Gravitational Dressing of 3D Conformal Galileon
Gregory Gabadadze Giorgi Tukhashvili
  















 











    [ 1/2r∂∂Min-kow-ski
    
    NYU-TH-12/13/18
    
    
    
     Gravitational Dressing of 3D  Conformal Galileon
    Gregory Gabadadze and Giorgi Tukhashvili
     Center for Cosmology and Particle Physics, Department of Physics, 
    
    New York University, 726 Broadway,  New York, NY, 10003
    
    
    
    Could a conformal  Galileon  be describing  a gauge mode of a  broader diffeomorphism invariant theory?
    We  answer this question affirmatively  in  3D   by using a coset construction  for a nonlinearly
    realized conformal symmetry. In particular, we show that the conformal Galileon  emerges  as a Stückelberg
    field of a local Weyl  symmetry. The coset construction  gives us a  diffeomorphism  and Weyl invariant 3D theory containing
    first, second, and third  powers of the curvatures, their couplings to certain tensors made of the Galileon and its derivatives, and the conformal
    Galileon terms. In a theory with a boundary  additional surface terms are required for the boundary Weyl anomaly cancellation.
    When gravity is switched off, the  theory  reduces to the  conformal Galileon plus a boundary term.
    On the other hand, it reduces to a generalization of “New Massive Gravity" by A. Sinha,  if the Weyl
    symmetry is gauge-fixed in a particular way. Last but not least,  there exists a parameter space for which the
    theory has no propagating  ghosts. Thus we show by an explicit construction that “New Massive Gravity"
    (and its generalization) is a unique scalar ghost free low energy effective field theory non-linearly realizing conformal symmetry in 3D.
    
    
    
    
    
    
    
    § INTRODUCTION AND SUMMARY
    
    
    
    
    A Lagrangian for a  free scalar field can be amended by the mass and nonlinear interaction terms  that are polynomial
    in the field.  Theories  with up to a quartic polynomial are motivated by the renormalisability in 4D (notwithstanding the
    Landau pole), and are often used in  particle physics and cosmology. For a more general polynomial such a theory is not renormalizable but can be quantized as an effective field theory; the quantum loop-induced  interactions with derivatives would  be suppressed by powers of a momentum as compared to the renormalized polynomial terms.
    
    Alternatively, one could think of a single scalar field theory that is made nontrivial by
    interaction terms  containing derivatives of the field,  while the polynomials
    would not appear due to symmetry reasons (e.g., field space shift symmetry or Galilean invariance).
    Generically such theories cannot be truncated  to  a finite   number of interaction terms as this would give rise to a ghost;
    instead, they are usually viewed as a low energy approximation to a certain microscopic  theory.
    A well known example is the chiral Lagrangian for the Nambu-Goldstone mesons
    of massless QCD,  that gets completed into the asymptotically free quark-gluon
    Lagrangian;  embedding of General Relativity (GR) into string
    theory is another example.
    
    However, there seems to be an exception from the above lore for the higher derivative theories  within
    a class  that yields only second
    order equations of motion  <cit.>. A subset of this class  contains a scalar theory that  emerged
    within the brane  induced gravity  <cit.> in a certain limit  <cit.>,  and then
    in a more general setting <cit.>, where astrophysical and  cosmological applications were discussed
    and the name Galileon introduced.
    
    
    The Galileon  action exhibits field space galilean invariance  while its   Lagrangian  transforms as a total derivative.
    Owing to this property, there are only a  finite number of  Galileon terms in any finite number of dimensions, and they
    contain less derivatives per field than a generic  galilean invariant Lagrangian term would.  Because of this, the
    generic Lagrangian terms – that are exactly Galilean invariant –  are suppressed  by powers of derivatives
    as compared to the Galileons, and hence one can focus on the latter in a low energy approximation[A trivial example:
    (∂π)^2 is a Galileon, while the galilean invariant term, (∂∂π)^2, is not.]. Importantly,
    the truncation of the theory to the Galileon terms  does not  lead to additional degrees of freedom.
    This is unusual but a perturbative quantum low energy effective field theory with a finite number of Galileons has a
    well defined range of validity,  and is predictive <cit.>.
    
