Eur. Phys. J. C (2011) 71: 1566DOI 10.1140/epjc/s10052-011-1566-9

Regular Article - Theoretical Physics

Hidden conformal symmetry of self-dual warped AdS3 black holesin topological massive gravity

Ran Li1,a, Ming-Fan Li2,b, Ji-Rong Ren3,c

1Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China2Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou 310027, China3Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, China

Received: 2 December 2010 / Revised: 31 January 2011 / Published online: 22 February 2011© Springer-Verlag / Società Italiana di Fisica 2011

Abstract We extend the recently proposal of hidden con-formal symmetry to the self-dual warped AdS3 black holesin topological massive gravity. It is shown that the waveequation of massive scalar field with sufficient small angu-lar momentum can be reproduced by the SL(2, R) Casimirquadratic operator. Due to the periodic identification in theφ direction, it is found that only the left section of hid-den conformal symmetry is broken to U(1), while the rightsection is unbroken, which only gives the left tempera-ture of dual CFT. As a check of the dual CFT conjectureof self-warped AdS3 black hole, we further compute theBekenstein–Hawking entropy and absorption cross sectionand quasinormal modes of scalar field perturbation and showthese are just of the forms predicted by the dual CFT.

1 Introduction

Topological massive gravity (TMG) is described by the the-ory of three dimensional Einstein gravity with a gravita-tional Chern–Simons correction and the cosmological con-stant [1, 2]. The well-known spacelike warped AdS3 blackhole [3] (previously obtained in [4–6]), which is a vacuumsolution of topological massive gravity, is conjectured to bedual to a two dimensional conformal field theory (CFT) withnon-zero left and right central charges [7–9]. The spacelikewarped AdS3 black hole is a quotient of warped AdS3 space-time, just as BTZ black hole is a quotient of AdS3 space-time. This leads to the breaking of the SL(2, R)× SL(2, R)

isometry of AdS3 to the SL(2, R) × U(1) isometry ofwarped AdS3 black hole. It is shown in [10] that, for a

a e-mail: liran05@lzu.cnb e-mail: 11006139@zju.edu.cnc e-mail: renjr@lzu.edu.cn

certain low energy limit, the wave equation of the mas-sive scalar field in the background of spacelike warpedAdS3 black hole can be written as the Casimir operator ofSL(2, R)L × SL(2, R)R Lie algebra, which uncovers thehidden SL(2, R) × SL(2, R) symmetry of the wave equa-tion of scalar field.

Recently, a new class of solutions in three dimensionaltopological massive gravity named self-dual warped AdS3

black holes has been proposed by Chen et al. in [11]. Itis conjectured that the self-dual warped AdS3 black holeis dual to a chiral CFT with only nonvanishing left centralcharge, which is very different from the spacelike warpedAdS3 black hole. The self-dual warped AdS3 black holeis locally equivalent to spacelike warped AdS3 spacetimevia a coordinate transformation. The isometry group is justU(1)L × SL(2, R)R , similar to the warped AdS3 blackhole. Under the consistent boundary condition, the U(1)Lisometry is enhanced to a Virasoro algebra with nonvanish-ing left central charge, while the SL(2, R)R isometry be-comes trivial with the vanishing right central charge, whichis similar to the case of the extremal Kerr/CFT correspon-dence [12, 13]. This suggests a novel example of a warpedAdS/CFT dual.

In this paper, motivated by the recently proposed hiddenconformal symmetry of the wave equation of scalar fieldpropagating in the background of the general rotating blackhole [14], we consider the case of self-dual warped AdS3

black holes in TMG. It is shown that the wave equation of amassive scalar field propagating in the background of a self-dual warped AdS3 black hole can be rewritten in the formof a Casimir operator of the SL(2, R)L × SL(2, R)R Lie al-gebra. Unlike the higher dimensional black holes where thenear-horizon limit should be taken into account to match thewave equation with the Casimir operator, in the present case,only the condition of the small angular momentum of scalarfield is imposed, which suggests that the hidden conformal

