcdð−16Rμνabecρfdσ þ 64eaμfbνecρfdσÞ

d4xL;

ð5Þ

which contains three independent fields ωμab; eaμ, and faμ. The fields eaμ and faμ appear symmetrically in action, so we treat them on equal footing. If one imposes the requirement that the action is invariant under proper conformal gauge transformations, one needs to require that the gauge generator of translations

RμνaðPÞ ¼ −ð∂μeaν − ωμbaebνÞ þ ð∂νeaμ − ωνbaebμÞ

þ ðeaμbν − eaνbμÞ

ð6Þ

vanishes. This constraint on the generator determines the gauge field ωμab identified with spin connection. The gauge field bν is a generator of dilatations, and it does not appear in the action. Action (5) is scale and proper conformal invariant for ω ¼ ωðeÞ. Keeping this spin connection, one
can also integrate out the nonpropagating field faμ to obtain the

Z

I ¼ α d4xCμνabCρσcdϵμνρσϵabcd

ð7Þ

8

conformal gravity action; here, Cμνab is a Weyl tensor. One more approach to consider action is without
background expectation value for the field faμ. One can integrate out faμ to obtain an action that depends on ωμab and eaμ. The action would be nonunitary and similar to the Weyl squared action but different from it since ωμab would be an independent field and not a function of eaμ.

A. Equations of motion
Varying the Lagrangian under action (5) with respect to ωμab, one obtains its equation of motion,
δωL ¼ ð−2ecν∂ρfdσ þ 2ecνωkρdfkσ − 2fcν∂ρedσ þ 2fcνωkρdekσÞϵμνρσϵabcd ¼ 0; ð8Þ
in terms of the eaμ and faμ gauge fields. Since the fields eaμ and faμ appear symmetrically, we can compute the equation of motion for one gauge field and know it for the other gauge field as well. If we assume that eaμ is invertible and has a nonzero determinant, we can determine its equation of motion from variation with respect to eiκ,

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δeL ¼ ϵμνκσϵabid½−Rμνabfdσ þ 8fbνfdσeaμŠ ¼ 0; ð9Þ
while for the analogous equation for fiκ we have to take analogous assumptions for faμ,

δfL ¼ ϵμνκσϵabid½−Rμνabedσ þ 8ebνedσfaμŠ ¼ 0; ð10Þ

which corresponds to [1]

1 1 

faμ ¼ − 4 Raμ − 6 Reaμ :

ð11Þ

Here, we have used the contractions

III. PERTURBATION AROUND vaμ = f (a0μ)
We choose the background with vaμ ¼ fðμ0Þa. In the leading order the solution for equation of motion (8) is

ωðν0aÞb ¼ − 12 ðvbβ∂βvaν þ vaαvβbvcνð−∂αvcβ þ ∂βvcαÞ − vaβ∂βvbν − vbβ∂νvaβ þ vaβ∂νvbβÞ;

ð18Þ

which agrees with the well-known spin connection for Einstein gravity. Leading order equations (9) and (10) will expectedly give an equal solution, which is an Einstein action with the cosmological constant

Rbμ ¼ Rμνabeaν;

R ¼ Raμeaμ;

ð12Þ

and fμν ¼ eaμfaν. Equation (10), inserted back, is known to give conformal gravity action for a vanishing of the translation generator [1,13,14]. However, we keep both of the gauge fields dynamical and perturbatively solve Eq. (8) for ωμab.
We introduce perturbations of the gauge fields

eaμ ¼ vaμ þ η χaμ þ η2ζaμ þ Á Á Á ;

ð13Þ

faμ ¼ fðμ0Þa þ ηθaμ þ η2ψ aμ þ Á Á Á ;

ð14Þ

and the perturbation of spin connection ωμab,

ωμab ¼ ωðμ0aÞb þ ηωðμ1aÞb þ η2ωðμ2aÞb þ Á Á Á ;

ð15Þ

Rðμ0νÞ − 4vμν ¼ 0:

ð19Þ

Here, we have defined vμν ¼ vbμvbν. For the analysis of the linear order, it is convenient to introduce the tensor

eaμfbν ¼ Qabμν;

ð20Þ

whose subleading order reads

Qða1bÞμν ¼ vbν χaμ þ vaμθbν;

ð21Þ

and we rewrite the subleading order of Eq. (8) in terms of it:

vd½νvkσωðρ1Š Þck − vc½νvkσωðρ1Š Þdk − ∂½ρQð1Þ½cdŠνσŠ ¼ 0: ð22Þ

with an η small perturbation parameter. In the expansion of curvatures in Eq. (10),

Rbμ ¼ Rðb0μÞÞ þ ηRðb1μÞ þ Á Á Á

ð16Þ

for Rðb0μÞ ¼ Rðμ0νÞabvaν, one needs to take into account the contractions Rðb1μÞ ¼ Rðμ1νÞabvaν þ Rðμ0νÞab χ˜ aν from Eq. (12).
Analogously, the expansion of the Ricci scalar is

R ¼ Rðb0μÞvbμ þ ηðRðb1μÞvbμ þ Rðb0μÞ χ˜ bμÞ þ Á Á Á : ð17Þ

The allowed vacuum points around which we can perturb the action and equations of motion need to be backgrounds with curvature. One would naively perturb the fields around the flat background; however, the choice of eaμ ¼ faμ would not satisfy the equation of motion for faμ or eaμ if both of them are flat. If one of them were flat, the other one would have to be zero. One could further analyze around which backgrounds is it allowed to perturb the solution by studying the allowed solutions, as was done for Einstein theory in Ref. [15].

