# Cosmological New Massive Gravity and Galilean Conformal Algebra in 2 Dimensions

Authors: M. R. Setare; V. Kamali
Journal: Advances in High Energy Physics
 (2011)

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## Abstract

We consider the realization of 2-dimensional Galilean conformal algebra (GCA2) on the boundary of cosmological new massive gravity. At first, we consider the contracted BTZ black hole solution. We obtain entropy formula for the GCA2 in terms of contracted scaling dimension Δ and central charge C1. This entropy formula exactly matches with the nonrelativistic limit of Bekenstein-Hawking entropy of BTZ black hole. Then, we extend our study to the contracted warped AdS3 black hole solution of CNMG. We obtain the entropy of dual GCA2 in terms of central charges and finite temperatures, T1, T2. Again, this expression coincides with the nonrelativistic limit of Bekenstein-Hawking entropy formula of warped AdS3 black hole.


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## Body

## 1. Introduction

Recently, there has been some interest in extending the AdS/CFT correspondence to nonrelativistic field theories [1, 2]. The Kaluza-Klein-type framework for nonrelativistic symmetries, used in [1, 2], is basically identical to the one introduced in [3, 4] (see also [5, 6]). The study of a different nonrelativistic limit was initiated in [7, 8], where the nonrelativistic conformal symmetry was obtained by a parametric contraction of the relativistic conformal group. Galilean conformal algebra (GCA) arises as contraction relativistic conformal algebras [7–10], where in 3+1 space-time dimensions the Galilean conformal group is a fifteen-parameter group which contains the ten parameter Galilean subgroup. Infinite-dimensional Galilean conformal group has been reported in [9, 10], the generators of this group are   Ln=-(n+1)tnxi∂i-tn+1∂t, Min=tn+1∂i, and Jijn=-tn(xi∂j-xj∂i)  for an arbitrary integer n, where i and j are specified by the spatial directions. There is a finite-dimensional subgroup of the infinite-dimensional Galilean conformal group which generated by  (Jij0,L±1,L0,Mi±1,Mi0). These generators are obtained by contraction (t→t,xi→ϵxi,ϵ→0,vi~ϵ) of the relativistic conformal generators. Recently the authors of [11] (see also [12, 13]) have shown that the GCA2 is the asymptotic symmetry of cosmological topologically massive gravity (CTMG) in the nonrelativistic limit. They have obtained the central charges of GCA2, and also a nonrelativistic generalization of Cardy formula. In the present paper we want to investigate similar problem for cosmological new massive gravity (CNMG).Recently, a new theory of massive gravity in three dimensions was proposed by Bergshoeff et al. [14, 15]. This theory, referred to as new massive gravity (NMG), seems to offer a possibility for formulating a consistent theory of quantum gravity in three dimensions. NMG is defined by adding higher derivative term to the EH term in action, with coupling 1/m2. If we add to it the negative cosmological constant we can refer to this three-dimensional gravity as cosmological new massive gravity (CNMG). In this theory, the linearized excitations about the anti de Sitter vacuum describe a propagating massive graviton. One can show that CNMG admits the BTZ black holes as solutions [14, 15], moreover in NMG regular warped black holes have been obtained by Clément [16] (see also  [17, 18]).In this paper we propose the contracted BTZ and warpedAdS3 black hole solution of CNMG as a gravity dual of 2d GCA in the context of the nonrelativistic AdS3/CFT2 correspondence. The rest of the paper is organized as follows: in Section 2 we give a brief review of 2d CFT and its contraction, the GCA parameters were realized in term of CFT parameters. In Sections 3 and 4 we study GCA realization of BTZ, and warped AdS3 black hole solutions of NMG, respectively, in these sections GCA parameters were constructed in terms of gravity parameters, and finally we obtained finite entropy in nonrelativistic limit. The last section is devoted to the conclusion.


## 2. Galilean Conformal Algebra in 2 Dimensions

Galilean conformal algebra in2d can be obtained from contracting 2d conformal symmetry [19]. 2d conformal algebra at the quantum level is described by two copy of Virasoro algebra. In two-dimensions space-time (z=t+x, z¯=t-x), the CFT generators (2.1)Ln=-zn+1∂z,L¯n=-z¯n+1∂z¯
obey the centrally extended Virasoro algebra (2.2)[Lm,Ln]=(m-n)Lm+n+cR12m(m2-1)δm+n,0,[L¯m,L¯n]=(m-n)L¯m+n+cL12m(m2-1)δm+n,0.
By taking the nonrelativistic limit (t→t, x→ϵx with ϵ→0), the GCA generators Ln and Mn are constructed from Virasoro generators by (2.3)Ln=lim⁡ϵ→0(Ln+L¯n),Mn=lim⁡ϵ→0ϵ(Ln-L¯n).
From (2.2) and (2.3), one obtains centrally extended 2d GCA (2.4)[Lm,Ln]=(m-n)Lm+n+C1m(m2-1)δm+n,0,[Lm,Mn]=(m-n)Mm+n+C2m(m2-1)δm+n,0,[Mn,Mm]=0.
Note that [Mn,Mm] cannot have any central extension. The GCA central charges (C1, C2) are related to CFT central charges (cL, cR) as (2.5)C1=lim⁡ϵ→0cL+cR12,C2=lim⁡ϵ→0(ϵcL-cR12).
Similarly, rapidity ξ and scaling dimensions Δ, which are the eigenvalues of M0 and L0 respectively, are given by (2.6)Δ=lim⁡ϵ→0(h+h¯),ξ=lim⁡ϵ→0ϵ(h-h¯),
where h and h¯ are eigenvalues of L0 and L¯0, respectively. So the 2d GCA was obtained by the nonrelativistic limit of the Virasoro CFT2.


