Spontaneously Broken 3d Hietarinta/Maxwell Chern-Simons Theory and Minimal Massive Gravity
aa r X i v : . [ h e p - t h ] J u l Spontaneously Broken 3 d Hietarinta/MaxwellChern-Simons Theory and Minimal Massive Gravity
Dmitry Chernyavsky a , Nihat Sadik Deger b,c and Dmitri Sorokin d,e a School of Physics, Tomsk Polytechnic University,634050 Tomsk, Lenin Ave. 30, Russia b Department of Mathematics, Bogazici University,Bebek, 34342, Istanbul, Turkey & c Max-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut)Am M¨uhlenberg 1, D-14476 Potsdam, Germany d I.N.F.N., Sezione di Padova & e Dipartimento di Fisica e Astronomia “Galileo Galilei”, Universit`a degli Studi di Padova,Via F. Marzolo 8, 35131 Padova, Italy
Abstract
We show that minimal massive 3 d gravity (MMG) of [1], as well as the topologicalmassive gravity, are particular cases of a more general ‘minimal massive gravity’ theory(with a single massive propagating mode) arising upon spontaneous breaking of a localsymmetry in a Chern-Simons gravity based on a Hietarinta or Maxwell algebra. Similarto the MMG case, the requirements that the propagating massive mode is neithertachyon nor ghost and that the central charges of an asymptotic algebra associatedwith a boundary CFT are positive, impose restrictions on the range of the parametersof the theory. ontents SL (2 , R ) × SL (2 , R ) × SL (2 , R ) CS theory as a degenerate case of MMGand HMCSG 135 Hamiltonian analysis 146
AdS background and the central charges of the asymptotic symmetryalgebra 16 AdS solution of the HMCSG field equations . . . . . . . . . . . . . . . 166.2 Asymptotic symmetries and central charges . . . . . . . . . . . . . . . . 17 AdS background 18 ρ =0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.3 ˜ α = ˜ β = ˜ ρ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Three-dimensional gravity theories have attracted great deal of attention since the early80s as simpler tools for studying features of General Relativity in higher dimensions,its possible consistent modifications and extensions to quantum gravity. Since then avariety of different 3 d gravity models with interesting geometric and physical propertieshave been constructed and analyzed. Among these is the minimal massive 3 d gravity(MMG) [1] which will be the focus of our attention in this paper. This gravity modelis a particular case of a class of Chern-Simons-like theories [2–4]. In contrast to thegenuine 3 d Chern-Simons theories which do not have local degrees of freedom in thebulk, the Chern-Simons-like gravities have propagating massive spin-2 modes coupledto a number of other spin-2 fields. One of the main motivations for constructing modifications of 4 d General Rela-tivity which include massive gravitons is to try to explain in this way the nature of By “spin-2 fields” we somewhat loosely mean 3 d Lorentz-vector-valued one-form fields a ra = dx µ a raµ ( x )( r = 1 , , . . . , N ) which include a dreibein e a ( x ) and a dualized spin connection ω a = ε abc ω bc . ark matter and dark energy. Three-dimensional massive gravities serve as useful toymodels for studying peculiar features and issues of these theories regarding e.g. theabsence of Ostrogradski ghosts etc. An open fundamental question regarding gravitytheories with massive gravitons is whether a spin-2 field mass can be attributed tospontaneous breaking of a space-time symmetry which in general can be an extensionof the Poincar`e group. To answer this question one should first individualize sucha symmetry and then, ideally, find a mechanism generating mass of a correspondingspin-2 field similar to that of Englert–Brout–Higgs–Guralnik–Hagen–Kibble. By now,such a mechanism is not known for gauge spin-2 fields. In this situation one can resortto old constructions called Phenomenological Lagrangians (see e.g. [5–7]) which haveproved useful for understanding the most general structure of symmetry breaking termswith the use of Goldstone fields on which these symmetries are realized non-linearly. Anotable example is the first construction of the supergravity action with non-linearlyrealized local supersymmetry [8] (see [9] for a review and further developments).In this paper we would like to address the above question for 3 d Chern-Simons-likeMMG of [1] and, in particular, to understand whether the presence of a massive spin-2 mode therein can be seen as an effect of (partial) spontaneous breaking of a localsymmetry containing the 3 d Poincar`e group as a subgroup. We will show that this isindeed the case. The MMG contains three ‘spin-2’ fields, the dreibein e a , the spin connection ω a and an additional one-form field h a . The first two are associated with gauge fields ofthe local 3 d Poincar`e group generated by the translations P a and Lorentz rotations J a .We would also like to treat h a as a gauge field associated with an additional vectorgenerator Z a that extends the Poincar`e group to a larger symmetry which is howeverbroken in the MMG action. We will restore this larger symmetry by coupling the gaugefields e a and h a to a St¨uckelberg-like spin-1 Goldstone field associated with spontaneousbreaking of Z a -symmetry. The symmetry algebra in question is the simplest amongalgebras constructed by Hietarinta [11], a class of finite-dimensional supersymmetry-like algebras containing higher-spin generators. The commutators of the generatorsof this algebra are[ J a , J b ] = ǫ abc J c , [ J a , P b ] = ǫ abc P c , [ J a , Z b ] = ǫ abc Z c , [ P a , P b ] = 0 , [ Z a , Z b ] = ǫ abc P c . (1.1)Note that the commutator of Z a closes on translations, somewhat similar to super-symmetry. Notice also that this algebra is isomorphic (dual) to the three-dimensionalMaxwell algebra [21,22] in which the role of the generators P a and Z a gets interchanged( P a ↔ Z a ), namely [ Z a , Z b ] = 0 , [ P a , P b ] = ǫ abc Z c . (1.2) The broken symmetry under consideration is not a 3 d Weyl symmetry which was assumed to be a sourceof the graviton mass in [10]. The most studied example of the Hietarinta algebras is the one in which the spin-1/2 generators of asupersymmetry algebra are replaced by their spin-3/2 