The combinations of the Qabμν tensor which appear in Eq. (22) allow us to rewrite the partial derivatives in terms of the general covariant derivative defined on the background space because the Christoffels and spin connections added and subtracted to form the covariant derivative exactly cancel. One obtains

ωðκ1aÞb ¼ 12 vcαvdβðvbκvaγ − vbγvaκÞ∇½αQð1Þ½cdŠβ㊠þ vdβðvaαηbc − vbαηacÞ∇½αQð1Þ½cdŠδκŠ:

ð23Þ

The subleading order of the spin connection consists of the background vielbeins which are a solution of Eq. (19), Einstein spaces, and fluctuations χaμ, θaμ, which will be defined through Eqs. (9) and (10). The subleading order of Eq. (9),





θbμ ¼ − 14 Rðb1μÞ − 16 Rð0Þ χbμ − 16 Rð1Þvbμ ;

ð24Þ

consists of

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Rðb1μÞ ¼ ð−∂μωðν1aÞb þ ∂νωðμ1aÞb þ ωcμða0Þωðν1cÞb − ωcνað0Þωðμ1cÞb þ ωcμða1Þωðν0cÞb − ωcνað1Þωðμ0cÞbÞvaν þ ð−∂μωðν0aÞb þ ∂νωðμ0aÞb þ ωcμða0Þωðν0cÞb − ωcνað0Þωðμ0cÞbÞ χ˜ aν ð25Þ

for

Rð1Þ ¼ Rðb1μÞvbμ þ Rðb0μÞ χ˜ bμ

ð26Þ

and Rð0Þ ¼ Rðb0μÞebμ, and it gives the dependence of χaμ and θaμ.

IV. AdS BACKGROUND
We set the background perturbation to the AdS metric, which is Weyl flat, allowing us to write

vaμ ¼ ρðxÞδaμ

ð27Þ

and the leading order spin connection

ωðν0aÞb ¼ −δ½aν∂bŠρðxÞ;

ð28Þ

here, we denote ∂b ¼ δμb∂μ. Equations (9) and (10) reduce to Rða1μÞ ¼ 4δμν. The subleading order of Eq. (8), just as Eq. (23) after a few technical manipulations, shows that the linear term in the ωμab perturbation can be rewritten in terms of the sum of two linear terms of Einstein spin
connections,

ωðκ1aÞk ¼ ωðκ1aÞkð χÞ þ ωðκ1aÞkðθÞ:

ð29Þ

Here,

ωðκ1aÞkð χÞ ¼ − 41ρ ðδαa∇α χkκ þ δαk∇κð χaαÞ þ δαkδbκ δβa∇β χbαÞ

−a↔k

ð30Þ

is the linearized spin connection for Einstein gravity, and ∇ denotes the Lorentz covariant derivative. For transparency, we keep the Lorentz covariant derivative, and do not evaluate it for background AdS. The expression for the linearized spin connection evaluated on AdS is given in the Appendix. This form of ωðμ1aÞb allows us to split the curvatures in parts depending only on χaμ or θaμ fluctuation. Therefore, we can write the subleading order of the Riemann tensor as sum of linearized Riemann tensors for Einstein gravity. The subleading order of the Ricci tensor, however, will not be possible to write in the form of two linearized Ricci tensors for Einstein gravity because of the term Rðμ0νÞab χ˜ aν (Rðμ0νÞabθ˜aν), which is visible from Eq. (25):

PHYS. REV. D 100, 084012 (2019)

Rðb1μÞ ¼ ðRðμ1νÞabð χÞ þ Rðμ1νÞabðθÞÞvaν þ Rðμ0νÞab χ˜ aν: ð31Þ
Here,
Rðμ1νÞabð χÞ ¼ −∂μωðν1aÞbð χÞ þ ∂νωðμ1aÞbð χÞ þ ωcμða0Þωðν1cÞbð χÞ − ωcνað0Þωðμ1cÞbð χÞ þ ωcμða1Þð χÞωðν0cÞb − ωcνað1Þð χÞωðμ0cÞb ð32Þ

is a linearized Riemann tensor for Einstein gravity. We contract Eq. (24) with vbσ and write





θbμvbσ ¼ − 14 Rðb1μÞvbσ − 16 Rð0Þ χbμvbσ − 16 Rð1Þvbμvbσ :

ð33Þ

In terms of the Einstein gravity perturbations in the fields χaμ and θaμ, using Eqs. (31) and (26), this is



θbμvbσ ¼ −1 4

Rðμ1νÞabðχÞ þ Rð1ÞμνabðθÞvaν



þ Rð0Þμνab χ˜ aν − 1Rð0Þ χbμ vbσ

6

− 1ððRðα1νÞacðχÞ þ Rðα1νÞacðθÞÞvaν þ Rðα0νÞac χ˜ aνÞvcαvbμvbσ

6



− 1Rðc0αÞ χ˜ cαvbμvbσ :