## 3. GCA Realization of BTZ Black Hole Solution of NMG

In this section we would like to propose that the contracted BTZ black hole solution of three-dimensional new massive gravity (NMG) is gravity dual of 2d GCA in the context of AdS/CFT correspondence. It is notable that the NMG (as a gravity dual) has to yield finite parameters (Δ,ξ,C1,C2, and entropy SGCA) for GCA. The action of the cosmological new massive gravity in three dimensions is  [14, 15] (3.1)S[gμν]=116πG∫-g(R-2λ+1m2LNMG),
where the NMG term is (3.2)LNMG=RμνRμν-38R2.
The Einstein equation of motion of this action is (3.3)Gμν+λm2gμν-12m2Kμν=0,
where (3.4)Kμν=-12∇2Rgμν-12∇μ∇νR+2∇2Rμν+4RμανβRαβ-32RRμν-RαβRαβgμν+32R2gμν,
where gμνKμν=ℒNMG. The parameter λ is dimensionless and characterizes the cosmological constant term while m has the dimension of mass and provides the coupling to the NMG term. The solution of BTZ black hole is given by (3.5)ds2=(-f(r)+16G2J2r2)dt2+dr2f(r)+r2dφ2+8GJdtdφ,
where (3.6)f(r)=(r2l2-8GM+16G2J2r2),l-2=2m2(-1±1-λm2).
The parameters M and J correspond to the mass and angular momentum in the case without the new massive gravity term, but their definitions in the case with NMG term are [16] (3.7)M1=(1-12m2l2)M,J1=(1-12m2l2)J.
Due to NMG term, the Bekenstein-Hawking entropy is renormalized by the same factor (1-1/2m2l2)(3.8)SBH=πrh2G(1-12m2l2),
here rh=2Gl(lM+J)+2Gl(lM-J). In the paper [20] Liu and Sun have analyzed the Brown-Henneaux boundary condition for the NMG. They have calculated the conserved charges corresponding to the generators of the asymptotical symmetry under the Brown-Henneaux boundary condition (see Appendix). Then they have obtained the central charges of the Virasoro algebra as [20–22] (3.9)cL=cR=3l2G(1-12m2l2).
For the BTZ black hole solution in NMG,h and h¯ are calculated as (3.10)h=12(lM1+J1)+cL24,h¯=12(lM1-J1)+cR24,
then the microscopic entropy is expressed by Cardy formula (3.11)SCFT=2π(cLh6+cRh¯6)=πrh2G(1-12m2l2),
which agrees with renormalized Bekenstein-Hawking entropy formula (3.8). Now let us consider nonrelativistic limit in three-dimensional gravity: (3.12)t⟶t,r⟶r,φ⟶ϵφ.
Accordingly, the parameters M and J in the BTZ solution must scale like (3.13)M⟶M,J⟶ϵJ.
As have been discussed in [11], the black hole metric (3.5) degenerates and looks singular in the Galilean limits (3.12), (3.13). This is similar to the usual Newtonian approximation c→∞. This situation in the bulk gravity is describe by the Newton-Cartan-like geometry for the geometry with the AdS2 base [9, 11].Since the Virasoro generators corresponds toℒn±=iξnR,L, from the gravity side using (A.3), (3.12), and (2.3) one can define the generators Ln and Mn as (3.14)Ln=einτ[(1-2e-2ρn2)∂τ+inϕ(1+2e-2ρn2)∂ϕ-in∂ρ],Mn=einτ(1+2e-2ρn2)∂ϕ,
these generators satisfy the centerless version of GCA algebra (2.4). From (2.5) and (3.9) the GCA central charges C1 and C2 in this case are (3.15)C1=l4G(1-12m2l2),C2=0.
So, one of the GCA central charges vanishes because of the parity invariance of the NMG.Similarly, from (2.6) and (3.10), scaling dimensions Δ and rapidity ξ, which are the eigenvalue of L0 and M0 are given by (3.16)Δ=lim⁡ϵ→0(lM1+cL+cR24)=(1-12m2l2)lM+C12,ξ=lim⁡ϵ→0ϵ(J+cL-cR24)=0.
When M is large enough, the last term C1/2 can be neglected.Now we want to obtain the entropy of the GCA. The scaling limit (M→M, J→ϵJ) requires that the event horizon of the BTZ black hole should scale  rh→2l2GM, so the black hole entropy (3.8) is given by (3.17)lim⁡ϵ→0SBH=lim⁡ϵ→0SCFT=π(1-12m2l2)2l2MG.
From the expressions of the central charge C1 (3.15) and scaling dimension Δ (3.16), we can rewrite the entropy as (3.18)SGCA=π8ΔC1.
This expression is the entropy for the GCA in two dimensions.