    In spite of this progress in  the perturbative quantum effective field theory, there are issues
    in the classical theory related to the lack of global hyperbolicity of the respective nonlinear equations of motion:
    while they contain only second derivatives acting on a field, π,
    they are also at least quadratic in π, leading in general
    to issues  of conceptual  and computational nature <cit.>.
    
    
    Understanding whether or not such issues  represent an impasse  is of
    great interest for both  a formal field theory of Galileons and  their potential applications in
    cosmology.  The present paper does not aim to study  hyperbolicity for Galileons,
    but rather to provide a possibly useful framework for addressing it as follows:
    It might be easier to tackle the issue if the Galileon is embedded into a diff invariant theory
    of gravity in which it would  just be a gauge mode of a tensor field that enters with no more derivatives than it does
     in GR.  For instance, in both the
    brane induced gravity <cit.> and the  diff invariant  massive gravity  <cit.>,
    the Galileon  appears as a gauge mode  describing the helicity-0
    state of a massive graviton in a certain limit. Only in that specific limit the Galileon acquires a
    gauge-invariant meaning.  Away from the limit  one could remove it by  turning to the unitary gauge, i.e., absorb
    it into the metric g_μν.  Its classical equations of motion are Einstein's  equations amended
     by a polynomial  in the metric and its inverse, but no extra derivatives. Moreover, the Bianchi identities lead to a constraint that is similar to a harmonic gauge-fixing condition of GR.  Hence, the issue of hyperbolicty in massive gravity in unitary gauge seems to be  at least somewhat similar to that
     in GR[The results of <cit.> might be interpreted
     as supporting a thought that the embedding  of the Galileons into a  gravitational
     theory helps with hyperboliciy.].
    
    Motivated by this, we set a goal to embed the conformal Galileon into a
    diffeomorphism invariant theory, where the conformal Galileon would be a gauge mode.
    In the present paper this will be done in 3D.  A sequel will address a 4D  construction.
    
    The paper is organized as follows: In section 2 we briefly collect the well known facts about
    the coset construction for a conformal group  that has  a Poincaré subgroup realized linearly, and the rest nonlinearly.
    
    Section 3 constitutes the bulk of the paper. Using the coset fields we first construct the effective Largangian in the absence
    of a dynamical gravitational field; we show that the inverse  Higgs constraint emerges  as a solution to the classical
    equations of motion,  and after solving it, reproduce the  Lagrangian  of the conformal Galileon field theory.
     En route we find that the conformal  Galileons can be obtained from a Lagrangian that is identical to that
    of the ghost free massive gravity  <cit.>,  but with  a different   Stückelberg sector.
    
    Next, we introduce a dynamical gravitational field  by gauging the tangent-space Poincaré symmetry.  We then  find the effective Lagrangian.
    It contains  four independent sets of terms that  enter with arbitrary coefficients not specified by the coset construction.
    The  key point is that  the conformal Galileon  naturally emerges  as a Stückelberg
    field of a local Weyl  symmetry. The obtained theory is  diffeomorphism  and Weyl invariant.  In addition
    to the conventional terms of a Weyl invariant 3D GR,  it contains second and third  powers of the curvatures, their couplings to certain
    tensors made of the Galileon and its derivatives, and the full set of the conformal Galileon terms. If a boundary is introduced,
    then additional surface terms are required  for the boundary Weyl anomaly  cancellation.
    When gravity is switched off, the  theory  reduces to the  conformal Galileon plus a boundary term. On the other hand, it reduces to a generalization of “New Massive Gravity" <cit.> by A. Sinha <cit.>,  if the Weyl symmetry is gauge-fixed in a particular way. Therefore we prove by construction that “New Massive Gravity" plus its curvature cubed generalization is a unique theory realizing  spontaneously broken conformal symmetry in 3D. Alternatively, one can gauge-fix a conformal  mode of the metric tensor by imposing, e.g.,
    unimodularity, det g=1, in which case one would  obtain the covariantized conformal Galileons coupled to unimodular 3D gravity.
    