ð34Þ

6

This way, one obtains the constraint on the χμν related to θμν. An analogous appearance of both equations of motions
for the faμ and eaμ gauge fields assuming them invertible implies that equation for χbμ is

χbμvbσ



¼ −1 4

Rðμ1νÞabðθÞvaν þ Rð1Þμνabð χÞvaν



þ Rð0Þμνabθ˜ aν − 1 Rð0Þθbμ vbσ

6

− 1 ðRðα1νÞacðθÞvaνvcα þ Rðα1νÞacð χÞvaνvcα

6



þ 2Rðα0νÞacθ˜ aνvcαÞvbμvbσ :

ð35Þ

If we subtract Eqs. (35) and (34), we obtain

ðθbμ − χbμÞvbσ





¼ − 1 Rð0Þμνabð χ˜ aν − θ˜ aνÞ − 1 Rð0Þð χbμ − θbμÞ vbσ

4

6



− 1 ð2Rðα0νÞacvcαÞvbμvbσð χ˜ aν − θ˜ aνÞ :

ð36Þ

6

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The equation does not contain any linearized curvatures
due to their cancellation. The reason for this is that
the terms with the linearized Riemann tensor can be
written as a sum of the linear Riemann tensor for Einstein gravity and can contain both perturbations, χaμ and θaμ, in both Eq. (34) and Eq. (35). Subtracting the equations will cancel these terms. Using the conventions Rðμ0νÞαβ ¼−λ˜ð−vμβvνα þvμαvνβÞ, Rαβ ¼ 3λ˜vαβ, χ˜ aν ¼ − χaν, and θ˜aν ¼ −θaν, we evaluate Eq. (36) and get

λ˜ðθμσ − χμσÞ ¼ 2ð2 þ λ˜Þðθσμ − χσμÞ

ð37Þ

for λ˜ ¼ −1; this is

χμσ − θμσ ¼ 2ðθσμ − χσμÞ

ð38Þ

or

θμσ þ 2θσμ ¼ 2 χσμ þ χμσ:

ð39Þ

Owing to Lorentz invariance, we can impose a gauge in which χaμ is a symmetric matrix, χaμ ¼ χμa. This would imply that χaμvaν ¼ χμavaν → χμν ¼ χνμ. This condition
requires that

θμσ þ 2θσμ ¼ 3 χσμ:

ð40Þ

Summing Eqs. (35) and (34) and using the same notation give
ðθσμ þ χσμÞ ¼ − 1 ðk:t: þ λ˜ð χμσ þ θμσÞ − 2λ˜ð χσμ þ θσμÞÞ 4 ð41Þ

for the k.t. kinetic term





k:t: ¼ 2Rðμ1νÞabð χ þ θÞvaνvbσ − 13 Rðα1νÞacð χ þ θÞvaνvcαvμσ :

ð42Þ

To evaluate the linear term Rðμ1νÞab ¼ δRμνab, we linearize the tensor in the metric formulation and use the projection to the tetrad formulation

δRμνcdð χÞ ≡ Rðμ1νÞcdð χÞ ¼ Rð1Þλσμνvλcvσdð χÞ − Rðμ0νÞabδac χbd − Rðμ0νÞab χacδbd: ð43Þ

We then obtain

k:t: ¼ 6λ˜ðhμσ þ qμσÞ − DσDμðh þ qÞ − D2ðhμσ þ qμσÞ

þ 2DðμDαðhα þ qα Þ − 1 ð3λ˜ðh þ qÞ − D2ðh þ qÞ σÞ σÞ 3

þ DαDβðhαβ þ qαβÞÞvμσ − 2λ˜ð χμσ þ θμσÞ:

ð44Þ

Here, we have defined hμν ¼ vaμ χaν þ vaν χaμ and qμν ¼ vaμθaν þ vaμθaν, their traces h and q, respectively, and we have not used any gauge conditions. The last term in
Eq. (44) comes from the two last terms in Eq. (43). For the
sum of the constraint equations on the linear term in the
perturbation of the gauge field, from Eq. (41) we obtain

0 ¼ −DσDμðh þ qÞ

− D2ðhμσ þ qμσÞ þ 2DðμDαðhασÞ þ qασÞÞ

− 1 ð−D2ðh þ qÞ þ DαDβðhαβ þ qαβÞÞvμσ 3
þ 6λ˜ðhμσ þ qμσÞ − λ˜ðh þ qÞvμσ

− λ˜ð χμσ þ θμσÞ − ð2λ˜ − 4Þð χσμ þ θσμÞ

ð45Þ

for 2DðμDαhσÞα ¼ DμDαhσα þ DσDαhμα. One can also

choose

the

de

Donder

gauge

D

α

ðhα

β

þ

q

α

β

Þ

¼

1 2

D

β

ðh

þ

qÞ,

which keeps in the equation Laplace operators acting on the

sum of the symmetrized linear terms in the expansion of the

gauge field, their traces, and the mass terms

0 ¼ −D2ðhμσ þ qμσÞ þ 1 D2ðh þ qÞvμσ þ 6λ˜ðhμσ þ qμσÞ 6
− λ˜ðh þ qÞvμσ − λ˜ð χμσ þ θμσÞ − ð2λ˜ − 4Þð χσμ þ θσμÞ:
ð46Þ