counterparts. This algebra underlies the so-calledHypergravity put forward in D = 2 + 1 by Aragone and Deser [12] (see e.g. [13–20] for further studies of thistheory). he Chern-Simons action for gravity with the local symmetry generated by the 3 d Maxwell algebra was constructed and studied in [23–26]. , while its Hietarinta counter-part was considered in [18, 30]. Since from the algebraic point of view the constructionof the action is the same for (1.1) and (1.2) and the only difference between the twois the choice of the physical interpretation of the generators and corresponding gaugefields, in what follows we will call the general model under consideration the Hietar-inta/Maxwell Chern-Simons Gravity (HMCSG).In Sections 2 and 3 we will show that augmenting the HMCSG action with termsthat break linearly realized symmetry (1.1) along Z a one gets an extension of theMinimal Massive Gravity. It has, in general, two more coupling terms in comparisonwith the MMG, but still has a single massive propagating degree of freedom, as we showby performing the Hamiltonian analysis in Section 5 and studying linear perturbationsof the fields around an AdS background in Section 7. In Section 4, as a side remark,we demonstrate that when the parameters of the HMCSG are restricted by a certaincondition which makes its equations of motion integrable, the model reduces to a pureChern-Simons theory with the gauge group SL (2 , R ) × SL (2 , R ) × SL (2 , R ). In Section6 we compute the central charges of an asymptotic symmetry algebra of the HMCSGwith AdS boundary conditions. As in the MMG case, the requirements that thepropagating massive mode is neither tachyon nor ghost and that the boundary CFTcentral charges are positive impose restrictions on the range of the parameters of theHMCSG theory. We analyze these restrictions for some particular cases for which theparameters of the HMCSG differ from those of the original MMG in Section 7, andconclude with comments and an outlook in Section 8. Let us start by reviewing the construction of a gravity action which enjoys local symme-try transformations associated to the algebra (1.1). The algebra (1.1) has the followinginvariant bilinear form h J a , Z b i = a η ab , h J a , P b i = h Z a , Z b i = − σ mη ab , h J a , J b i = η ab , (2.1)where m is a parameter of the dimension of mass, a has the dimension of m , while( − σ ) is an arbitrary dimensionless constant . The dimensions of the coefficients reflectthe canonical dimensions of [ J a ] = m , [ P a ] = m and [ Z a ] = m .In the case of the Maxwell algebra (1.2) the dimension of Z a changes to m andthe corresponding bilinear form is h J a , P b i = a mη ab , h J a , Z b i = h P a , P b i = − σm η ab , h J a , J b i = η ab , (2.2) Higher-spin extensions of the Maxwell algebra and corresponding gravity models were considered in [27].See also [28,29] for a detailed study of the 3 d Maxwell group, its infinite-dimensional extensions, applicationsand additional references. The minus sign in front of σ was chosen to make our convention closer to that of [1]. We will also setthe value of the gravitational constant as 16 πG = 1. here now the parameter a is dimensionless.The bilinear form (2.1) is used to construct the Chern-Simons action (in which thewedge product of the differential forms is implicit) S = 12 m Z M h A d A + 23 A i , (2.3)for the gauge field one-form A taking values in the algebra (1.1) A = e a P a + ω a J a + h a Z a . (2.4)Explicitly, for the components of (2.4) the action (2.3) takes the following form S HCS = 12 Z M (cid:20) a m h a R a − σ (2 e a R a + h a ∇ h a ) + 1 m (cid:18) ω a dω a + 13 ε abc ω a ω b ω c (cid:19)(cid:21) , (2.5)where ∇ h a = dh a + ε abc ω b h c , R a = dω a + 12 ε abc ω b ω c . (2.6)The Hietarinta Chern-Simons (HCS) action (2.5) is invariant (up to a boundary term)under the infinitesimal gauge transformations δe a = ∇ ε aP + ε abc ( h b ε cZ + e b ε cJ ) ,δh a = ∇ ε aZ + ε abc h b ε cJ ,δω a = ∇ ε aJ . (2.7)Note that the term a m h a R a in (2.5) can be absorbed by the term 2 e a R a upon the fieldredefinition e a → e a − a σm h a . So, without loss of generality, instead of (2.5) we willdeal with the action S HCS = 12 Z M (cid:20) − σ (2 e a R a + h a ∇ h a ) + 1 m (cid:18) ω a dω a + 13 ε abc ω a ω b ω c (cid:19)(cid:21) . (2.8)Also note that if instead of the Hietarinta algebra (1.1), we had used the Maxwellalgebra (1.2) and the corresponding bilinear form (2.2) as the basis for constructingthe action (2.3), instead of (2.8) we would get S MCS = 12 Z M (cid:20) a e a R a − m σ (2 h a R a + e a ∇ e a ) + 1 m (cid:18) ω a dω a + 13 ε abc ω a ω b ω c (cid:19)(cid:21) . (2.9)In this action the role of the dreibein e a (associated with the Poincar`e translations)and of the additional spin-2 field h a get interchanged in comparison to (2.8). Nowwe can absorb the first term of (2.9) into its second term by redefining h a → h a − a σm e a and get S MCS = 12 Z M (cid:20) − m σ (2 h a R a + e a ∇ e a ) + 1 m (cid:18) ω a dω a + 13 ε abc ω a ω b ω c (cid:19)(cid:21) . (2.10) Notice that in the Maxwell case the dimension of h a gets changed in comparison with the Hietarintacase in accordance with the change of the dimension of Z a in (2.2). o, if one insists on associating the genuine graviton field with the Poincar´e generator P a , one concludes that the Maxwell Chern-Simons (MCS) gravity based on (2.10)actually does not have the standard Einstein term e a R a . In this respect the MaxwellChern-Simons gravity (2.10) can be regarded as a deformation of the “exotic” Einsteingravity considered e.g. in [31]. The parity-odd first order action of the latter is obtainedfrom (2.10) by removing its first term.From the Chern-Simons structure of the action (2.3) it follows that the modelsunder consideration do not have propagating degrees of freedom in the 3 d bulk. We would now like to generate non-trivial bulk dynamics (and mass) of fields in theabove Hietarinta/Maxwell Chern-Simons model by adding to the action (2.8) termswhich can be associated with a spontaneous breaking of the Hietarinta symmetry (1.1)down to its Poincar´e subalgebra.