Page 2 of 8 Eur. Phys. J. C (2011) 71: 1566

symmetry is valid for the scalar field with arbitrary energy.So we have uncovered the hidden SL(2, R)L × SL(2, R)Rsymmetry of the wave equation of a massive scalar field ina self-dual warped AdS3 black hole. Then, we show that,due to the periodic identification in the φ direction, only onecopy of hidden SL(2, R) symmetry is broken to U(1), whilethe another copy is unbroken. This only gives the left tem-perature TL of dual CFT, while the right temperature TR can-not be read from this approach. This point is also differentfrom the higher dimensional black holes [14–26]. Despitethat the right temperature cannot be directly read from theperiodic identification in the φ direction, one can still con-juncture that a self-dual warped AdS3 black hole is holo-graphically dual to a two dimensional CFT with the left tem-perature TL = α

2πand the right temperature TR = x+−x−

4π,

which is exactly matches with the warped AdS/CFT corre-spondence suggested in [11].

As a check of this conjecture, we also show the entropy ofthe dual conformal field theory given by the Cardy formulamatches exactly with the Bekenstein–Hawking entropy ofself-dual warped AdS3 black hole. Furthermore, the absorp-tion cross section of scalar field perturbation calculated fromthe gravity side is in perfect match with that for a finitetemperature 2D CFT. Finally, we present an algebraic cal-culation of quasinormal modes for scalar field perturbationfirstly proposed by Sachs et al. in [27]. It is shown that thequasinormal modes coincide with the poles in the retardedGreen’s function obtained in [11], which is a prediction ofthe AdS/CFT duality.

This paper is organized as follows. In Sect. 2, we give abrief review of self-dual warped AdS3 black hole in topo-logical massive gravity. In Sect. 3, we study the hidden con-formal symmetry of this black hole by analysing the waveequation of massive scalar field. In Sect. 4, we give someinterpretations of the dual conformal field description of aself-warped AdS3 black hole by computing the entropy, ab-sorption cross section and quasinormal modes of the scalarfield and comparing the results from both the gravity and theCFT side. The last section is devoted to a discussion and ourconclusion.

2 Self-dual warped AdS3 black hole

In this section, we will give a brief review of self-dualwarped AdS3 black hole in topological massive gravity. Theaction of topological massive gravity with a negative cos-mological constant is given by

ITMG = 1

16πG

∫M

d3x√−g

(R + 2

l2

)

+ l

96πGν

∫M

d3x√−gελμν

× Γ αλσ

(∂μΓ σ

αν + 2

3Γ σ

μτΓτνα

). (1)

Varying the above action with respect to the metric yieldsthe equation of motion, which is given by

Gμν − 1

l2gμν + l

3νCμν = 0, (2)

where Gμν = Rμν − 12Rgμν is the Einstein tensor and Cμν

is the Cotton tensor,

Cμν = ε αβμ ∇α

(Rβν − 1

4Rgβν

). (3)

Recently, a new class of solutions of topological massivegravity named as the self-dual warped AdS3 black hole isinvestigated by Chen et al. in [11]. The metric is given by

ds2 = 1

ν2 + 3

(−(x − x+)(x − x−)dτ 2

+ 1

(x − x+)(x − x−)dx2

+ 4ν2

ν2 + 3

(αdφ + 1

2(2x − x+ − x−)dτ

)2), (4)

where x+ and x− are the location of the outer and inner hori-zons respectively, and we have set l = 1 for simplicity. Themass M and angular momentum J of this black hole aregiven by

M = 0, J = (α2 − 1)ν

6G(ν2 + 3). (5)

The Hawking temperature TH , angular velocity of the eventhorizon ΩH and the Bekenstein–Hawking entropy SBH ofthis solution are, respectively, given by

TH = x+ − x−4π

,

ΩH = −x+ − x−2α

,

SBH = 2παν

3G(ν2 + 3).

