7 June 2001

Physics Letters B 509 (2001) 157–162www.elsevier.nl/locate/npe

Vibration modes of giant gravitons in the backgroundof dilatonic D-branes

Jin Young Kima, Y.S. Myungb

a Department of Physics, Kunsan National University, Kunsan 573-701, South Koreab Department of Physics, Inje University, Kimhae 621-749, South Korea

Received 2 March 2001; received in revised form 9 April 2001; accepted 11 April 2001Editor: M. Cvetic

Abstract

We consider the perturbation of giant gravitons in the background of dilatonic D-branes whose geometry is not of aconventional form of AdSm × Sn. We use the quadratic approximation to the brane action to investigate their vibrations aroundthe equilibrium configuration. We found the normal modes of small vibrations of giant gravitons and these vibrations are turnedout to be stable. 2001 Elsevier Science B.V. All rights reserved.

Recently stable extended brane configurations in some string theory background, called giant gravitons, attractedinterests in connection with the stringy exclusion principle. Myers [1] found that certain D-branes coupled to RRpotentials can expand into higher-dimensional branes. McGreevy, Susskind and Toumbas [2] have shown that amassless particle with angular momentum on the S part of AdSm ×Sn spacetime blows up into a spherical brane ofdimensionalityn− 2. Its radius increases with increasing angular momentum. The maximum radius of the blown-up brane is equal to the radius of the sphere that contains it since the angular momentum is bounded by the radius ofSn. This is a realization of the stringy exclusion principle [3] through the AdS/CFT correspondence [4]. Later it wasshown that the same mechanism can be applied to spherical branes on the AdS part [5,6]. However, they can growarbitrarily large since there is no upper bound on the angular momentum. To solve this puzzle, instanton solutionsdescribing the tunneling between the giant gravitons on the AdS part and on the S part were introduced [5,7]. Amagnetic analogue of the Myers effect was investigated by Das, Trivedi and Vaidya [8]. They suggested that theblowing up of gravitons into branes are also possible on some backgrounds other than AdSm × Sn spacetime.

Perturbations of the giant gravitons around their equilibrium configuration were studied by Das, Jevicki andMathur [9]. Using quadratic approximation to the action, they computed the natural frequency of the normal modefor giant gravitons in AdSm × Sn spacetime for both cases when gravitons are extended in AdS space and they areextended on the sphere. Frequencies are related to the curvature scale of the background and are independent ofthe radius of the brane itself. These modes are believed to play an important role in the study of black holes withinthe string theory context. We cannot explain the microscopic properties of a black hole if it is regarded as a point

E-mail addresses: jykim@ks.kunsan.ac.kr (J.Y. Kim), ysmyung@physics.inje.ac.kr (Y.S. Myung).

0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0370-2693(01)00491-9

158 J.Y. Kim, Y.S. Myung / Physics Letters B 509 (2001) 157–162

singularity. But in string theory, a pointlike singularity can be replaced by an extended brane, and vibration modesof the brane can be used to study the microscopic entropy [10] and Hawking radiation [11].

In this Letter, we will study the vibration mode of giant gravitons in the near-horizon geometry of dilatonicD-brane background. As a concrete example, we will consider the case when a probep-brane wraps the transversedirection of Sp+2 in the near-horizon geometry of D(6 − p) branes [8]. Since the background geometry isnot exactly of the form AdSm × Sn, it would be very interesting to study the fluctuation analysis around thisconfiguration.

Consider a probe Dp-brane moving in the near-horizon geometry of extremalN D(6 − p)-branes. Thebackground metric is given by

(1)ds2 = gtt dt2 +

6−p∑k=1

gkk(dxk)2 + grr dr

2 + f (r)r2dΩ2p+2,

where

(2)gtt = gkk =(r

R

) (p+1)2

, grr = f (r)=(R

r

) (p+1)2

,

and R can be expressed in terms ofN and the tension of a probe Dp-brane asRp+1 = N/TpVp , where

Vp = 2πp+1

2 /Γ(p+1

2

)is the volume of unit Sp . dΩ2

p+2 can be parametrized as

(3)dΩ2p+2 = 1

1− ρ2 dρ2 + (

1− ρ2)dφ2 + dΩ2p,

with

(4)dΩ2p = dθ2

1 + sin2 θ1(dθ2

2 + sin2 θ2(· · · + sin2 θp−1

)dθ2

p

).