By the Goldstone’s theorem, the spontaneous breaking of a rigid (global) continuoussymmetry is characterized by the appearance of massless Nambu-Goldstone fields asso-ciated with broken symmetry generators. In the case under consideration these are thevector generators Z a and the corresponding Goldstone field is a vector field A a ( x ) ofmass dimension m [18] which should not be confused with the Chern-Simons one-form(2.4). The Goldstone vector field appears in the Cartan one-form Ω = g − dg = E a P a + H a Z a ,E a = dx a − f − ε abc A b dA c , (2.11) H a = f − dA a ( x ) , where g = e x a P a e f − A a ( x ) Z a , (2.12)is a Hietarinta group element with x a being a flat 3 d space-time coordinate and f beinga symmetry breaking parameter of mass-dimension m . The subscript 0 indicates that,at this moment, we are dealing with a rigid symmetry with respect to which the one-form (2.11) is invariant under the transformation g → e ε aJ J a e ε aP P a e ε aZ Z a g , (2.13)where the parameters are x -independent. The spontaneously broken symmetry asso-ciated with ε aZ Z a is realized on x a and the Goldstone field A a ( x ) infinitesimally as anon-linear transformation [18] δx a = f − ε abc ε b Z A c ( x ) , δA a = f ε aZ − f − ε dbc ε bZ A c ∂ d A a . (2.14) For the details of the model see [18] which in turn is based on the Volkov-Akulov construction [32, 33]of Lagrangians with spontaneously broken and non-linearly realized supersymmetry. he unique Lagrangian for A a ( x ) with the minimal number of derivatives (up to two)which is invariant under (2.14) is of the Volkov-Akulov type and has the following form S = µ f Z ε abc E a E b E c (2.15)= µ Z d x (cid:18) f + 12 ε abc A a ∂ b A c − f − ε abc ε def A a A d ∂ e A b ∂ f A c (cid:19) , where µ is a dimensionless constant parameter.Note that a would be third-order derivative term in (2.15) vanishes. Interestingly,the action (2.15) contains the Abelian Chern-Simons term for A a , while the presence ofthe quartic term breaks U (1) gauge invariance of the CS action and makes propagatinga scalar mode of A a which happens to be of a Galileon type (see [18] for details).Therefore, the spontaneous breaking of the Hietarinta symmetry produces the vectorGoldstone field which has only one dynamical degree of freedom.Using the components of the Cartan form (2.11) one can also construct a Hietarinta-invariant term which is of the third order in derivatives of A a ( x ) S = µ f Z ε abc H a E b E c , (2.16)where µ is a dimensionless parameter. Modulo total derivatives, it has the followingexplicit form S = − µ f − Z ε abc dA a dA b dA c A . Also note that two more possible contributions to the Goldstone field action are actuallytotal derivatives S , = Z ε abc (cid:16) µ f H a H b E c + µ f H a H b H c (cid:17) (2.17)= Z ε abc (cid:16) µ f − d ( A a dA b E c ) + µ f − dA a dA b dA c (cid:17) . To recapitulate, the actions (2.15)-(2.17) are manifestly invariant under Lorentz rota-tions, Poincar´e translations and rigid Hietarinta symmetry (2.14). The last one actsas a (non-linear) shift on the Goldstone field A a and thus is spontaneously broken bythe vacuum solution A a = 0. To couple the Goldstone field A a ( x ) to the gauge fields (2.4), we should covariantizethe Cartan form (2.11) which makes it invariant under the transformation (2.13) whoseparameters are promoted to functions of the space-time coordinates x µ . The result isΩ = g − ( d + A ) g = E a P a + ω a J a + H a Z a , (2.18)where now g = e φ a ( x ) P a e f − A a ( x ) Z a , (2.19) ith φ a ( x ) being an arbitrary 3 d vector function and E a = e a + ∇ φ a + f − ε abc h b A c − f − ε abc A b ∇ A c ,H a = h a + f − ∇ A a . (2.20)The gauge group acts on φ a and A a as follows δφ a = − ε aP − ε abc ( ε Zb h c + ε Jb φ c ) ,δA a = − f ε aZ − ε abc ε Jb A c . (2.21)Combined with the variations of the gauge fields (2.7), the action of the gauge trans-formations on (2.20) reduces to their Lorentz rotations δ J E a = − ε abc ε Jb E c , δ J H a = − ε abc ε Jb H c , (2.22)leaving the one-forms (2.20) invariant under the action of the transformations along P a and Z a .The following comment is now in order. The vector φ a ( x ) might be thought of asa Goldstone (St¨uckelberg) field associated with breaking of the local Poincar´e transla-tions. However, this “breaking” does not result in changing the number of the physical(on-shell) degrees of freedom of the dreibein e a . The reason is that, in addition to theinvariance under local Hietarinta symmetry, the Chern-Simons gravity action (2.5) isinvariant under the 3 d diffeomorphisms x µ → x µ + ζ µ ( x ) . (2.23)Under the diffeomorphisms the dreibein transforms as follows δe a = ∇ ( ξ µ e aµ ) − ε abc ( ξ µ ω µb ) e c + i ξ ∇ e a . (2.24)Comparing (2.24) with (2.7) we see that the first and the second term in (2.24) canbe associated, respectively, with local Poincar´e translations and Lorentz rotations.Regarding the third term, since on the mass shell ∇ e a = − ε abc h b h c this term can beassociated with an ε aZ variation of e a . Therefore, on the mass shell, the local Poincar´etranslations are a redundant symmetry and can be completely substituted with the 3 d diffeomorphisms, while off the mass shell the local Poincar´e translations can be usedto set φ a = 0. Note that once this is done the flat space one-forms (2.11) are obtainedfrom (2.20) by simply setting e a = dx a and h a = 0.We are now ready to generalize the actions (2.15)-(2.17) to describe gauge-invariantcouplings of the Goldstone field A a ( x ) to the spin-2 fields e a , h a and ω a by replacing E a and H a with E a and H a defined in (2.20). We thus get the following symmetrybreaking action S sym.br. = 12 Z M ε abc (cid:18) Λ E a E b E c + ˜ βE a E b H c + ˜ αE a H b H c + ˜ ρ H a H b H c (cid:19) , (2.25) here Λ , ˜ β , ˜ α and ˜ ρ are arbitrary coupling constants whose dimensions are determinedby appropriate powers of the symmetry breaking parameter f . Note that the firstVolkov-Akulov-like term in (2.25) generates a cosmological constant. Note also that,in contrast to (2.17), the last two terms in (2.25) are not total derivatives.We will now show that the theory described by the sum of the actions (2.8) and(2.25) contains the Minimal Massive Gravity of [1]. The MMG and HMCSG actionsare related to each other by a linear transformation of the three spin-2 fields whencertain parameters in the latter are set to zero. The action (2.25) contains the Goldstone fields φ a and A a which, as usual, can begauge fixed to zero by the corresponding local symmetry transformations (2.21) withthe parameters ε aP = − φ a and ε aZ = − A a . In this (unitary) gauge the one-forms E A and H a reduce, respectively, to e a and h a , and we get the gauge-fixed action S HMCSG = 12 Z M (cid:18) − σ (2 e a R a + h a ∇ h a ) + 1 m ( ω a dω a + 13 ε abc ω a ω b ω c ) (cid:19) + 12 Z M ε abc (cid:18) Λ e a e b e c + ˜ β e a e b h c + ˜ αe a h b h c + ˜ ρ h a h b h c (cid:19) , (3.1)whose residual symmetries are the 3 d local Lorentz transformations and the diffeomor-phisms.On the other hand, in our conventions and notation the MMG action [1] has thefollowing form S MMG = 12 Z M (cid:18) − σe a R a + 2 h a ∇ e a + 1 m ( ω a dω a + 13 ε abc ω a ω b ω c ) (cid:19) + 12 Z M ε abc (cid:18) Λ e a e b e c + α e a h b h c (cid:19) , (3.2)where again σ = ±