## 4. GCA Realization on WarpedAdS3 Black Hole Solution of NMG

In this section we consider the warped black holes in NMG. The metric of warped NMG can be written as  [16–18] (4.1)ds2l2=dt2+dr2ζ2η2(r-r+)(r-r-)+|ζ|(r+ηr+r-)dtdφ+ζ24r((1-η2)r+η2(r++r-)+2ηr+r-)dφ2.
The ranges of the coordinates are t∈(-∞,+∞), r∈[0,+∞), and φ∈[0,2π]. There are two horizons which are located at r+ and r-. The parameters η, ζ are given by (4.2)η2=214+2m2l2ζ2,ζ2=4m2l221(-6±3(5-7λ)).
The Bekenstein-Hawking entropy in this case is (4.3)SBH=πlζ32m2G(r++ηr+r-).
The microscopic entropy of the dual CFT can be computed by the Cardy formula which matches with the black hole Bekenstein-Hawking entropy  [17, 18] (4.4)SCFT=π2l3(cLTL+cRTR).
The left and right moving temperatures introduced in  [17, 18]: (4.5)TL=η2ζ28πl(r++r-+2ηr+r-),TR=η2ζ28πl(r+-r-).
By the calculation such as we review briefly in the Appendix, one can obtain the central charges of warped AdS3 black hole solution of NMG as (see also  [17, 18]) (4.6)cL=cR=6lζGη2m2.
From the above central charges and temperatures (4.5) we have (4.7)SCFT=πlζ32m2G(r++ηr+r-),
which agrees precisely with the gravity result presented in  [17, 18]. Now we consider nonrelativistic limit in three-dimensional new massive gravity: (4.8)t⟶t,r⟶r,φ⟶ϵφ.
GCA central charges C1 and C2 are defined in terms of CFT central charges (2.5) (4.9)C1=lζGη2m2,C2=0.
CFT entropy (4.4) with the limit (ϵ→0) is converted to Galilean conformal entropy: (4.10)SGCA=lim⁡ϵ→0(π23[6C1(TL+TR)+6C2(TL-TRϵ)]).
We define Galilean conformal temperatures as follows: (4.11)T1=lim⁡ϵ→06(TL+TR),T2=lim⁡ϵ→06TL-TRϵ,
the GCA entropy is (4.12)SGCA=π23(C1T1+C2T2).
To make the GCA entropy finite, TL+TR~O(1) and TL-TR~O(ϵ)(4.13)TL+TR=η2ζ24πl(r++ηr+r-),TL-TR=η2ζ24πl(r-+ηr+r-).
The parameters r+ and r- in the nonrelativistic warped NMG must scale like (4.14)r+⟶r+,r-⟶ϵ2r-.
Then, TL+TR=(η2ζ2/4πl)r+~O(1) and TL-TR=(η2ζ2/4πl)ϵ~O(ϵ), and the GCA entropy is finite (4.15)SGCA=πlζ32m2Gr+,
which agrees with Bekenstein-Hawking entropy of warped CNMG in nonrelativistic limit (4.14).


## 5. Conclusion

In the present paper we have considered the BTZ and warpedAdS3 black hole solution of CNMG in the nonrelativistic limit. The BTZ solution of CNMG has been obtained from a BTZ solution of pure Einstein gravity only by redefinition of mass and angular momentum parameter of metric, as (3.7), [16]. We have shown that the contracted BTZ solution has a dual description in term of a 2-dimensional GCA. We have obtained the central charges C1, C2 and also the scaling dimension Δ and rapidity ξ as (3.15) and (3.16), respectively. Using these results and scaling limits (M→M,J→ϵJ), we have obtained the entropy of GCA2 by (3.18), this expression exactly agrees with the nonrelativistic limit of Bekenstein-Hawking entropy of the black hole. After that we studied the warped AdS3 black hole solution in nonrelativistic limit. Using the central charges C1, C2 for GCA2, we obtained the entropy of nonrelativistic CFT2, as (4.10). Then we have defined Galilean conformal temperatures by (4.11). Using these results we have obtained final form of GCA entropy by (4.15), which is exactly the nonrelativistic limit of entropy formula (4.3).



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