Also we have the dilaton (Φ) and RR potential (Ap+1) given by

(5)eΦ =(R

r

) (p−3)(p+1)4

,

(6)Ap+1φθ1...θp

=Rp+1ρp+1εθ1...θp ,

whereεθ1...θp is the volume form of the unitp-sphere. In the case ofr = R, Φ = 0. For 0< r < R, we have anon-zero dilaton. In this sense, Eq. (1) is called the dilatonic D-brane background.

We consider an equilibrium configuration in which a probe Dp-brane wraps the Sp . The action for this case,ignoring the fermions, is given by

(7)S = SDBI + SCS = −Tp

∫dp+1ξ e−Φ

√−detGαβ + Tp

∫dp+1ξ Ap+1,

whereGαβ andAp+1 are pullbacks of the metric and the RR(p + 1)-form potential, respectively:

(8)Gαβ = ∂xM

∂ξα

∂xM

∂ξβgMN,

(9)A(p+1)ξ0ξ1...ξp

=A(p+1)M1...Mp

∂xM1

∂ξ0

∂xM2

∂ξ1

∂xMp+1

∂ξp.

If we choose a static gauge: the time parameter of the worldvolumeξ0 = t ; thep angular parametersξi are setto be the angles onSp , ξi = θi , then the dynamical variables are given byr(t, θi), xk(t, θi), ρ(t, θi) andφ(t, θi).

J.Y. Kim, Y.S. Myung / Physics Letters B 509 (2001) 157–162 159

We choose an equilibrium configuration such that these quantities do not depend onθi so that there are no braneoscillations. Since there exist translational symmetries alongxi , the corresponding momenta are conserved. Wewill study the motion whose conserved momenta are identically zero. Then, the dynamical variables take the formof r(t), ρ(t) andφ(t). With an appropriate choice of gauge, we get the full probe brane action as

(10)S = −TpVp

∫dt e−Φ

(f (r)ρ2r2)p/2√gtt − grr r2 − gρρρ2 − gφφφ2 +N

∫dt ρp+1φ,

where dot denotes derivative with respect tot andVp stands for the volume of unitp-sphere. Its conjugate momentaand Hamiltonian are:

(11)Pr = ∂L

∂r= TpVpe

−Φ√gtt − grr r2 − gρρρ2 − gφφφ2

(fρ2r2)p/2grr r,

(12)Pρ = ∂L

∂ρ= TpVpe

−Φ√gtt − grr r2 − gρρρ2 − gφφφ2

(fρ2r2)p/2gρρρ,

(13)Pφ = ∂L

∂φ= TpVpe

−Φ√gtt − grr r2 − gρρρ2 − gφφφ2

(fρ2r2)p/2gφφφ +Nρp+1,

(14)H = Pr r + Pρρ + Pφφ −L= √gtt

[(TpVpe

−Φ)2(

f (r)ρ2r2)p + P 2r

grr+ P 2

ρ

gρρ+ (Pφ −Nρp+1)2

gφφ

]1/2

.

Note that if we impose a condition

(15)TpVpe−Φ

(f (r)r2) p+1

2 = N,

and usegφφ = f (r)r2(1−ρ2), we can combine the first and last terms within the square brackets of Eq. (14). Thenthe Hamiltonian can be expressed as

(16)H = √gtt

[P 2φ

f (r)r2+ P 2

r

grr+ P 2

ρ

gρρ+ (ρPφ −Nρp)2

gφφ

]1/2

.