1, and m , Λ and α are arbitrary (dimensionful) parameters, andthe spin-2 fields are formally denoted in the same way as in (3.1) to simplify notation,though now h a is dimensionless. Note that when α = 0 in (3.2), the action reduces tothe first-order action for the Topologically Massive Gravity (TMG) [34] for which therequirement of positive energy of the massive spin-2 mode singles out the sign σ = − σ = 1 (see the discussion in [1]).The difference between the actions (3.1) and (3.2) is obvious. However, we willnow show that the MMG action is a particular case of (3.1) with three independentparameters. To this end, it is useful to notice that the actions are of a Chern-Simons-like type [1, 2, 4], i.e. they can be written in the following form S = Z M (cid:18) g rs a r · da s + 16 f rst a r · a s × a t (cid:19) , (3.3) here a ra = ( e a , h a , ω a ) (i.e. r = 1 , , r = e, h, ω ), and g rs and f rst are symmetric tensors with constant components. In (3.3) we used the convenient3 d Lorentz-vector algebra notation [2]( a r × a s ) a = ε abc a rb a sc , a r · a s = η ab a ra a sb . (3.4)In the case of (3.1) g rs and f rst have the following non-zero components g eω = − σ, g ωω = 1 m , g hh = − σ,f eee = Λ , f ωωω = 1 m , f hhh = ˜ ρ,f eeh = ˜ β, f eωω = − σ, f ehh = ˜ α, f ωhh = − σ, (3.5)while for MMG (3.2) g eω = − σ, g ωω = 1 m , g eh = 2 ,f eωω = − σ, f ωωω = 1 m , f ehω = 1 ,f eee = Λ , f ehh = α. (3.6)The matrix of the linear transformation of the fields˜ a p = T pq a q , (3.7)which relates (modulo a total derivative) the HMCSG tensor g pr in (3.5) to the MMGone in (3.6) g MMG = T T g HMCSG T, (3.8)has the following form T pq = − m √− mσ − σ . (3.9)Note that the form of the matrix T requires mσ to be negative. This is related to thesign of g hh = − σ in the HMCSG case. This sign can be flipped by performing theparity transformation e a → − e a and σ → − σ in the HMCSG action (3.1).Thus, upon performing the transformation (3.7) one brings the action (3.1) to thefollowing form (in which, for simplicity, we remove ‘tilde’ over the redefined fields) S HMCSG = 12 Z M (cid:18) − σe a R a + 2 h a ∇ e a + 1 m ( ω a dω a + 13 ε abc ω a ω b ω c ) (cid:19) + 12 Z M ε abc (cid:18) Λ e a e b e c + α e a h b h c + β e a e b h c + ρ h a h b h c (cid:19) , (3.10)where β = ˜ β √− mσ − Λ m , α = − βm √− mσ − ˜ ασm + Λ m − σ,ρ = − ˜ ρσm √− mσ + 3 ˜ βm √− mσ + 3 ˜ ασm − Λ + m σm (3.11) nd the values of g rs and f rqs are g eω = − σ, g ωω = 1 m , g eh = 1 ,f eωω = − σ, f ωωω = 1 m , f ehω = 1 ,f eee = Λ , f ehh = α, f eeh = β , f hhh = ρ . (3.12)The action (3.10) reduces to the MMG action (3.2) when β = ρ = 0.The equations of motion which follow from (3.10) are − σR + 2 ∇ h + Λ e × e + αh × h + 2 β e × h = 0 , ∇ e + 2 αe × h + βe × e + ρh × h = 0 , (3.13) − σ ∇ e + 2 m R + 2 e × h = 0 . Note that in order to have three independent dynamical equations, the coefficient of thegravitational Chern-Simons term, i.e. 1 /m , should be non-zero. A linear combinationthereof brings the above equations to the form2 R + 2 m (1 + σα ) e × h + σmβe × e + σmρ h × h = 0 , ∇ h + 2( mσ (1 + σα ) + β ) e × h + ( mβ + Λ ) e × e + ( mρ + α ) h × h = 0 , ∇ e + 2 αe × h + βe × e + ρh × h = 0 . (3.14)Upon the redefinition of the connectionΩ = ω + αh + β e, (3.15)we have 2 R (Ω) + C e × e + C e × h + ρ C h × h = 0 , ∇ (Ω) h + C e × h + (Λ + mβ ) e × e + ( mρ − α ) h × h = 0 , ∇ (Ω) e + ρh × h = 0 , (3.16)where C = 14 ( β + 2 mσ ) + α (Λ + mβ ) − m ,C = 2 (cid:0) α ( β + 2 mσ ) + m (1 + α ) (cid:1) (3.17) C = ( β + 2 mσ ) + 2 mα. Note that in (3.16) (and (3.17)) β always appears in the combinations Λ + mβ and β + 2 mσ . So effectively β shifts Λ and promotes σ = ± aking the covariant derivative of these equations and comparing the results onefinds that for consistency either α ( β + 2 mσ ) + m (1 + α ) − ρ (Λ + mβ ) = 0 , (3.18)or h · e = 0 . (3.19)The latter implies that h aµ e aν is a symmetric tensor as in the MMG theory [1], for whichthe first option (3.18) reduces to 1 + ασ = 0 . (3.20)For ρ = 0 the equations (3.16) take the form2 R (Ω) + C e × e + C e × h = 0 , ∇ (Ω) h + C e × h + (Λ + mβ ) e × e − αh × h = 0 , ∇ (Ω) e = 0 , (3.21)Note that now among the five coefficients C i ( i = 1 , , mβ and α only fourare functionally independent and expressed in terms of four independent parametersΛ + mβ , α , β + 2 mσ and m .We see that when ρ = 0 the geometry is torsionless and, in addition, the first equa-tion can be solved for h as in MMG (provided that C = α ( β + 2 mσ ) + m (1 + α ) = 0and hence (3.19) is satisfied), but in our case there is still one more independent cou-pling constant β like in [35] (equations (A5)-(A8) therein). Alternatively, if we wouldlike to treat h as the dreibein, we can arrive at the torsionless condition by modifyingthe connection starting from the second equation in (3.14) and setting Λ = 0.As in the MMG case [1], solving the first equation in (3.21) for h we get h µν = h aµ e bν η ab = − C (cid:18) S µν + C g µν (cid:19) , g µν = e aµ e bν η ab , (3.22)where S µν = R µν − g µν R is the 3 d Schouten tensor. Substituting this solution intothe second equation of (3.21) and expressing Ω a through e a by solving the torsionlesscondition in (3.21) we get C µν + (cid:18) C αC C (cid:19) G µν − (cid:18) C C − (Λ + mβ ) C + αC C (cid:19) g µν = 2 αC J µν , (3.23)where G µν is the Einstein tensor, C µν = √− det g ε τρµ ∇ τ S ρν is the Cotton tensor and J µν =