(6)

This solution is asymptotic to the spacelike warped AdS3

spacetime. It is shown in [11] that the self-dual warped AdS3

black hole is locally equivalent to spacelike warped AdS3

spacetime. Under a coordinate transformation,

v = tan−1[

2√

(x− x+)(x− x−)

2x− x+ − x−sinh

(x+ − x−

2τ

)],

σ = sinh−1[

2√

(x− x+)(x− x−)

2x− x+ − x−cosh

(x+ − x−

2τ

)],

u = αφ + tan−1[

2x− x+ − x−x+ − x−

coth

(x+ − x−

2τ

)],

(7)

Eur. Phys. J. C (2011) 71: 1566 Page 3 of 8

the metric of self-dual warped AdS3 black hole solutioncan be transformed to the metric of spacelike warped AdS3

spacetime

ds2 = 1

ν2 + 3

(− cosh2 σdv2 + dσ 2

+ 4ν2

ν2 + 3(du + sinhσdv)2

). (8)

It will be shown that this coordinate transformation is justthe appropriate coordinate transformation to uncover thehidden conformal symmetry. The isometry group of thissolution is U(1)L × SL(2, R)R , which is generated by theKilling vectors

J2 = 2∂u, (9)

and

J̃1 = 2 sinv tanhσ∂v − 2 cosv∂σ + 2 sinv

coshσ∂u,

J̃2 = −2 cosv tanhσ∂v − 2 sinv∂σ − 2 cosv

coshσ∂u,

J̃0 = 2∂v.

(10)

It is also shown in [11] that, under the consistent bound-ary condition, the U(1)L isometry is enhanced to a Virasoroalgebra with the central charge

cL = 4ν

ν2 + 3, (11)

while the SL(2, R)R isometry becomes trivial with the van-ishing central charge cR = 0, which is similar to the case ofextremal Kerr/CFT dual [12, 13]. The entropy of the self-dual warped AdS3 black hole can be reproduced by theCardy formula. So it is conjectured that the self-dual warpedAdS3 black hole is holographically dual to a two dimen-sional chiral conformal field theory with nonvanishing leftcentral charge.

3 Hidden conformal symmetry

In this section, we study the hidden conformal symmetry byanalyzing the massive scalar field propagating in the back-ground of a self-dual warped AdS3 black hole. Firstly, it isfound that the scalar field equation can be exactly solvedby the hypergeometric function. Then, by introducing theSL(2, R)L × SL(2, R)R generators and using the coordi-nate transformation (7), we show that the wave equation canbe reproduced by the SL(2, R) Casimir operator.

3.1 Scalar field perturbation

Let us consider the scalar field Φ with mass m in the back-ground of a self-dual warped AdS3 black hole, where thewave equation is given by the Klein–Gordon equation

(1√−g

∂μ

(√−ggμν∂ν

) − m2)

Φ = 0. (12)

The scalar field wave function Φ(τ, x,φ) can be expandedin eigenmodes as

Φ = e−iωτ+ikφR(x), (13)

where ω and k are the quantum numbers. Then the radialwave equation can be written as

[∂x

((x − x+)(x − x−)∂x

) +(ω + x+−x−

2αk)2

(x − x+)(x+ − x−)

−(ω − x+−x−

2αk)2

(x − x−)(x+ − x−)

]R(x)

=(

−3(ν2 − 1)

4ν2

k2

α2+ 1

ν2 + 3m2

)R(x). (14)

This radial wave equation can be exactly solved by thehypergeometric function. In order to solve the radial equa-tion, it is convenient to introduce the variable

z = x − x+x − x−

. (15)

Then, the radial equation can be rewritten in the form ofhypergeometric equation

z(1 − z)d2R(z)

dz2+ (1 − z)

dR(z)

dz

+(

A

z+ B + C

1 − z

)R(z) = 0, (16)

with the parameters

A =(

k

2α+ ω

x+ − x−

)2

,

B = −(

k

2α− ω

x+ − x−

)2

,

C = 3(ν2 − 1)

4ν2

k2

α2− 1

ν2 + 3m2.

(17)

For later convenience, we consider the solution with the in-going boundary condition at the horizon which is given bythe hypergeometric function

R(z) = zα(1 − z)βF (a, b, c, z), (18)

Page 4 of 8 Eur. Phys. J. C (2011) 71: 1566

where

α = −i√

A, β = 1

2−

√1

4− C, (19)

and

c = 2α + 1,

a = α + β + i√−B,

b = α + β − i√−B.