    
    
    
    
    
    
    
    
    § THE SO(N,2)/ISO(N-1,1) COSET AND CONFORMAL GALILEONS
    
    
    
    We start by a  brief summary of  the coset construction for spacetime symmetries <cit.> and
    in particular  for a  conformal group spontaneously broken down to the Poincaré group <cit.>.
    The method  has been  nicely reviewed in <cit.> and  was used
    to construct the Lagrangians of conformal Galileons in a flat space. In the next section we will generalize the
    method to arbitrary gravitational backgrounds by gauging the tangent-space Poincaré symmetry (see <cit.> and references therein).
    
    The n dimensional conformal algebra is realized through the n(n+1)/2 generators of the Poincaré algebra with the generators conventionally denoted by J_ab,P_a, plus n generators of special conformal transformations, K_a,  and a generator of dilatations, D,  with the well-known  commutation relations:
    
        [ P_a, D ] = P_a,           [ D, K_a ] = K_a, 
        [ J_ab, K_c ] = η_a c K_b - η_b c K_a,     [ J_ab, P_c ] = η_a c P_b - η_b c P_a , 
        [ K_a, P_b ] = 2 J_ab- 2 η_a b D ,    [ J_ab, J_cd] = η_ac J_bd - η_bc J_ad + η_bd J_ac
        - η_ad J_bc .
    
    Following  the coset construction method  we should look at a  Maurer-Cartan form built out of the SO(n,2)/ISO(n-1,1) coset element Σ. A convenient parametrization for Σ is:
    Σ = e^π D e^ξ^c K_c .
    
    Since the Maurer-Cartan form is an element of the Lie algebra we may expand it in the basis of the generators:
    Σ^-1 ( d  +  dx^a P_a  ) Σ =
    E^a P_a + ω_K^a K_a + ω_D D - 1/2ω_J^ab J_ab ,
    
    where[Some clarifications: δ^a ≡δ^a_μ dx^μ. We are distinguishing the Latin indices from
    the Greek indices for the later convenience; when we introduce gravity
    the former will refer to a Local Lorentz group, while the latter to a space-time group.
    ξ^a is a zero form in space-time but a vector of the Local Lorentz group.]:
    
    E^a = e^πδ^a ,
    ω_D = d π + 2 E^a ξ_a,
    ω_K^a = d ξ^a - ξ^2 E^a + ξ^a ω_D,
    ω_J^ab = - 2 E^a ξ^b + 2 E^b ξ^a .
    
    Here π and ξ^a are the coset fields of  nonlinearly realized dilatations and special conformal transformations,
    which may or may not  end up giving rise to the respective dynamical  Nambu-Goldstone fields.  Indeed,
    since [ K_a, P_b ] ∼η_a b D, not all the Nambu-Goldstone fields parametrizing the coset
    should be physical and it  should be possible
    to eliminate the ones associated with the generator K_a. This could be achieved by imposing the inverse Higgs constraint (IHC)
    ω_D = 0<cit.>.
    
    In the present  paper we will take a different path, instead of imposing the IHC by hand at the level of the algebra
    we will  show that it emerges  as a consequence of the classical equation of motion for a properly built action.
    
    
    
    
    
    
    
    
    
    
    
    § THE EFFECTIVE LAGRANGIAN
    
    
    
    
     §.§ Flat background metric
    
    
    
    Simplest actions in 3D constructed via the coset elements of the last section are <cit.>:
    
        S_0     = ∫ε_abc E^a ∧ E^b ∧ E^c ;
        	 δ_ξ S _0 = 0  , 
        
        	S_1     = ∫ε_abc E^a ∧ E^b ∧ω_K^c ;
        	δ_ξ S_1 ≃ - ∫ε_ab f ω_D ∧ E^a ∧ E^b  δξ^f ,
        
        	S_2     = ∫ε_abc E^a ∧ω_K^b ∧ω_K^c ;
        	δ_ξ S_2 ≃ 2 ∫ε_ab f ω_D ∧ E^a ∧ω_K^b  δξ^f  ,
        
        	S_3     = ∫ε_abcω_K^a ∧ω_K^b ∧ω_K^c ;
        	δ_ξ S_3 ≃ 9 ∫ε_ab f ω_D ∧ω_K^a ∧ω_K^b  δξ^f .
    