12 det g ε ρσµ ε την S ρτ S ση . The above equation has the same form as the MMGmetric field equation [1] containing three coefficients, which are now composed of fourcontinuous parameters Λ + mβ , α , m and β + 2 mσ . Also, when ρ is non-zero, one can make a shift e → e + ch (with an appropriate constant c ) such that fora certain range of the parameters the term h × h disappears from the first equation of (3.16). Thus, one cansolve it for h , but the geometry, in general, remains torsionful, due to the structure of the last two equationsin (3.16). So it is not possible, in general, to solve these equations for Ω in terms of the dreibein e . Still, aswe will see below, also in the case with ρ = 0 the theory has a single propagating bulk degree of freedomand can be studied perturbatively around an AdS vacuum, like the MMG. SL (2 , R ) × SL (2 , R ) × SL (2 , R ) CS theory as adegenerate case of MMG and HMCSG
Though the main subject of this paper is the massive gravity theory whose fields satisfythe consistency condition (3.19), in this Section we would like to elucidate the structureof the model for which the equation (3.18) holds, so the model has only five independentparameters. Then the equations (3.14) (or (3.16)) are integrable in the sense that theircovariant derivatives are identically zero without imposing the additional constraint(3.19) on the fields. This means that (3.14) become the Maurer-Cartan equations forthe one-forms e a , h a and ω a which should thus be the components of a Cartan formassociated with a gauge group of rank 9. This group is semi-simple and should containthe 3 d Lorentz group SL (2 , R ) as a subgroup. As such, the most reasonable candidateis SL (2 , R ) × SL (2 , R ) × SL (2 , R ). A Chern-Simons gravity based on this group wasconsidered in [36–38].To show that this is indeed so, let us consider a simpler case in which ρ = 0. Then ineq. (3.14), in which the remaining parameters satisfy the condition (3.18), we redefinethe fields e a and ω a as follows h → α h + mα (cid:0) α − (cid:1) (cid:0) α − m (1 + ασ ) (3 ασ − (cid:1) − / e,ω → ω + 2 m (1 + ασ ) (cid:0) α − m (1 + ασ ) (3 ασ − (cid:1) − / e − h, (4.1) e → α (4Λ α − m (1 + ασ ) (3 ασ − − / e , where we assume, without loss of generality, that the expression under the square rootis positive. Then the equations (3.14) (with ρ = 0 and β = − mα (1 + σ α ) ) take thefollowing form R + 12 e × e = 0 , ∇ e = 0 , (4.2) ∇ h − h × h + 12 e × e = 0 . As one can easily check, these are the Maurer-Cartan equations for the one-form A = ω a J a + e a P a + h a Z a associated with the following linear combinations of the threesets T , T and T of the generators of SL (2 , R ) × SL (2 , R ) × SL (2 , R ), respectively: J = T + T + T , P = T − T , Z = − T . (4.3)In the general case (i.e. when ρ = 0) the transformation of the fields to the form whichresults in eq. (4.2) is much more cumbersome and we will not give it here.We have thus found that the action (3.10) which produce the equations of motion(3.14) with the parameters satisfying the condition (3.18) is similar to that of [38].Therefore in this case all the bulk degrees of freedom are pure gauge, as e.g. in thecase of Gravity based on SL (2 , R ) × SL (2 , R ). Here we just have an additional SL (2 , R )field. Of course, the physical content of the theory depends on the boundary conditions hich can be imposed on the components of A . These boundary conditions determinefor us which is the true dreibein and connection and asymptotic symmetries. Forinstance, we can associate them with those belonging to SL (2 , R ) × SL (2 , R ) andthen the third SL (2 , R ) gauge field completely decouples (see [36–38] for more details).In summary, the particular choice of the parameters (3.18) in the HMCSG ac-tion does not break the Hietarinta/Maxwell symmetry but deforms it to SL (2 , R ) × SL (2 , R ) × SL (2 , R ). This is similar to how the Poincar´e symmetry gets deformedto the (A)dS symmetry by adding the cosmological term to the Einstein gravity ac-tion. On the other hand, since the Hietarinta/Maxwell algebra is a contraction of the sl (2 , R ) × sl (2 , R ) × sl (2 , R ) algebra, the HMCS action (2.5) can be obtained as thecontraction limit of the SL (2 , R ) × SL (2 , R ) × SL (2 , R ) Chern-Simons action. We shall now sketch, following [1–3], the Hamiltonian analysis of the system describedby the action (3.1) and show that it has one propagating degree of freedom as in theparticular case of the MMG model.Let us assume that the manifold M on which the theory is defined can be pre-sented as the product R × Σ, where Σ is a two-dimensional manifold with boundaryparametrized by the coordinates x i , i = 1 ,
2, while R defines the temporal directionparametrized by x . Upon this splitting the general Chern-Simons-like action (3.3)takes the following form S = Z R dx Z Σ d xε ij (cid:20) g rs ˙ a ri · a sj + a r · (cid:16) g rs ∂ i a sj + 12 f rst a si × a tj (cid:17)(cid:21) , (5.1)where dot denotes the derivative with respect to x and ε ij ≡ ε ij .From the form of this action we see that the canonical momenta p iar associated to a ari are constrained to be linear combinations of the fields themselves p iar = ε ij g rs a asj . Upon solving these constraints, one gets the equal-time Poisson (actually Dirac) brack-ets for the fields a ari { a ari ( x ) , a bqj ( y ) } = ε ij η ab g rq δ ( x − y ) , where g rq is the inverse of g rq .From the structure of (5.1) we also see that a s plays the role of a Lagrange multipliergiving rise to 9 constraints ϕ ar = ε ij (cid:16) g rs ∂ i a sj + 12 f rst a si × a tj (cid:17) a . The corresponding constraint functional for arbitrary fields χ ra with well defined vari-ation has the following form ϕ [ χ ] = Z Σ d xχ r · ε ij (cid:16) g rs ∂ i a sj + 12 f rst a si × a tj (cid:17) + Z ∂ Σ dx i χ r · a r . (5.2) he Poisson brackets of these constraints have the following structure { ϕ [ χ ] , ϕ [ ξ ] } = ϕ [[ χ, ξ ]] + Z Σ d xχ ra ξ sb P abrs − Z ∂ Σ dφχ r · ( g rs ∂ φ ξ s + f rst a sφ × ξ t ) , (5.3)with [ χ, ξ ] t = f rst χ r × ξ s and P abrs = f tq [ r f s ] pt η ab ∆ pq + 2 f tr [ s f q ] pt V ab,pq , V pqab = ε ij a pia a qjb , ∆ pq = V pqab η ab . (5.4)The integration variable φ parametrises a (compact) boundary ∂ Σ.The number of first- and second-class constraints for the model under considerationcan be read off from the rank of the matrix P , eq. (5.4), in which one should insert theexplicit expressions (3.12) for the tensors g rs and f rqs . Note that in (5.4) the indicesare raised with the matrix g rs If we assume that (3.19) holds, we have an additional(secondary) constraint ∆ eh = 0 . (5.5)Taking this into account, a straightforward computation shows that the first term in(5.4) vanishes and P becomes degenerate P = (cid:0) αβ − ρ (Λ + mβ ) + m (1 + ασ ) (cid:1) − V hhab V heab V ehab − V eeab