(20)

It should be noted that, generally, the wave equation can-not be analytically solved and the solution must be obtainedby matching solutions in an overlap region between the near-horizon and asymptotic regions. But in the present case wehave shown that the radial equation can be exactly solvedby hypergeometric functions. As hypergeometric functionstransform in the representations of SL(2, R), this suggeststhe existence of a hidden conformal symmetry. In the nextsubsection, we will try to explore this hidden conformalsymmetry.

3.2 SL(2, R)L × SL(2, R)R

Now, we will uncover the hidden conformal symmetry byshowing that the radial equation can also be obtained by us-ing the SL(2, R) Casimir operator. Let us define the vectorfields

H0 = − i

2J̃2,

H1 = i

2(J̃0 + J̃1),

H−1 = i

2(J̃0 − J̃1),

(21)

and

H̃0 = i

2J2,

H̃1 = 1

2(J1 + J0),

H̃−1 = 1

2(J1 − J0),

(22)

with

J1 = −2 sinhu

coshσ∂v − 2 coshu∂σ + 2 tanhσ sinhu∂u,

J0 = 2 coshu

coshσ∂v + 2 sinhu∂σ − 2 tanhσ coshu∂u.

(23)

Note that (J1, J2, J0) and (J̃1, J̃2, J̃0) are the SL(2, R)L ×SL(2, R)R Killing vectors of AdS3 spacetime. The vectorfields (H1,H0,H−1) obey the SL(2, R) Lie algebra

[H0,H±1] = ±iH±1, [H−1,H1] = 2iH0, (24)

and similarly for (H̃1, H̃0, H̃−1). According to the coordi-nate transformation (7), the SL(2, R) generators can be ex-pressed in terms of the black hole coordinates (τ, x,φ) as

H0 = i

2πTR

∂τ ,

H−1 = ie−2πTRτ

[√(x − x+)(x − x−) ∂x

− 1√(x − x+)(x − x−)

· TR

TL

∂φ

+(x − x++x−

2

)√

(x − x+)(x − x−)· 1

2πTR

∂τ

],

H1 = ie2πTRτ

[−√

(x − x+)(x − x−) ∂x

− 1√(x − x+)(x − x−)

· TR

TL

∂φ

+(x − x++x−

2

)√

(x − x+)(x − x−)· 1

2πTR

∂τ

],

(25)

and

H̃0 = i

2πTL

∂φ,

H̃−1 = ie−2πTLφ

[√(x − x+)(x − x−) ∂x

+(x − x++x−

2

)√

(x − x+)(x − x−)· 1

2πTL

∂φ

− 1√(x − x+)(x − x−)

∂τ

],

H̃1 = ie2πTLφ

[−√

(x − x+)(x − x−) ∂x

+(x − x++x−

2

)√

(x − x+)(x − x−)· 1

2πTL

∂φ

− 1√(x − x+)(x − x−)

∂τ

],

(26)

where TL and TR are defined by

TL = α

2π, TR = x+ − x−

4π. (27)

The SL(2, R) quadratic Casimir operator is defined by

H2 = H̃2 = −H 20 + 1

2(H1H−1 + H−1H1). (28)

In terms of the (τ, x,φ) coordinates, the SL(2, R) quadraticCasimir operator becomes

H2 = ∂x

((x − x+)(x − x−)∂x

)

Eur. Phys. J. C (2011) 71: 1566 Page 5 of 8

− x+ − x−x − x+

[1

4πTR

∂τ − 1

4πTL

∂φ

]2

+ x+ − x−x − x−

[1

4πTR

∂τ + 1

4πTL

∂φ

]2

. (29)

For the case of ν = 1, the first term of the right hand sideof (14) is vanishing. It should be noted that, unlike the caseof higher dimensional black holes [14–26], where the near-region limit for the radial wave equation is taken into ac-count, in the present case no extra approximation is neededto match the wave equation of scalar field with the Casimiroperator. So for the case of ν = 1, the self-dual warped blackholes exhibit the local SL(2, R)L × SL(2, R)R symmetryjust like the BTZ black hole.