    We did not include derivative terms as they'd give  additional  degrees of freedom.
    ω_D  has no Latin indices and does not   enable to construct  any nontrivial  coset
    term  (see, however,  the section on the Wess-Zumino terms). We've also written out above
    the variations with respect to ξ (the sign ≃ denotes
    equality up to a total derivative); these expressions
    confirm  that IHC, ω_D = 0, is a solution of the equations of motion.
    
    The total action is a sum of the above four actions each weighted with some coefficient which
    cannot be determined by the effective action formalism but instead should be constrained
    by various  consistency requirements and experimental/observational results whenever available.
    
    Here, we note that  one can rewrite
    the above actions in a form equivalent to that of  3D massive gravity potentials <cit.>
    but with a different  matrix 𝒦 entering those potentials.
    Indeed, define the matrix 𝒦  as follows:
    ω_K^a = E^a_ν𝒦_μ^ν dx^μ ,
    
    then use IHC, ω_D=0,  to deduce  the expression for 𝒦 in terms of π:
    𝒦_μ^ν = - 1/2 e^- 2 π( ∂_μ∂^νπ -
    ∂_μπ∂^νπ + 1/2( ∂π)^2 δ^ν_μ) ,
    
    and finally, substitute this expression into the above four actions, to get the four conformal  Galileon terms:
    
        S_0 =     - 6 ∫ d^3 x  e^3 π ,
        
        S_1 =     - 2 ∫ d^3 x  e^3 π[ 𝒦]
        ≃ - 1/2∫ d^3 x  e^π( ∂π)^2  ,
        
        S_2 =     - ∫ d^3 x  e^3 π( [ 𝒦]^2 - [ 𝒦^2 ] )  ≃ - 1/8∫ d^3 x  e^- π( ∂π)^2 L_1^TD , 
        
        S_3 =     - ∫ d^3 x  e^3 π( [ 𝒦]^3 - 3 [ 𝒦] [ 𝒦^2 ] + 2 [ 𝒦^3 ] ) ≃
        ≃   3/16∫ d^3 x  e^-3 π( ( ∂π)^2 L_2^TD - 3/2( ∂π)^4 L_1^TD + ( ∂π)^6 ) ,
    
    where,  L_1,2^TD are the total derivative terms of the respective order, L_n^TD∼ (∂∂π)^n.
    
    
    
    
    
    
     §.§ Dynamical metric
    
    
    The goal of this section is to dress the conformal Galileons with a dynamical gravitational field.
    In particular, we'd like to arrive at a theory
    in which a conformal Galileon field describes a gauge mode of a certain local symmetry. For this, we
    will follow a method used in <cit.>; it consists of gauging the Poincaré group by  introducing new gauge fields associated with the translations, e^a (vielbein),  and rotations ω^ab (spin connection). The exterior derivative in (<ref>) must be replaced with the  exterior covariant derivative. As before,
    we use the expansion of the Maurer-Cartan form in the basis of generators:
    Σ^-1( d + e^a P_a - 1/2ω^ab J_ab) Σ =
    E^a P_a + ω_K^a K_a + ω_D D - 1/2ω_J^ab J_ab .
    
    The covariantized counterparts  of (<ref>)-(<ref>) are:
    
    E^a = e^π e^a ,
    ω_D = d π + 2 E^a ξ_a ,
    ω_K^a = D ξ^a - ξ^2 E^a + ξ^a ω_D ,
    ω_J^ab = ω^ab - 2 E^a ξ^b + 2 E^b ξ^a .
    
    It should be emphasized that e^a = e^a_μ dx^μ is the vielbein of the pure gravitational field, while D is the covariant derivative with respect to e^a, and ω^ab is the spin connection of e^a. Using ω_J^ab we can build another coset element - the Riemann curvature two form:
    ℛ^ab = d ω_J^ab + ω_J^a_c ∧ω_J^cb = R^ab + 2 E^a ∧ω_K^b + 2 ω_K^a ∧ E^b ,
    
    where
    
    R^ab = d ω^ab + ω^a_c ∧ω^cb .
    
    The curvature two form   can  be used, alongside with the  fields utilized in the last section,
    to construct the effective Lagrangian terms.
    