00 0 0 . (5.6)Now one should also compute the Poisson brackets of the constraint (5.5) with ϕ ( χ ). Using a general formula of [2] one gets { ∆ eh , ϕ [ χ ] } = ε ij (cid:16) ∇ i χ e · h j − ∇ i χ h · e j (cid:17) + ε ij e i × h j · (cid:16) mρχ h + mσ (1 + ασ ) χ e (cid:17) − ε ij h i × h j · (cid:16) αχ e + ρχ h (cid:17) + ε ij e i × e j · (cid:16) (Λ + mβ + mσ (1 + ασ )) χ e + βχ h (cid:17) (5.7)with ∇ i χ = ∂ i χ + ω i × χ. (5.8)As in the MMG [1] we thus have the (10 ×
10) matrix of the Poisson brackets of 10constraints, i.e. ϕ ar and ∆ eh , which has rank four. This implies that, if the coefficientin front of the matrix (5.6) is non-zero, system has 6 first-class and 4 second classconstraints which reduce the number of the phase-space physical degrees of freedomin a ari to 2, i.e the system has a single bulk degree of freedom in the Lagrangianformulation.When the coefficient in (5.6) is zero, which is equivalent to the choice (3.18), theconstraint (5.5) is absent and one has 9 first-class constraints ϕ ar which reduce thenumber of bulk physical degrees of freedom to zero. In this case, as we discussed inSection 4, the considered system reduces to the Chern-Simons theory with the gaugegroup SL (2 , R ) × SL (2 , R ) × SL (2 , R ). AdS background and the central charges ofthe asymptotic symmetry algebra We shall now study properties of the HMCSG theory for field configurations whosegeometry is asymptotically
AdS and compute the corresponding centrally extendedasymptotic symmetry algebra which underlies a dual CF T . AdS solution of the HMCSG field equations For the
AdS background to satisfy the field equations (3.16) we take the followingansatz for the vevs of e , h and Ω h e i := ¯ e , h h i := mC ¯ e , h Ω i := ¯Ω − ρm C e , (6.1)where ¯ e and ¯Ω are AdS dreibein and connection, and C is a real dimensionless pa-rameter.Substituting this ansatz into equations (3.16) we find that, provided that C satisfiesthe cubic equation ρm C − ( mρ − α ) m C − ( β + 2 mσ (1 + ασ )) mC − (Λ + mβ ) = 0 , (6.2)which always has at least one real root, eqs. (3.16) reduce to those describing the AdS space ¯ R ( ¯Ω) + l − e × ¯ e = 0 , ¯ ∇ ¯ e = 0 , (6.3)where l − ≡ − Λ = ρ m C ρm C β + 2 mσ (1 + ασ )) + 2 mC ( βα + m (1 + ασ ) )+ (cid:18) β α + mβσ (1 + ασ ) (cid:19) = 14 (cid:0) ρm C + β + 2 mσ (1 + ασ ) (cid:1) + 2 mC ( βα + m (1 + ασ ) )+Λ α − m (1 + ασ ) . (6.4) l − is assumed to be positive so that the cosmological constant Λ is negative. A more general class of vacuum solutions in MMG including those with a positive cosmological constantwere considered e.g in [39–46], in particular at a specific point called “merger point”. The merger point isa point in the space of the parameters of the theory at which for all values of C defined by the equation(6.2) the cosmological constant Λ (6.4) has a unique value. It would be of interest to study a similar classof vacuum solutions also in the HMCSG context. .2 Asymptotic symmetries and central charges In [47] Brown and Henneaux studied asymptotic symmetry properties of the pure 3 d GR with
AdS boundary conditions. The local 3 d Lorentz symmetry and 3 d diffeo-morphisms of GR give rise to six first-class constraints generating these symmetries.These can be split into two mutually commuting sets of generators corresponding tothe SL (2 , R ) × SL (2 , R ) group of the Chern-Simons formulation of the theory. Whenevaluated on an asymptotically AdS space, each set was shown to generate the Vira-soro algebra with a nontrivial central extension. This analysis was generalized to 3Dmassive theories of gravity in [1, 4, 48–50] and to the Maxwell-Chern-Simons gravityin [51].We will now carry out the computation of the centrally extended asymptotic sym-metry algebra for the HMCSG theory, following closely the steps explained in detailin [48] and [1]. Consider the following combination of the constraints (5.2) L ± [ χ ] = ϕ e [ χ µ e µ ] + ϕ h [ χ µ h µ ] + a ± ϕ ω [ χ µ e µ ] , (6.5)in which the parameters in the brackets are field-dependent and χ µ ( x ) are associatedwith the parameter of 3 d diffeomorphisms.For convenience we have defined ϕ ω for the spin connection ω in (3.10). Theconstant parameters a ± should be properly tuned in order to make the Poisson bracketof L + and L − vanish. It can be shown that (6.5) are a combination of the first-classconstraints, corresponding to the local 3 d Lorentz transformations and diffeomorphisms[1, 48]. Using the general formula (5.3) one finds that for the
AdS solution (6.1) thePoisson bracket of L + and L − reduces to { L + [ χ ] , L − [ η ] } = ϕ ω [[ χ, η ]] (cid:0) a + a − + 2 m C (1 + ασ ) + mβσ + m C ρσ (cid:1) + ( ϕ e [[ χ, η ]] + mCϕ h [[ χ, η ]]) (cid:0) a + + a − + 2 αmC + β + m C ρ (cid:1) . (6.6)Note that on the AdS solution (6.1) the second term in (5.3) vanishes. Also theboundary contribution (the last term in (5.3)) vanishes. To see this, one should takeinto account the linear redefinition which relates Ω with ω (3.15), the vacuum valueof the Ω spin connection (6.1), and the corresponding AdS asymptotic symmetryparameters χ and η (see [48] for details). Requiring that the Poisson bracket (6.6)vanishes, we find that the parameters a ± should have the following values a ± = ± l − αmC − β − C m ρ , (6.7)where l is the radius of the AdS background defined in (6.4). Using the generalexpression (5.3) once again, one also finds { L ± [ χ ] , L ± [ η ] } = ± l L ± [[ χ, η ]] (6.8) ± l (cid:18) σ ± ml + αC + β m + mρC (cid:19) Z ∂ Σ dφχ · (cid:18) ∂ φ η + ¯Ω φ × η ± l ¯ e φ × η (cid:19) , here in order to get the boundary term expressed via the AdS spin connection ¯Ω, wemade use of (3.15) and (6.1). After expanding the asymptotic symmetry parameters η and χ in Fourier modes, the commutation relations above represent two copies of theVirasoro algebra with central charges c ± = 3 l G (cid:18) ± ml + σ + β m + αC + mρC (cid:19) , (6.9)where to be in agreement with the Brown–Henneaux central charge expression [47] wehave included the Newton’s constant by restoring 1 / πG in the action (3.10).For the boundary CFT associated with (6.8) to be unitary both central chargesshould be positive, which implies σ + β m + αC + mρC − | ml | > . (6.10)For certain choices of the parameters α , β , ρ and σ = ±
1, the above expressions reduceto those of pure GR [47], TMG [48] and MMG [1].