We consider the nontrivial case of ν2 > 1, when thesesolutions are free of naked CTCs. An additional condi-tion must be imposed to match the wave equation with theCasimir. In order to neglect the first term of right hand sideof (14), we impose the condition that the angular momentumk of scalar field is sufficient small

3(ν2 − 1)

4ν2

k2

α2� 1. (30)

Then, we find that the wave equation of massive scalar fieldwith the sufficient small angular momentum k can be rewrit-ten as the Casimir operator

H2Φ = H̃2Φ = 1

ν2 + 3m2Φ, (31)

and the conformal weights of dual operator of the massivescalar field Φ are given by

(hL,hR) =(

1

2+

√1

4+ m2

ν2 + 3,

1

2+

√1

4+ m2

ν2 + 3

). (32)

So we have found that, similar to the case of higher di-mensional black holes, the hidden SL(2, R)L × SL(2, R)R

symmetry of a self-dual warped AdS3 black hole is uncov-ered by investigating the wave equation of scalar field inits background. Note that the hidden conformal symmetryis not derived from the conformal symmetry of spacetimegeometry itself.

It is also interesting that the hidden conformal symmetryof the self-dual warped AdS3 black hole is locally the isom-etry of AdS3 spacetime, which means that scalar fields withsufficient small angular momentum k satisfying the condi-tion (30) do not feel the warped property of spacetime, whilefor the case of a spacelike warped AdS3 black hole as inves-tigated in [10], this observation is valid for the scalar fieldswith sufficiently low energy.

4 CFT interpretation

4.1 Temperature and entropy

As pointed out in [14], for the case of higher dimensionalblack hole, the vector fields of SL(2, R) generators are notglobally defined. Because of the periodic identification inthe φ direction, the hidden SL(2, R)L × SL(2, R)R sym-metry is spontaneously broken to U(1)L × U(1)R subgroup,which gives rise to the left and right temperatures of dualCFT.

The story is somewhat different in the present case. Thegenerators of SL(2, R)L presented in (26) are affected bythe periodic identification in the φ direction, while the gen-erators of SL(2, R)R in (25) are not affected because thegenerators of SL(2, R)R symmetry are just the Killing vec-tors associated to SL(2, R) isometry of the backgroundgeometry. This means that only the left sector of hiddenconformal symmetry is broken to U(1) symmetry, while theright sector is unbroken, which only gives the left temper-ature TL of dual CFT. The right temperature TR cannot beobtained from this approach. The periodical identificationin the φ direction makes no contribution to the right tem-perature. This point can also be reached from the coordinatetransformation (7), which indicates that the self-dual warpedAdS3 black hole cannot be obtained as a quotient of warpedAdS3 vacuum.

As discussed in [11], the left and right temperatures ofdual CFT can be defined with respect to the Frolov–Thornevacuum [28]. Consider the quantum field with eigenmodesof the asymptotic energy ω and angular momentum k. Af-ter tracing over the region inside the horizon, the vacuum isa diagonal density matrix in the energy-angular momentum

eigenbasis with a Boltzmann weighting factor e− ω−kΩ

TH . Theleft and right charges ωL, ωR associated to ∂φ and ∂t are k

and ω, respectively. In terms of these variables, the Boltz-mann factor is

e− ω−kΩ

TH = e− ωL

TL− ωR

TR , (33)

which gives the definition of left and right temperatures (27).The definition of the left and right charges ωL and ωR issomewhat ambiguous. The consistency can be checked inthe following by reproducing the black hole entropy and ab-sorption cross section from the CFT side.

So one can conjecture that the self-dual warped AdS3

black hole is holographically dual to a two dimensional CFTwith the left temperature TL = α

2πand the right tempera-

ture TR = x+−x−4π

. As a check of this conjecture, we nowwant to calculate the microscopic entropy of the dual CFT,and compare it with the Bekenstein–Hawking entropy of aself-dual warped AdS3 black hole. By imposing the consis-tent boundary condition, Chen et al. [11] have calculated the

Page 6 of 8 Eur. Phys. J. C (2011) 71: 1566

central charge of the asymptotic symmetry group, where theconclusion is presented in (11). So the microscopic entropyof the dual conformal field can be calculated by using theCardy formula,

SCFT = π2

3(cLTL + cRTR) = 2παν

3G(ν2 + 3)= SBH, (34)

which matches with the Bekenstein–Hawking entropy ofself-dual warped AdS3 black hole.