    Before proceeding to this task, we recall that  a gravitationally covariantized conformal unitary field theory must be Weyl invariant <cit.>. To incorporate this property, we define the Weyl transformations as:
    
    e^a → e^σ e^a , π→π - σ , ξ^a →ξ^a + 1/2 e^-π∂^a σ .
    
    It is easy to verify that E^a,  ω_D, and ℛ^ab are all
    invariant under these transformations. Moreover,
    since ω_K^a is part of ℛ^ab it would be logical to construct the
    effective Weyl invariant actions using the above three elements only.
    
    In 3D the Weyl tensor is identically zero and R^ab is entirely determined in terms of the Ricci tensor and scalar,
    so it is more convenient to work in terms of these quantities rather than the Riemann two form R^ab itself:
    
    R^ab = e^a ∧S̅^b + S̅^a ∧ e^b  , S̅^a = R^a - 1/4 R e^a ,
    ℛ^ab = E^a ∧Ω^b + Ω^a ∧ E^b , Ω^a = 1/2 e^-πS̅^a + ω_K^a .
    
    Here R^a ≡ e^a_ν R^ν_μ dx^μ and R are the Ricci one form and scalar respectively. S̅^a is the first order analogue of the Schouten tensor, we will refer it as a Schouten one form. This quantity has interesting properties in 3D which are inherited from the curvature two form. In particular, from (<ref>) we see that it satisfies both Bianchi identities:
    
    D S̅^a = 0 ,  e_a ∧S̅^a = 0 .
    
    Expressions (<ref>) and (<ref>) enable  us to conclude that, in 3D, S̅^a fully replaces the curvature two form, R^ab. Following the same logic, from (<ref>) we also see that Ω^a can replace ℛ^ab as a coset element. Thus there are three Weyl invariant coset elements that can be used to build effective actions: ω_D,  E^a and Ω^a. The rest is straightforward, building the actions follows the same recepe as in the case of the flat metric. Actions and their variations with respect to ξ are given by:
    
        𝒮_0     = ∫ε_abc E^a ∧ E^b ∧ E^c ;
        δ_ξ𝒮_0 = 0  , 
        𝒮_1     = ∫ε_abc E^a ∧ E^b ∧Ω^c ;
        δ_ξ𝒮_1 ≃ - ∫ε_ab f ω_D ∧ E^a ∧ E^b  δξ^f  ,  
        𝒮_2     = ∫ε_abc E^a ∧Ω^b ∧Ω^c ;
        δ_ξ𝒮_2 ≃ 2 ∫ε_ab f ω_D ∧ E^a ∧Ω^b  δξ^f  ,  
        𝒮_3     = ∫ε_abcΩ^a ∧Ω^b ∧Ω^c ;
        δ_ξ𝒮_3 ≃ 9 ∫ε_ab f ω_D ∧Ω^a ∧Ω^b  δξ^f  .
    
    The actions, 𝒮_0,  𝒮_1,  𝒮_2, and 𝒮_3,  are respectively: the  Weyl invariant analogues of the cosmological term, the   Einstein-Hilbert kinetic term, the  Bergshoeff, Hohm & Townsend term <cit.> (usually referred as “New Massive Gravity"),  and the curvature cubed term <cit.>. It is clear  that IHC remains to be a solution of the equations of motion. Substituting that solution,
    the actions  can be rewritten  in the following  form:
    𝒮_0 = - 6 ∫ d^3 x √(g) e^3 π ,
    𝒮_1 = -2 ∫ d^3 x √(g) e^π( 1/8 R + e^2 π[ 𝒦] ) ,
    
        𝒮_2     = - ∫ d^3 x √(g) e^- π[ - 1/4( R_μν R^μν - 3/8 R^2 ) - e^2 π( R^μ_ν - 1/2δ^μ_ν R ) 𝒦_μ^ν + 
                                     + e^4 π( [ 𝒦]^2 - [ 𝒦^2 ] ) ] ,
        