AdS background We shall now study, following [1, 52], the conditions on the parameters of our modelfor which the propagating mode is neither a tachyon nor a ghost. To this end let usconsider perturbations around the
AdS vacuum solution which are convenient to takeas follows e = ¯ e + k, Ω = ¯Ω − ρm C e + k ) − mCρ p + v, h = mC (¯ e + k ) + p, (7.1)where k , v and p denote infinitesimal excitations of the fields. Then, using the relation(6.2) and the definition (6.4) of l − we get the linearized equations for (3.16) as¯ ∇ v + l − ¯ e × k + ¯ e × p (cid:0) βα + m (1 + ασ ) − ρ (Λ + mβ ) (cid:1) = 0 , ¯ ∇ p + M ¯ e × p = 0 , ¯ ∇ k + ¯ e × v = 0 , (7.2)where M = 12 (cid:0) β + 2 mσ (1 + ασ ) + 2 mC ( mρ − α ) − m C ρ (cid:1) . (7.3)The integrability condition (3.19) for the above equations reduces to¯ e · p = 0 . Making the redefinition (assuming that | ℓM | 6 = 1) f ± = ± l − k + βα + m (1 + ασ ) − ρ (Λ + mβ )( ± l − − M ) p + v, (7.4) Note that in the MMG case (i.e. when β = ρ = 0) the value M = 0 defines the merger point [39] for thevalues of the cosmological constant. This, however, is not the case anymore for ρ = 0. ne diagonalizes two of the equations (7.2) and gets¯ ∇ f ± ± l − ¯ e × f ± = 0 , ¯ ∇ p + M ¯ e × p = 0 . (7.5)The first two equations in (7.5) describe the linearized 3 d Einstein gravity with acosmological constant and the third equation describes the propagation of the spin-2mode p with the mass M given by M = M − l − . In accordance with the general Hamiltonian analysis we thus see that the HMCSGmodel has exactly the same field content as the MMG. The no-tachyon condition is [1] M − l − > . (7.6)Let us now find the form of the action (3.10) up to the second order in perturbations.Upon taking into account the form of the transformation (3.15), the excitations (7.1)and the linear redefinition (7.4) one gets S = Z M λ + (cid:0) f + ¯ ∇ f + + l − ¯ e · f + × f + (cid:1) + λ − (cid:0) f − ¯ ∇ f − − l − ¯ e · f − × f − (cid:1) + Z M m (1 − C ) (cid:0) p ¯ ∇ p + M ¯ e · p × p (cid:1) , (7.7)where λ ± = 12 m ∓ l m (cid:0) mσ + β + 2 mCα + m C ρ (cid:1) . (7.8)The first two terms are two linearized SL (2 , R ) Chern–Simons terms. Comparing (6.9)with (7.8) we see that c ± = ± λ ∓ /G .The product of λ + and λ − is λ + λ − = − l − C ) . (7.9)If the product is negative, the first two terms describe the linearized pure GR as thedifference of two SL (2 , R ) Chern-Simons terms. In the general case, however, theproduct may also have the positive sign, then the resulting theory can be interpretedas a kind of “exotic” GR with additional terms. However, − λ + and λ − (7.8) areproportional to the central charges c + and c − (6.9) in the asymptotic algebra and if werequire both central charges to be positive (6.10), then the product of λ + and λ − (7.9) Note that the CS action for GR corresponds to the SO (2 ,
2) bilinear form h J a , P a i = h J a + , J a + i −h J a − , J a − i = η ab , where J a ± are two copies of SO (1 ,
2) generators, related to that of SO (2 ,
2) as J a = J a + + J a − and P a = J a + − J a − . One can use the additional bilinear form of the SO (2 ,
2) algebra given by h J a , J a i = h P a , P a i = c η ab with a constant c to extend the GR action by the topological and torsion terms c ( ωdω + ω ) + ce ∇ e . At the linearized level the sign of the product (7.9) depends on the value of theconstant parameter c . ust be negative and hence (1 − C ) >
0. Note that at the chiral point of the theory,at which one of the boundary central charges vanishes, 1 − C = 0 and equation (7.7)becomes singular.The last term in (7.7) describes the propagating massive spin-2 mode. The no-ghostcondition implies (see [1] for details)(1 − C ) mM < . (7.10)We shall now consider in more detail three particular cases in which the values of theparameters differ from the original MMG. In this case the equation (6.2) reduces to the following relationΛ + mβ = 0 ⇒ Λ = − mβ, (7.11)while (6.4) and (7.3) respectively simplify to l − = 14 ( β + 4 mβσ ) = m (cid:18) β m + σ (cid:19) − m > , (7.12)and M = m (cid:18) β m + σ + α (cid:19) . (7.13)From (7.12) we have β m + σ > β m + σ < − . (7.14)The no-tachyon condition (7.6) takes the form2 α (cid:18) β m + σ (cid:19) + 1 + α > . (7.15)In the action (7.7) we now have λ + λ − = − l / <
0. Hence, the first two terms arethe difference of two linearized SL (2 , R ) Chern–Simons terms describing the linearized3 d Einstein gravity. The last term describes the propagating massive spin-2 modewhose no-ghost condition (7.10) requires mM < ⇒ β m + σ + α < . (7.16)The positive central charge condition in the case C = 0 is σ + β m − | ml | > . (7.17) ow we would like to analyze consequences of the conditions (7.14)-(7.17). From(7.17) we see that σ + β m > α < − , (cid:18) α + β m + σ (cid:19) > (cid:18) β m + σ (cid:19) − . (7.18)So finally, the range of the parameters which satisfies the conditions (7.14)-(7.17) is β m + σ > , α < − s(cid:18) β m + σ (cid:19) − − (cid:18) β m + σ (cid:19) , Λ = − mβ , (7.19)and ρ is arbitrary. ρ =0 In this case we have α m C − ( β + 2 mσ (1 + ασ )) mC − (Λ + mβ ) = 0 ,l − = 2 mC ( αβ + m (1 + ασ ) ) + (cid:18) β α Λ + mβσ (1 + ασ ) (cid:19) , (7.20) M = β mσ (1 + ασ ) − mαC . The solution for C is (assuming α = 0) C = β + 2 mσ (1 + ασ )2 mα ∓ r Λ + mβm α + ( β + 2 mσ (1 + ασ )) m α , (7.21)so M = ± mα r Λ + mβm α + ( β + 2 mσ (1 + ασ )) m α . (7.22)and l − = − m (1 − C ) (cid:18) αβm + (1 + ασ ) (cid:19) + M > , The no-tachyon condition is M − l − = m (1 − C ) (cid:18) αβm + (1 + ασ ) (cid:19) > , (7.23)and the no-ghost condition is as in (7.10). Note that C is real iff M ≥
0. Hence,the no-tachyon condition also guarantees C to be real. For M = 0, the C equation(7.20) has a double root, but this case is un-physical since the no-tachyon conditionis violated. Also note that unlike the original MMG, for which β = 0, the no-tachyoncondition does not in general imply (1 − C ) > ollecting the positive central charge, the no-tachyon and the no-ghost conditionstogether we have αC + σ + β m − | ml | > ⇒ − C > αβm + (1 + ασ ) > , mM < . (7.24)The positivity of l − in (7.23) also requires that0 < − C < M m (cid:16) αβm + (1 + ασ ) (cid:17) . Due to the definition of M in (7.20), the condition mM < α ( C − − β m − σ > , (7.25)which when summed up with the first condition in (7.24) gives α (1 − C ) + 1 | ml | < ⇒ α < − | ml | (1 − C ) < . From (7.25) and the fact that α is negative it follows, that in the solution (7.21) weshould choose the minus sign in front of the square root and plus one in (7.22).We have thus identified a range of the parameters compatible with the conditions(7.24). One can proceed further with the analysis and specify this range in more detail.Namely, from the third condition in (7.24), as in the case C = 0, it also follows that (cid:18) β m + σ + α (cid:19) > (cid:18) β m + σ (cid:19) − . (7.26) • Iff | β m + σ | ≥ , we have α < − s(cid:18) β m + σ (cid:19) − − (cid:18) β m + σ (cid:19) or α > s(cid:18) β m + σ (cid:19) − − (cid:18) β m + σ (cid:19) . These are compatible with α < β m + σ ≥ • Another brunch of (7.26) is − < β m + σ < , − < α < , for which a particularly