4.2 Absorption cross section

In this subsection, we will calculate the absorption probabil-ity for the scalar filed perturbation and compare it with resultfrom the CFT side. For the spacelike warped AdS3 blackhole, this aspect has been investigated in [29] and [30].

Under the condition (30), the solution to the radial equa-tion of scalar field perturbation with the ingoing boundarycondition is explicitly given by

R(x) =(

x − x+x − x−

)−i( k2α

+ ωx+−x− )(

x+ − x−x − x−

) 12 −

√14 + m2

ν2+3

× F

(1

2−

√1

4+ m2

ν2 + 3− i

2ω

x+ − x−,

1

2−

√1

4+ m2

ν2 + 3− i

k

α,

1 − i

(k

α+ 2ω

x+ − x−

),x − x+x − x−

). (35)

At asymptotic infinity, the solution behaves as

R(x → ∞) ∼ Ax− 1

2 +√

14 + m2

ν2+3 , (36)

with

A=Γ (1 − i( k

α+ 2ω

x+−x− ))Γ (2√

14 + m2

ν2+3)

Γ ( 12 +

√14 + m2

ν2+3− i 2ω

x+−x− )Γ ( 12 +

√14 + m2

ν2+3− i k

α)

.

(37)

The absorption cross section is then proportional to

Pabs ∼ |A|−2

∼ sinh

(πk

α+ 2πω

x+ − x−

)

×∣∣∣∣Γ

(1

2+

√1

4+ m2

ν2 + 3− i

2ω

x+ − x−

)∣∣∣∣2

×∣∣∣∣Γ

(1

2+

√1

4+ m2

ν2 + 3− i

k

α

)∣∣∣∣2

. (38)

To compare it with the result from the CFT side, we needto find the related parameters. From the first law of blackhole thermodynamics,

δSBH = δM − ΩH δJ

TH

, (39)

one can calculate the conjugate charges

δSBH = δEL

TL

+ δER

TR

. (40)

The solution is given by

δEL = δJ, δER = δM. (41)

Identifying δM = ω and δJ = k, one can find the left andright conjugate charges as

ωL ≡ δEL = k, ωR ≡ δER = ω, (42)

which are in coincidence with the left and right chargesgiven in the last subsection. Finally, the absorption cross sec-tion can be expressed as

Pabs ∼ T2hL−1L T

2hR−1R sinh

(ωL

2TL

+ ωR

2TR

)

×∣∣∣∣Γ

(hL + i

ωL

2πTL

)∣∣∣∣2∣∣∣∣Γ

(hR + i

ωR

2πTR

)∣∣∣∣2

, (43)

which is precisely in coincidence with the absorption crosssection for a finite temperature 2D CFT.

4.3 Quasinormal modes

In this subsection, we will compute the quasinormal modesof scalar field perturbation by using the algebraic methodfirstly proposed by Sachs et al. in [27] and compare the re-sults with that presented in [31]. This method strongly de-pends on the observation of hidden conformal symmetry inthe last section. This method has also been employed to in-vestigate the quasinormal modes of vector and tensor pertur-bation by Chen et al. in [32].