        𝒮_3      = - ∫ d^3 x √(g) e^- 3 π[ 1/4 R_μ^ν R_ν^ρ R_ρ^μ - 9/32 R R_μν R^μν +17/256 R^3 + 
                                      + 3/4 e^2 π( 5/8 R^2 δ^μ_ν - 3/2 R R^μ_ν - R_αβ R^αβδ^μ_ν +2 R^μ_ρ R^ρ_ν) 𝒦_μ^ν + 
                                     + 3 e^4 π( R^μ_ν - 3/8 R δ^μ_ν) ( 𝒦_μ^ρ𝒦_ρ^ν - [ 𝒦] 𝒦_μ^ν) + 
                                     + e^6 π( [ 𝒦]^3 - 3 [ 𝒦] [ 𝒦^2 ] + 2 [ 𝒦^3 ] ) ] ,
    
    with
    𝒦_μ^ν = - 1/2 e^- 2 π( ∇_μ∇^νπ - ∂_μπ∂^νπ + 1/2( ∂π)^2 δ^ν_μ)  .
    
    Thus we've arrived at the action in which  the conformal galileon is  a  Stuckelberg field for the local Weyl invariance. It is also clear that in the limit when
    gravity is switched off  h_μν→ 0 (where h_μν = g_μν - η_μν; g_μν = e^a_μ e^b_νη_a b being the dynamical metric) the actions 𝒮_i, i=0,1,2,3 reduce to their flat space counterparts S_n,n=0,1,2,3.
    
    
    
    
    
    
    
    
    
    
    
    
     §.§ Wess-Zumino terms and anomalies
    
    
    It is known that there are no conformal anomalies in odd dimensions <cit.>.
    We can verify this in the case at hand  by explicitly building the Wess-Zumino terms.
    This is done by invoking  ω_D, which does not have a Lorentz index and can be used
    to create a four form in 3D. From the expressions below we see that the WZ terms
    coincide with (<ref>)-(<ref>) and therefore do not contain any new information:
    
        3 ∫ε_abc ω_D ∧ E^a ∧ E^b ∧ E^c     =  𝒮_0  , 
        ∫ε_abc ω_D ∧ E^a ∧ E^b ∧Ω^c     =  𝒮_1  , 
        
        	-∫ε_abc ω_D ∧ E^a ∧Ω^b ∧Ω^c     =  𝒮_2  , 
        
        	-3 ∫ε_abc ω_D ∧Ω^a ∧Ω^b ∧Ω^c     =  𝒮_3 .
    
    The absence of anomalies allows us to  eliminate the gauge mode π from the actions, therefore the obtained  theory in the unitary gauge coincides with “New Massive Gravity" <cit.> and its extensions <cit.>.
    
    What happens if we include a 2D boundary?  In that case and for classically scale invariant
    degrees of freedom there is a quantum trace anomaly on
    the boundary <cit.>:
    ∫ d^3 x √(g) T^μ_μ = ∫ d^3 x √(g)δ (x_⊥) ( c_1 r + c_2 k^μ_ν k^ν_μ) ,
    
    where the coefficients c_1  and c_2 depend on a field content and boundary conditions considered;
     δ (x_⊥) has support only on the boundary, r and k^μ_ν are its intrinsic and traceless part of extrinsic curvature respectively.
    Under the Weyl transformation r transforms, while the √(g) k^μ_ν k^ν_μ is invariant. The effective action that reproduces the trace
    anomaly can be expressed as <cit.>:
    𝒮_anomaly = ∫ d^3 x √(g)δ (x_⊥) [ c_1 ( π r +  (∂π)^2 )+ c_2 π k^μ_ν k^ν_μ] .
    
    This action should be included, with the opposite sign,  for  the Weyl symmetry to remain exact at the quantum level. If so then
    the  π  mode can  be eliminated by a gauge fixing. Alternatively,
    one could choose to  gauge fix a conformal mode of the metric tensor, by imposing,  e.g.,  a unimodularity condition,
    g=1, and thus obtaining unimodular gravity coupled to covariantised conformal Galileons. The boundary terms would be relevant if one were to study the holographic dual of the model in AdS space. Investigations in this interesting
    direction are beyond the scope of the present work.
    