simple case is β m + σ = 0. • One more simple case is Λ = − mβ for which either C = 0 and we are back toSubsection 7.1, or αC = 2 (cid:16) β m + σ + α (cid:17) and hence, due to (7.25) and (7.23), β m + σ + α > , − < α < . et us now consider the case in which α = 0 (but β = 0). Then we have C = − Λ + mβm ( β + 2 mσ ) , M = β mσ , ℓ − = − m (Λ + mβ ) β + 2 mσ + β mσβ . (7.27)So, the no-tachyon condition is m + 2 m (Λ + mβ ) β + 2 mσ > , (7.28)and the no-ghost condition is (cid:18) m + 2(Λ + mβ ) β + 2 mσ (cid:19) ( β + 2 mσ ) < . (7.29)From (7.28) and (7.29) we see that β m + σ < . (7.30)Note that if β = 0, the model under consideration (3.10) is TMG [34] for which theabove no-ghost condition requires σ = − β = 0, the model with α = ρ = 0 is equivalentto the TMG [34]. Indeed, upon making the redefinition of the connection (3.15) in theaction (3.10) with α = ρ = 0, we get S = 12 Z M (cid:18) − mσ + βm e a R a + 1 m (Ω a d Ω a + 13 ε abc Ω a Ω b Ω c ) (cid:19) (7.31)+ 16 Z M (cid:18) Λ − β m (cid:19) ε abc e a e b e c + 12 Z M (cid:18) h + β + 4 mβσ m e a (cid:19) ∇ e a . This coincides with the first-order action for the topological massive gravity upon theredefinition ˜ h a = h a + (cid:16) β m + βσ (cid:17) e a and appropriate rescalings of e a and ˜ h a . ˜ α = ˜ β = ˜ ρ = 0 Let us now consider the case in which in the action (3.1) we have ˜ α = ˜ β = ˜ ρ = 0.This is the situation in which the spontaneous breaking of the Hietarinta/Maxwellsymmetry occurs only due to the contribution associated with the classical lower-derivative Volokov-Akulov-like Goldstone (“cosmological”) term (2.15). In this casethe parameters (3.11) in the MMG-like action (3.10) are β = − Λ m , α = Λ m − σ, ρ = − Λ + m σm . (7.32)Hence, similar to the case of Subsection 7.1 we have Λ + mβ = 0 but with particularexpressions for α and ρ in terms of Λ and m . Then the equation (6.2) for C reducesto C (cid:2) (Λ + m σ ) C − C + Λ (cid:3) = 0 . (7.33) or the solution C = 0 of this equation from (6.4) and (7.3) we get ℓ − = Λ m − Λ σ , M = Λ m . We see that to satisfy the no-tachyon (7.6) and the no-ghost (7.10) conditions togetherwith the requirement ℓ − > σ = − < − m which are in agreementwith (7.19). Note that if Λ = 0, then C = 0 and hence this possibility is ruled outby the last inequality. Actually, in this case the background is flat not AdS , whilethe model reduces to the Chern-Simons theory with the unbroken Hietarinta (2.8) orMaxwell symmetry (2.9).When ρ = 0, i.e. Λ = − m σ , then from (7.33) we see that either C = 0 or C = 1 /
2. In the both cases the no-tachyon condition (7.6) is not satisfied.Finally, let us consider the case in which C = 0, Λ = 0 and ρ = 0. Then from(7.33) we get C = Λ ± √− Λ σm Λ + m σ (7.34)So the existence of the real solutions (associated with AdS vacua) requires thatΛ σ < . (7.35)In this case from (6.4) and (7.3) we find ℓ − = m C , M = Λ ( C − m . Note that C = 1 is not a solution of (7.33). The no-tachyon condition (7.6) becomes − (Λ σ + m ) C > ⇒ Λ σ < − m , and the no-ghost condition (7.10) using (7.33) implies(Λ + m σ )(1 − C ) < . Combining the last two inequalities we see σ (1 − C ) > , (7.36)which shows that we should take the plus sign in the solution of C in (7.34).From (7.9) the positivity of the central charges implies 1 − C > − Λ σ >
0, the no-ghost condition takes the formΛ ( C − < ⇒ σ ( − Λ σ )(1 − C ) < ⇒ σ (1 − C ) < , which is incompatible with (7.36). Therefore, for the choice of the parameters consid-ered in this Subsection the no-ghost and no-tachyon conditions, and the positivity ofcentral charges are satisfied only for C = 0. Conclusion
In this paper we have shown that both the TMG [34] and the MMG [1] can be treatedas spontaneously broken phases of the Chern-Simons theory based on the Hietar-inta/Maxwell algebra. In general, the spontaneous symmetry breaking in the HMCSGtheory leads to a more general class of minimal massive gravities propagating a sin-gle massive spin-2 mode and having two more coupling parameters with respect tothe MMG. For a certain range of the parameters these models have neither tachyonsnor ghosts and their asymptotic algebra has positive central charges thus giving riseto unitary boundary CFTs. A further more detailed analysis of these models in theAdS/CFT context might be of interest.As a generalization of the results of this paper, it would be interesting to identifythe group-theoretical structure of Chern-Simons theories whose symmetry breakinggives rise e.g. to “New”, “General” [53, 54], “Zwei-Dreibein” [50, 55] and “Exotic”Massive Gravities [35, 56, 57], for more references see [4]. And of course the most chal-lenging issue is to find an Englert–Brout–Higgs–Guralnik–Hagen–Kibble mechanismwhich might lead to such a symmetry breaking.Another interesting direction is to look for a relation of the HMCSG to a “simple”theory of 3 d massive gravity constructed and studied in [58, 59]. The simplicity ofthis model is due to the fact that it contains only two one-form fields, a dreibein anda would-be Lorentz spin connection, but the local Lorentz symmetry in this model isbroken. For a certain choice of the parameters in the letter its field equations reproducethose of the MMG. A question is whether for a more general range of the parametersthe simple massive gravity may also reproduce the equations of motion of the HMCSGtheory constructed in this paper (upon solving for h a in (3.16)).It might also be of interest to consider supersymmetric and higher-spin extensionsof these models elaborating on the constructions obtained e.g. in [27–30, 37, 60–63].Regarding supersymmetric generalizations, let us make the following final remark.The simplest extension of the Maxwell algebra (1.2) by a two-component Majoranaspinor generator Q α [64] is such that [ Q α , P a ] = [ Q α , Z a ] = 0 and { Q α , Q β } = 2 γ aαβ Z a ,i.e. the anti-commutator of Q can only close on Z due to Jacobi identities. Hence, thissimplest super-Maxwell algebra is not an extension of the conventional N = 1 , D = 3super-Poincar´e algebra. On the contrary, the similar supersymmetric extension of theHietarinta algebra (1.1) is the extension of the simple 3 d super-Poincar´e algebra since inthis case the Jacobi identities allow { Q α , Q β } = 2 γ aαβ P a . This gives one more evidenceto the fact that the physical models based on the two versions of the same algebra area priori different. Acknowledgements
The authors are thankful to Eric Bergshoeff, Axel Kleinschmidt, Wout Merbis and PaulTownsend for interest to this work and useful comments. DC and NSD are gratefulto INFN, Padova for hospitality and financial support during an intermediate stageof this work. NSD wishes to thank Albert Einstein Institute, Potsdam for hospitality uring the final phase of this paper. Work of DC was supported by the Foundation forthe Advancement of Theoretical Physics and Mathematics “BASIS”. NSD is partiallysupported by the Scientific and Technological Research Council of Turkey (T¨ubitak)Grant No.116F137. References [1] E. Bergshoeff, O. Hohm, W. Merbis, A. J. Routh, and P. K. Townsend,
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