For λ˜ ¼ −1 Eq. (46) becomes

vaσTð1Þaμ ≡−D2ðhμσ þ qμσÞþ 1D2ðh þ qÞvμσ 6
−6ðhμσ þqμσÞþðhþqÞvμσ

þðχμσ þθμσÞ þ6ðχσμ þθσμÞ

¼ −D2ðhμσ þqμσÞ þ1D2ðhþqÞvμσ −5ðhμσ þqμσÞ 6

þ5ðχσμ þθσμÞþðhþqÞvμσ;

ð47Þ

we call this equation “vaσTð1Þaμ.” From Eqs. (38) and (47) one can notice that fluctuations cannot be fixed independently; they appear as a sum, which implies that there is an extra symmetry.
Highly symmetric equations (38) and (47) are pointing out the degeneracy of the perturbations around the maximally symmetric background. This becomes obvious when one tries to symmetrize Eq. (38). One obtains the equality χμσ þ χσμ ¼ θσμ þ θμσ, which inserted into symmetrized equation (47) leads to two equal equations for χμσ þ χσμ

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and θμσ þ θσμ. One could further analyze symmetrized equation (47) as

0 ¼ −2D2ðhμσ þ qμσÞ þ 1 D2ðh þ qÞvμσ − 5ðhμσ þ qμσÞ 3

þ 2ðh þ qÞvμσ;

ð48Þ

rewriting the perturbations in the transverse traceless split,
and one could consider its one loop partition function;
however, one would have to keep in mind the implications
of Eq. (38).
Independently, one can antisymmetrize Eq. (47), which
will lead to cancellation of the derivatives and χμσ − χσμ ¼ −θμσ þ θσμ. With the Lorentz invariance requirement that χσμ is symmetric, antisymmetrizing Eq. (40), one obtains that θσμ is also symmetric. Equation (40) will then lead to θμσ ¼ χμσ.
Equation (48), however, cannot be compared to the
known linearized equations of EG or CG. As shown in
subsection C of the Appendix on the example of Einstein gravity, projection of the general perturbed tensor Tμν ¼ Tðμ0νÞ þ ηTðμ1νÞ is Tðμ1νÞ ¼ vaμTð1Þaν þ χaνTða0μÞ. We can recognize Eq. (47) as the vaμTð1Þaν part of the equation. To be able to compare the equation with linearized EG and CG from the literature, we have to obtain Tðμ1νÞ, i.e., we have to add χaνTða0μÞ to the vaμTð1Þaν tensor. After that, Eq. (47) becomes

T

ð1Þ σμ

¼

−8ð

χ

σμ

þ

θσμ

Þ

−

5ðhμσ

þ

qμσ

Þ





− D2ðhμσ þ qμσÞ − vμσ −ðh þ qÞ − 1 D2ðh þ qÞ ¼ 0;

6

ð49Þ

which can be symmetrized to give

− 9ðhμσ þ qμσÞ − D2ðhμσ þ qμσÞ





− vμσ −ðh þ qÞ − 1 D2ðh þ qÞ ¼ 0: ð50Þ

6

One can compare this to the linearized minimal bimetric gravity model where, for the massless spin-2 particle hμν and a massive spin-2 particle uμν of mass m, one has [3]

Z S ¼ d4xðhμνϵˆμναβhαβ þ uμνϵˆμναβuαβÞ

m2 Z

− 4 d4xðuμνuμν − uμμuννÞ:

ð51Þ

Here, ϵˆμναβ denotes the Einstein-Hilbert (EH) kinetic operator. One can notice that Eq. (49), as well as linear equations that would come from Eq. (51), has the form of

two equal operators acting on two separate fields and a mass term. In Eq. (49) the kinetic operator is not EH. One could think of the equation as consisting of two EH operators and additional mass terms. When Eq. (49) is symmetrized and one obtains Eq. (50), there are two equal kinetic operators for two degenerate fields, which can be thought of as two EH operators and mass terms. Upon lifting the degeneracy between the fields, one should be able to diagonalize the resulting equation such that there are two EH operators, one for each field, and remaining terms which belong only to one massive field, as in Eq. (51).
Analysis of the spin-2 massive graviton has been done in tetrad formulation for the dRGT model using similar methods [16]. A possibly convenient area of further consideration might be in terms of the field Qμναβ. If we express the subleading order equation (38) in terms of this tensor, it reads

Qðβ1μÞνσ − Qðν1σÞβμ ¼ 2ðQðσ1νÞμβ − Qðμ1βÞσνÞ;

ð52Þ

while symmetrized equation (48) is

0 ¼ −2D2ðQðμ1βÞσν þ Qðβ1μÞνσ þ Qðσ1νÞμβ þ Qðν1σÞβμÞ þ 13D2Qð1Þvμσvβν þ 2Qð1Þvμσvβν − 5ðQðμ1βÞσν þ Qðβ1μÞνσ þ Qðσ1νÞμβ þ Qðν1σÞβμÞ: ð53Þ