    
    
    § DEGREES OF FREEDOM
    
    Last but not least, important comments concerning the degrees of freedom are in order here.
    The total  bulk action
    𝒮_tot = ∑_i=0^3 α_i  𝒮_i ,
    
    is a linear combination of the four actions with arbitrary coefficients denoted by α_i.
    These coefficients cannot be calculated  within the low energy effective field theory approach.
    However, various consistency conditions can be imposed on them.
    
    Since (<ref>) in general propagates ghosts,  we  impose the condition that any
    type of ghost to be absent  from the theory. Note that these ghosts are neither due to the
    conformal Galileons, nor the nonlinear terms of a conformal
    mode of the 3D tensor field  <cit.>, which are both ghost free. Instead,   the ghosts
    arise because  of the tensor in the  terms quadratic and cubic in the curvatures.
    
    To reveal the ghosts let us begin with  small perturbations above the Minkowski space in the unitary gauge
    π=0.  In that case the graviton propagator received contributions from  α_1  𝒮_1 and
    α_2  𝒮_2, while the cubic curvature terms  give rise to the interactions but not to a modification
    of the graviton propagator. Schematically, the graviton propagator is proportional to
    1α_1 ∂^2 + α_2 ∂^4 ,
    
    where  we have ignored the tensorial structure for simplicity. The second derivative
    arises from 𝒮_1, while the quartic derivative  comes form 𝒮_2.
    
     The above expression has two poles, describing  two states,  one massless and one massive, as it can
     also be seen from the equivalent rewriting of  (<ref>):
     1α_1 ∂^2   - 1α_1 (∂^2 + α_1/α_2) .
    
    The latter shows  that either  the first  or the second pole has to describe a ghost.  The situation seems hopeless, except
    that in 3D a massless tensor does not propagate any dynamical degrees of freedom, hence one  could
    afford it to have a  ghost-like sign of the kinetic term. Thus, we choose the coefficients α_1 and α_2  as in New Massive Gravity,  α_1=-1, α_2>0<cit.>. This guarantees that the massive
    tensor mode with two dynamical degrees of freedom  has a  good kinetic term and non-tachyonic mass;
    it also guarantees that the Galileon has a “right-sign" kinetic term.
    
    To reiterate,  the above choice corresponds
    to a “wrong"  sign of the Einstien-Hilbert term in the action (<ref>); as a result,
    the massive tensor   mode has good kinetic and mass terms,  while the massless one
    has a ghost-like kinetic term. Since a massless
    tensor field in 3D does not propagate any dynamical degrees of freedom, such a ghost-like term would not
    lead to the usual uncontrollable instabilities; it would  instead manifest itself  as “antigravity" for
    certain classical solutions in 3D <cit.>.
    
    Regarding the α_3  𝒮_3 term, as mentioned before it gives new interactions on Minkowski space, however
    would yield  further four-derivative modifications of the propagator on  a generic curved background;  hence we  will
    also put α_3=0 to avoid possible ghosts on generic curved backgrounds.
    
    With the above choice of the coefficients the  theory
    is just a Weyl invariant generalization  of New Massive Gravity, and could have been
    obtained from the latter by a Weyl transformation of the metric field. It is interesting that
    this very theory is derived via the coset construction  for a nonlinearly realized conformal
    symmetry.  Moreover, it  can be rewritten  as a theory  with  two
    tensor fields, one being dynamical and another algebraically determined <cit.> so that only
    linear terms in curvature appear in the action; the issue of hyperbolicity that we alluded to in the first section might
    be easier to study in this formulation than it is for conformal Galileons.
    
    Alternatively, the theory provides a covariant formulation of the conformal Galileon  in which the conformal mode of
    the tensor field can be gauged away instead of the Galileon. In that case, the dynamics of the gravitational
    conformal mode is entirely encoded in the conformal Galileon.
    
    If  a boundary is present, then the boundary terms should be introduced
    as in (<ref>) to maintain the Weyl invariance of the full theory at the quantum level.
    
    
    
    
    
    
    
     §.§ Acknowledgements
    
    
    We'd like to thank David Pirtskhalava and Siqing Yu for helpful comments.
    The work of GG was supported in part by NSF grant PHY-1620039. GT was supported by the NYU James Arthur Graduate Fellowship.
    
    
    
    
    
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