Here, we consider the general case without imposing thesmall angular momentum condition (30). The radial equa-tion of scalar field perturbation can be written using theSL(2, R) generators as

[1

2(H1H−1 + H−1H1) − H 2

0 + 3(ν2 − 1)

4ν2H̃ 2

0

]Φ

= m2

ν2 + 3Φ. (44)

Firstly, we consider the chiral highest weight modes satisfy-ing the condition

H1Φ = 0. (45)

Eur. Phys. J. C (2011) 71: 1566 Page 7 of 8

Under the ansatz (13) for scalar field, this condition impliesthe equation

d

dxlnR(x) = − i

(x − x+)(x − x−)

×(

TR

TL

k + [(x − x+) + (x − x−)]4πTR

ω

), (46)

which gives the solution

R(x) = (x − x+)−i( ω

4πTR+ k

4πTL)(x − x−)

−i( ω4πTR

− k4πTL

). (47)

Then the operator equation (44) can be transformed into analgebra equation

(ω

2πTR

)2

+ i

(ω

2πTR

)− C = 0. (48)

The solution of this equation gives the lowest quasinormalmode in the right-moving sector

ω

2πTR

= −ih′R, h′

R = 1

2+

√1

4− C. (49)

It should be noted that this conformal weight is slightly dif-ferent from that given by (32). If the small angular momen-tum limit is taken into account, it recovers the conformalweight for scalar field given by (32). Then, the descendentsof the chiral highest weight mode Φ

Φ(n) = (H−1)nΦ, (50)

give the infinite tower of quasinormal modes

ω

2πTR

= −i(n + h′

R

). (51)

This is just the result obtained in [31, 32], and coincides withthe poles in the retarded Green’s function obtained in [11].

The left sector of quasinormal scalar modes cannot be ob-tained analogously. The solution constructed from the chi-ral highest weight condition, H1Φ = 0, falls off in time aswell as at infinity, while for the other highest weight condi-tion, H̃1Φ = 0, one cannot obtain the solution with the sameproperty.

5 Conclusion

We have investigated the hidden conformal symmetry ofself-dual warped AdS3 black holes in topological massivegravity. The wave equation of massive scalar field propa-gating in this background with sufficient small angular mo-mentum can be rewritten in the form of SL(2, R) Casimiroperator. Interestingly, unlike the higher dimensional blackholes, where the near-horizon limit should be taken into ac-

count to match the wave equation with the Casimir operator,in the present case only the condition of the small angularmomentum of the scalar field is imposed, which suggeststhat the hidden conformal symmetry is valid for the scalarfield with arbitrary energy.

Despite that the right temperature cannot be directly readfrom the periodic identification in the φ direction, one canstill conjecture that the self-dual warped AdS3 black hole isdual to a 2D CFT with non-zero left and right temperatures.As a check of this conjecture, we also show that the en-tropy of the dual conformal field given by the Cardy formulamatches exactly with the Bekenstein–Hawking entropy of aself-dual warped AdS3 black hole. Furthermore, the absorp-tion cross section of a scalar field perturbation calculatedfrom the gravity side is in perfect match with that for a fi-nite temperature 2D CFT. At last, an algebraic calculation ofquasinormal modes for scalar field perturbation is presentedand the correspondence between quasinormal modes and thepoles in the retarded Green’s function is found.

Acknowledgement RL would like to thank Shi-Xiong Song andLin-Yu Jia for helpful discussions. The work of J.R.R. was supportedby the Cuiying Programme of Lanzhou University (225000-582404)and the Fundamental Research Fund for Physics and Mathematic ofLanzhou University (LZULL200911).

References

1. S. Deser, R. Jackiw, S. Templeton, Topologically massive gaugetheories. Ann. Phys. 140, 372 (1982)

2. S. Deser, R. Jackiw, S. Templeton, Three-dimensional massivegauge theories. Phys. Rev. Lett. 48, 975 (1982)

3. D. Anninos, W. Li, M. Padi, W. Song, A. Strominger, WarpedAdS3 black holes. J. High Energy Phys. 0903, 130 (2009)

4. K.A. Moussa, G. Clement, C. Leygnac, Class. Quantum Gravity20, L277 (2003)

5. A. Bouchareb, G. Clement, Class. Quantum Gravity 24, 5581(2007)

6. K.A. Moussa, G. Clement, H. Guennoune, C. Leygnac, Phys. Rev.D 78, 064065 (2008)

7. G. Compere, S. Detournay, Semi-classical central charge in topo-logically massive gravity. Class. Quantum Gravity 26, 012001(2009)

8. G. Compere, S. Detournay, Boundary conditions for spacelike andtimelike warped AdS3 spaces in topologically massive gravity.J. High Energy Phys. 0908, 092 (2009)