It can be useful to notice the property

Qðν1μÞβσ þ Qðβ1σÞνμ ¼ Qðβ1μÞνσ þ Qðν1σÞβμ:

ð54Þ

V. eaμ PERTURBED AROUND THE FLAT BACKGROUND AND f aμ AROUND ZERO
The linearized equations of motion when eaμ is perturbed around the flat background and faμ around zero in Eqs. (13) and (14) imply δaμ and zero, respectively, for leading order terms, and the subleading terms remain to be determined. The equation of motion for ωμab in the leading order vanishes because it is multiplied by the leading order term in the expansion of faμ. This naturally makes Eqs. (9) and (10) identically zero.
The subleading order of ωμab,

ωðμ1aÞb ¼ 16 ðδbρð−∂μ χaρ þ ∂ρ χaμÞ þ δaρð∂μ χbρ − ∂ρ χbμÞ

þ δaρδbαeð0Þdμð∂α χdρ − ∂ρ χdαÞÞ;

ð55Þ

agrees with a subleading term of ωμab in Einstein gravity, while the subleading order of Eq. (10) is





θðb1μÞ ¼ − 14 Rðb1μÞ − 16 Rð1Þδbμ :

ð56Þ

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The curvature terms in expansion are Rðb1μÞ ¼ ð−∂μωðν1aÞb þ ∂νωðμ1aÞbÞδaν and Rð1Þ ¼ Rðb1μÞδbμ. Following the procedure of
the previous chapter,

Rðb1βÞ ¼ 12 ð∂α∂γhγβ − ∂β∂αh þ ∂β∂γhγα − ∂γ∂γhαβÞδαb and

ð57Þ

Rð1Þ ¼ ∂β∂αhαβ − ∂β∂βh:

ð58Þ

Using the de Donder gauge and writing the derivatives with D,





θσμ ¼ 1 DαDαhσμ − 1 DαDαhδσμ :

ð59Þ

8

3

We can notice that there is dependency only on χ on the
right-hand side of Eq. (59), which is a result of the fact that, in ωðμ1aÞb, we have only χaμ appearing. The subleading order of ωðμ1aÞb does not depend on θaμ because, in the equation of motion that determines ωðμ1aÞb, fields faμ appear in pairs, which will make such terms vanish in the subleading order when fða0μÞ is vanishing.
The leading order of the Eq. (9) will vanish because the
perturbation of the faμ field is expanded around zero. The subleading order will also vanish because the first term of Eq. (9) is given by Rðμ0νÞabθaμ þ Rðμ1νÞabfða0μÞ, both of which vanish. The second term in Eq. (9) will be multiplied by the vanishing background fða0μÞ.

VI. eaμ IS PROPORTIONAL TO f aμ Taking the condition

faμ ¼ ρðxÞeaμ

ð60Þ

in the equation for ωμab, Eq. (8), with faμ ¼ ρðxÞeaμ; fμa ¼ ρðxÞ−1eμa; faμ ¼ ρðxÞ−1eaμ; ð61Þ

one obtains

2ρðxÞe½cj½νekσωρŠkjdŠ ¼ 2ρðxÞe½c½ν∂ρedŠσŠ þ e½c½ν∂ρρðxÞedŠσŠ: ð62Þ

To find ωμab, we multiply Eq. (62) with eνkηdiηρβ, δβδeνjeρaηdi, and eβkeρaηdi, respectively, and solve the
system of equations for ωμab:

ωνab ¼ 1 ðeaνebβ∂βρðxÞ − eaβebν∂βρðxÞÞ 2ρðxÞ
− 12 ðebβ∂βeaν þ eαaeβbecνð−∂αecβ þ ∂βecαÞ − eaβ∂βebν − ebβ∂νeaβ þ eaβ∂νebβÞ:

ð63Þ

This form of ωμab has been expected based on the known solution from Kaku et al. [1], where agreement is obtained by setting ρðxÞ as a constant. The condition of proportionality (60) would give the action

Z

Z

I ¼ d4xLs ¼ 8α d4xρðxÞðR þ 24ρðxÞÞe; ð64Þ

which is equal to Einstein gravity for ρðxÞ ¼ 1. Here, we used contractions

Rbμ ¼ Rðμ0νÞabeaν;

R ¼ Rμaeaμ:

ð65Þ

Obtaining Einstein gravity from Weyl gravity has been

studied from different angles [17,18]. In Ref. [18] the

relation between the Weyl and Einstein gravities have been

studied via breaking conformal gauge symmetries. After

imposing the relation between the gauge fields fμν and eμν

which breaks the conformal gauge symmetries, the

obtained Lagrangian agrees with the Lagrangian in

Eq.

(64)

when

ρðxÞ

→

−

1 4

ρð

x0

Þ

;

i.e.,

ρðxÞ

is

taken

to

be

−

1 4

ρðx0Þ

constant.