9. M. Blagojevic, B. Cvetkovic, Asymptotic structure of topologi-cally massive gravity in spacelike stretched AdS sector. J. HighEnergy Phys. 0909, 006 (2009)

10. R. Fareghbal, Hidden conformal symmetry of the warped AdS3black hole. arXiv:1006.4034 [hep-th]

11. B. Chen, G. Moutsopoulos, B. Ning, Self-dual warped AdS3 blackholes. arXiv:1005.4175 [hep-th]

12. M. Guica, T. Hartman, W. Song, A. Strominger, The Kerr/CFTcorrespondence. Phys. Rev. D 80, 124008 (2009)

13. T. Hartman, K. Murata, T. Nishioka, A. Strominger, CFT duals forextreme black holes. J. High Energy Phys. 0904, 019 (2009)

14. A. Castro, A. Maloney, A. Strominger, Hidden conformal symme-try of the Kerr black hole. arXiv:1004.0996 [hep-th]

Page 8 of 8 Eur. Phys. J. C (2011) 71: 1566

15. C. Krishnan, Hidden conformal symmetries of five-dimensionalblack holes. arXiv:1004.3537 [hep-th]

16. C.M. Chen, J.R. Sun, Hidden conformal symmetry of theReissner–Nordstrom black holes. arXiv:1004.3963 [hep-th]

17. Y.Q. Wang, Y.X. Liu, Hidden conformal symmetry of the Kerr–Newman black hole. arXiv:1004.4661 [hep-th]

18. B. Chen, J. Long, Real-time correlators and hidden conformalsymmetry in Kerr/CFT correspondence. arXiv:1004.5039 [hep-th]

19. R. Li, M.-F. Li, J.-R. Ren, Entropy of Kaluza–Klein black holefrom Kerr/CFT correspondence. arXiv:1004.5335 [hep-th]

20. D. Chen, P. Wang, H. Wu, Hidden conformal symmetry of rotatingcharged black holes. arXiv:1005.1404 [gr-qc]

21. M. Becker, S. Cremonini, W. Schulgin, Correlation functions andhidden conformal symmetry of Kerr black holes. arXiv:1005.3571[hep-th]

22. B. Chen, J. Long, On holographic description of the Kerr–Newman-AdS-dS black holes. arXiv:1006.0157 [hep-th]

23. H. Wang, D. Chen, B. Mu, H. Wu, Hidden conformal sym-metry of the Einstein–Maxwell–Dilaton–Axion black hole.arXiv:1006.0439 [gr-qc]

24. C.-M. Chen, Y.-M. Huang, J.-R. Sun, M.-F. Wu, S.-J. Zou, Onholographic dual of the dyonic Reissner–Nordstrom black hole.arXiv:1006.4092 [hep-th]

25. C.-M. Chen, Y.-M. Huang, J.-R. Sun, M.-F. Wu, S.-J. Zou,Twofold hidden conformal symmetries of the Kerr–Newman blackhole. arXiv:1006.4097 [hep-th]

26. C. Krishnan, Black hole vacua and rotation. arXiv:1005.1629[hep-th]

27. I. Sachs, S.N. Solodukhin, Quasi-normal modes in topologicallymassive gravity. J. High Energy Phys. 0808, 003 (2008)

28. V.P. Frolov, K.S. Thorne, Renormalized stress-energy tensor nearthe horizon of a slowly evolving, rotating black hole. Phys. Rev.D 39, 2125 (1989)

29. J.J. Oh, W. Kim, Absorption cross section in warped AdS3 blackhole. J. High Energy Phys. 0901, 067 (2009)

30. H.-C. Kao, W.-Y. Wen, Absorption cross section in warped AdS3black hole revisited. J. High Energy Phys. 0909, 102 (2009)

31. R. Li, J.-R. Ren, Quasinormal modes of self-dual warpedAdS3 black hole in topological massive gravity. arXiv:1008.3239[hep-th]

32. B. Chen, J. Long, Hidden conformal symmetry and quasi-normalmodes. arXiv:1009.1010 [hep-th]