VII. DISCUSSION
We have studied linearized equations of motion of the parity conserving action constructed from curvatures of conformal group. Since we have not imposed additional constraints by hand, the result is highly symmetric. One can notice that the symmetry which appears between the linearized fields χμν and θμν is a consequence of the symmetry which appears in the action, and one can speculate on whether its origin reaches the relations among the generators of special conformal transformations (SCTs) and translations (Ts) in the conformal group. The difference between the SCTs and Ts in a conformal group is due to a minus sign that, if absorbed in the SCT generator, reemerges in a change of sign of different commutation relation.
We have obtained the constraint equations on the fluctuations in the expansion of the gauge fields eaμ and faμ around the background AdS. When the constraint equations are symmetrized, one obtains two equal linearized expressions for both fields. The reason for this degeneracy, besides the conformal group, is in the perturbation around AdS space. For comparison, EG describes a massless graviton, and CG describes one massless and one partially massless mode. Here, the perturbations are

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IVA LOVREKOVIC
linearly dependent on each other, and the system has degeneracy. In order to count precisely the number of degrees of freedom, one would have to perform a canonical analysis of the theory. Based on current results, one may expect one massless and partially massless or massive mode. Inspecting the linearized equations and comparing them with the linearized equations of EG and CG, it is possible to speculate that the original effective theory consists of two Ricci scalars each for one metric and an additional potential. The exact form of the potential is yet to be studied. The parameter of the theory is an α dimensionless parameter inherited from the starting action. This is similar to the theory with CG, but unlike in CG there are two dynamical gauge fields, which is similar to dRGT theory.
It would be interesting to compute observables such as the one loop partition function for this theory and compare it to Einstein and conformal gravity, and possibly to look for generalizations to higher spins. If the generalization were to arbitrary dimensions, one could consider the general d-dimensional conformal algebra and its implications, which one could relate and motivate with multimetric theories [19]. One could also look into the implications of the gauge (40) and obtain symmetric vielbeins, as was done in Ref. [20].
ACKNOWLEDGMENTS
We are grateful to Arkady Tseytlin for the discussions and comments on the draft. The work was supported by Project No. J 4129-N27 in the framework of the ErwinSchrödinger Program of the Austrian Science Fund (FWF), and by Grant No. ST/P000762/1 of the Science and Technology Facilities Council (STFC).

APPENDIX A: INVERSE GAUGE FIELDS

To

obtain

the

inverse

of

the

perturbed

gauge

field

f

a μ

,

one

starts with the general form of the inverse gauge field f˜ μa.

The expansion of the latter,

f˜ μb ¼ f˜ ðb0Þμ þ ηθ˜ ðb1Þμ þ η2θ˜ ðb2Þμ þ η3θ˜ ðb3Þμ; ðA1Þ

in Oð0Þ order requires one to satisfy f˜ ðb0Þμfðμ0Þa ¼ δab. Multiplication of the two expansions in the leading order
gives that f˜ ðb0Þμ ¼ fðb0Þμ. The subleading order Oð1Þ gives the condition

fðα0Þaθ˜ ðb1Þα þ f˜ ðb0Þμθaμ ¼ 0;

from which it follows that θ˜ðb1Þα ¼ −fðb0Þμθðμ1Þafða0Þα. The order Oð2Þ leads to

θ˜ ðb2Þα ¼ −fðb0Þμθað2Þμfða0Þα þ θðγ1Þafða0Þαfðb0Þβθcβfðc0Þγ: ðA2Þ

PHYS. REV. D 100, 084012 (2019)

APPENDIX B: AdS BACKGROUND
When we consider above computation of the linear ωμab on the AdS background, it is most convenient to start from the equations of motion for ωμab. We can notice that Eq. (8) can be written as

αðecνRρσdðKÞ þ fcνRρσdðPÞÞϵμνρσϵabcd ¼ 0 ðB1Þ

for

RμνaðPÞ ¼ −ð∂μeaν − ωbμaebνÞ þ ð∂νeaμ − ωbνaebμÞ; ðB2Þ

RμνaðKÞ ¼ −ð∂μfaν − ωbμafbνÞ þ ð∂νfaμ − ωbνafbμÞ: ðB3Þ

In the leading order Eq. (B1) reads

αðvcνRðρ0σÞdðKÞ þ fðc0νÞRðρ0σÞdðPÞÞϵμνρσϵabcd ¼ 0; ðB4Þ

where we have used index (0) in Rðμ0νÞa to accent the order of perturbation. Since we use fðc0νÞ ¼ vcν, the equation reduces to

2αvcνRðρ0σÞdðPÞϵμνρσϵabcd ¼ 0;

ðB5Þ

where we can recognize the appearance of the no torsion condition, which corresponds to the requirement that the covariant derivative of the AdS vielbein vanishes. That means that in the subleading order

α½vcνðRðρ1σÞdðKÞ þ Rðρ1σÞdðPÞÞ þ χcνRðρ0σÞdðPÞ þ θcνRðρ0σÞdðPފϵμνρσϵabcd ¼ 0;

ðB6Þ

the second and the third term may be taken to zero due to the no torsion condition, so one obtains

αvcνðRðρ1σÞdðKÞ þ Rðρ1σÞdðPÞÞϵμνρσϵabcd ¼ 0 ðB7Þ

for

Rðρ1σÞdðPÞ ¼ −ð∂μ χaν − ωðμ0Þba χbν − ωðμ1ÞbavbνÞ þ ∂ν χaμ − ωðν0Þba χbμ − ωðν1Þbavbμ;

ðB8Þ

and Rðρ1σÞdðKÞ gives the same expression with θaμ on the place of χaμ in Eq. (B8).
Analogous to the procedure for Eq. (8), we can dualize Eq. (B7) to obtain the equation for ωðμ1aÞb:
vc½νðRðρ1σފdðKÞ þ Rðρ1σފdðPÞ − vd½νðRðρ1σފcðKÞ þ Rðρ1σފcðPÞÞ ¼ 0: ðB9Þ

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To solve Eq. (B9) for ωðμ1aÞb, we obtain three tensorial equations whose manipulation leads to the expression for ωðμ1aÞb. The simplification that can be taken for the AdS background is that the AdS background is Weyl flat, and
one can define

vaμ ¼ ρðxÞδaμ:

ðB10Þ

Here, ρðxÞ denotes the function of the coordinates on the manifold. The multiplication for obtaining the tensorial
equations is therefore also done by using Eq. (B10). To express ωðμ1aÞb, we use the Mathematica package xAct [21] and classify the terms as follows:
(1) Terms ωðμ1aÞbðω; χ; θÞ with ωμab, χða1μÞ, and θða1μÞ. (2) Terms ωðμ1aÞbð∂ χÞ with ∂μ χaν. (3) Terms ωðμ1aÞbð∂θÞ with ∂μθaν. There are no terms that involve the partial derivative acting
on the background vielbein. The reason for this becomes
clear from Eq. (B8). In the linear order we can have the
partial derivative of the background vielbein only from ωðμ0aÞb, while the remaining terms vanished due to the no torsion condition. (Below we omit writing (0) in ωð0Þ for simplicity.)
For the terms in 1, we obtain

ω˜ ðκ1aÞkðω; χ; θÞ ¼ − 41ρ ½ðωkbκ þ ωκbkÞðθba þ χbaÞ − ωabkðθbκ þ χbaފ;

ðB11Þ

here,

ωðκ1aÞkðω; χ; θÞ ¼ ω˜ ðκ1aÞkðω; χ; θÞ − ω˜ ðκ1kÞaðω; χ; θÞ: ðB12Þ The terms in 2 are ωðμ1aÞbð∂ χÞ ¼ ω˜ ðμ1aÞbð∂ χÞ − ω˜ ðμ1bÞað∂ χÞ and ω˜ ðμ1aÞbð∂ χÞ ¼ − 41ρ δkαδκb∂a χbα − 41ρ ∂a χkκ − 41ρ δkα∂κ χaα;
ðB13Þ

and the terms in 3 are equal to the terms in 2, with θaμ in place of χaμ: ωðμ1aÞbð∂θÞ ¼ ω˜ ðμ1aÞbð∂θÞ − ω˜ ðμ1bÞað∂θÞ:
ω˜ ðμ1aÞbð∂θÞ ¼ − 41ρ δkαδκb∂aθbα − 41ρ ∂aθkκ − 41ρ δkα∂κθaα: ðB14Þ

To identify the covariant derivatives, let us rewrite the θaμ part of Eq. (B11) with indices on ωμab not contracted:
− 41ρ ½ðδβkδαaδcκωβbc þ δαaωκbkÞθbα − δαaωαbkθbκŠ: ðB15Þ

Combining the third term from Eq. (B15) and the appropriate term from Eq. (B11), we have

δαað∂αθkκ − ωαckθcκÞ ¼ δαa∇αθkκ:

ðB16Þ

The remaining terms from Eq. (B15) analogously combine
with the antisymmetric pairs of the terms in Eq. (B11) to form covariant derivatives. Taking into account χaμ, θaμ, and Eqs. (B11)–(B14), we obtain

ωðκ1aÞk ¼ − 41ρ ðδαa∇αðθkκ þ χkκÞ þ δαk∇κðθaα þ χaαÞ þ δαkδbκ δβa∇βðθbα þ χbαÞÞ − a ↔ k:

ðB17Þ

For the EG spin connection it holds that

ωEGμab ¼ −ebνDμeaμ;

ðB18Þ

which is equal to Eq. (18) in the leading order, where we denote the covariant derivative with D. In the linearized order this is

ωEGð1Þμabð χÞ ¼ − χ˜ bνDμvaνvbνDðμ1Þvaν − vbνDμ χ˜ aν: ðB19Þ

We can write Eq. (29) as

ωðμ1aÞbð χ þ θÞ ¼ ωEμaGbð1Þð χÞ þ ωEμaGbð1ÞðθÞ: ðB20Þ

Linearizing Eq. (B20) around AdS, we can write the terms

ωðμ1aÞbEðGAdSÞð χÞ ¼ 21ρ2 ðð χbμ − χμbÞ∂aρ þ ð− χaμ þ χμaÞ∂bρ

þ ð− χab þ χbaÞ∂μρ

ðB21Þ

þ ðð−ηbμ − ημbÞ χνa þ ðηaμ þ ημaÞ χνbÞ∂νρÞ ðB22Þ

þ 1 ð∂a χbμ − ∂b χaμ þ δbν∂μ χaν − δaν∂μ χbν 2ρ

þ δμcðδbλ∂a χcλ − δaλ∂b χcλÞÞ

ðB23Þ

and ωðμ1aÞbEðGAdSÞðθÞ analogously. We can notice that the choice of symmetric perturbation χμb ¼ χbμ, χab ¼ χba reduces Eq. (B23) to

ð1ÞEG

ð−ημb χaν þ ημa χbνÞ∂νρ

ωμabðAdSÞsymmetric ¼

ρ2

ðB24Þ

1 þ 2ρ ð∂a χbμ − ∂b χaμ þ δμcðδbλ∂a χcλ − δaλ∂b χcλÞÞ:

ðB25Þ

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IVA LOVREKOVIC

Equation (B20) also requires that

− 41ρ ðvαa∇αð χkκÞ þ vαk∇κð χaαÞ þ vαkvbκ vβa∇βð χbαÞÞ − a ↔ k ¼ ð− χ˜ kνDκvνd þ vνkΓð1Þακνvαd − vνkDκ χdνÞηdc; ðB26Þ

where Γðκ1νÞα is Christoffel Γακν ¼ 12eαβð∂κeβν þ ∂νeβκ −∂βeκνÞ expanded for eμν ¼ eaμeaν, its expansion eμν ¼ vμν þ hμν, and

hμν ¼ vaμ χaν þ χaμvaν:

ðB27Þ

We have defined hμν as a symmetric term in the perturbation of eμν. Expansion is analogous for θaμ,

qμν ¼ vaμθaν þ θaμvaν:

ðB28Þ

Proving that Eq. (B20) holds makes it possible to write
the perturbation as a sum of perturbations in Einstein gravity. We can consider the linearized projection of the
Riemann tensor from the vielbein to metric formulation.
For the projection of the Riemann tensor, we know that Rλσμν ¼ eaλebσRμνab. When we rewrite the definition of Rabμν, Eq. (12), in terms of Eq. (B18), ωμab ¼ eaαebβΓαμβ − ebα∂μeaα, the projection gives us Rλσμν. The terms in the computation that contain one partial derivation ∂μ, ∂ν, and their combination fð∂μ; ∂νÞ (for the f function in ∂μ and ∂ν) at leading order separately cancel. Analogously, we consider them in linearized order.
We write the projection

Rλσμν ¼ eλaebσ ð−∂μðeaρeτbΓρντÞ þ ∂νðeaρeτbΓρμτÞ þ ∂μeτb∂νeaτ − ∂νeτb∂μeaτ

ðB29Þ

− ðeaρeτcΓρμτ − eτc∂μeaτ Þðecρ0 eτb0 Γρντ0 0 − eτb0 ∂νecτ0 Þ ðB30Þ

þ ðeaρeτcΓρντ − eτc∂νeaτ Þðecρ0 eτb0 Γρμ0τ0 − eτb0 ∂μecτ0 ÞÞ ðB31Þ

and linearize it. The linearized order projection is

Rð1Þλσμνð χÞ ¼ vaλvbσ Rð1Þμνabð χÞ þ Rð0Þμνabvaλ χbσ þ Rð0Þμνab χ˜ aλvbσ ;

ðB32Þ

PHYS. REV. D 100, 084012 (2019) the subleading order of Rλσμν ¼ −∂μΓλνσ þ ∂νΓλμσ − ΓλμαΓανσ þ Γλνα Γαμσ .

APPENDIX C: COMPARISON WITH EINSTEIN GRAVITY
An analogous consideration of Einstein gravity would lead to equations of motion in the subleading order

Gða1μÞ ¼ Rða1μÞ − 1 Rð1Þeaμ − 1 R χaμ ¼ 0:

2

2

ðC1Þ

Using the above method and the de Donder gauge leads to the constraint on χaμ

−λ˜ χμν − D2 χμν þ 1 ð2λ˜ χ þ D2 χÞvμν ¼ 0: 2

ðC2Þ

To compare this with the familiar result for the linearized Einstein operator, we have to consider hμν ¼ 2 χμν, which is symmetric, and

Gðμ1νÞ ¼ Gða1μÞvaν þ Gða0μÞ χaν ;

ðC3Þ

where Gða0μÞ χaν ¼ −3λ˜ χμν. We also need to take into account the cosmological constant, which is 6λ˜ χμν for four dimen-
sions. Adding this to Eq. (C2), we obtain a familiar result,

2λ˜ χμν − D2 χμν þ 1 ð2λ˜ χ þ D2 χÞvμν ¼ 0: 2

ðC4Þ

APPENDIX D: RELATIONS USED IN TEXT

Here we list several equations that were used in text

1 ϵabcdϵμνρσeaμebνecρedσ ¼ e; 4

ðD1Þ

1 ϵabcdϵμνρσecρedσ ¼ eðeμaeνb − eνaeμbÞ; 2!

ðD2Þ

δe ¼ eeμaδeaμ;

ðD3Þ

where e is a determinant.

[1] M. Kaku, P. K. Townsend, and P. van Nieuwenhuizen, Phys. Lett. 69B, 304 (1977).
[2] S. W. MacDowell and F. Mansouri, Phys. Rev. Lett. 38, 739 (1977); 38, 1376(E) (1977).

[3] S. F. Hassan and R. A. Rosen, J. High Energy Phys. 02 (2012) 126.
[4] S. F. Hassan and R. A. Rosen, J. High Energy Phys. 04 (2012) 123.

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