Since the Lagrangian does not depend onφ, Pφ is a constant of motion. For a givenPφ , the lowest energyconfiguration satisfiesPρ = 0 for all time becauseρ does not appear in the first two terms. Here we find a relationbetween the equilibrium value ofρ0 andPφ

(17)Pφ =Nρp−10 ,

which is obviously independent of the radial coordinater. The corresponding Hamiltonian is

(18)H = √gtt

[P 2φ

f (r)r2+ P 2

r

grr

]1/2

.

Actually this is the Hamiltonian of a massless particle with angular momentumPφ on Sp+2 sphere. Unlike theusual massless particle, the angular momentum of a probe brane is bounded because of 0 ρ 1. A possiblemaximum value of its angular momentum isN . This reminds us the stringy exclusion principle. Eq. (15) is animportant condition for the brane to behave like a massless particle.

So far we reviewed how the configuration of giant graviton appears in the near-horizon background of dilatonicD-brane. Now we consider the small vibration of giant graviton around the stable equilibrium configuration. Ingeneral, the brane can move in any direction. We can consider the brane motion along any of ther, ρ andφ. Ifone considers the case when the brane moves along ther direction, one can get the cosmological model known as

160 J.Y. Kim, Y.S. Myung / Physics Letters B 509 (2001) 157–162

the mirage cosmology [12]. The motion of the probe (or universe) brane in ambient space induces cosmologicalexpansion (or contraction). And if we consider the brane motion along theρ andφ directions, we get the giantgraviton picture as in Ref. [8]. Since we are interested in the perturbation from stable configuration of the giantgraviton, we neglect the motion of the probe brane along the transverse direction (r) of the background D(6− p)-branes. So we setr = 0, i.e.,r = r0 = constant. We find the angular velocity of this configuration from Eqs. (13)and (17)

(19)φ ≡ ω0 = ± 1

f0r0

with f0 = f (r0)= (r0/R)(p+1)/2.

A small vibration of the brane can be described by defining spacetime coordinates(ρ,φ, xk) as a function of theworldsheet coordinatesξ0, ξ1, . . . , ξp . In the static gauge, where

(20)ξ0 = t = τ, ξi = θi, i = 1, . . . , p,

perturbation of the remaining coordinates can be written as

(21)ρ = ρ0 + εδρ(τ, ξ1, . . . , ξp),

(22)φ = ω0τ + ε δφ(τ, ξ1, . . . , ξp),

(23)xk = εδxk(τ, ξ1, . . . , ξp), k = 1, . . . ,6− p.

First we expand the action (7) to linear-order inε

S(ε)= −εTp

∫dτ dpξ

√gξ e

−Φ0(f0r

20

) p2 ρ

p−10

×[(p+ 1)f0r

20ω

20ρ

20 + p

(1/f − f0r

20ω

20

)√1/f − f0r

20

(1− ρ2

0

)ω2

0

δρ −(1− ρ2

0

)f0r

20ω0ρ0√

1/f − f0r20

(1− ρ2

0

)ω2

0

δφ

]

(24)+ εN

∫dτ ρ

p

0

[(p + 1)ω0δρ + ρ0δφ

].

Using Eq. (15), we have

(25)

S(ε)= −εNρp−10

∫dτ

[1√f0r

20

(p + 1)f0r20ω

20ρ

20 + p

(1/f − f0r

20ω

20

)√1/f − f0r

20

(1− ρ2

0

)ω2

0

− (p + 1)ρ0ω0

δρ

+

1√f0r

20

−(1− ρ2

0

)f0r

20ω0ρ0√

1/f − f0r20

(1− ρ2

0

)ω2

0

− ρ20

δφ

].

Clearly, the coefficient ofδρ vanishes if one usesω0 = ± 1f0r0

. Also the coefficient ofδφ is a constant and thus thisterm does not contribute to the variation of the action upon the integration overτ .

On the other hand, the second order term inε is calculated as

S(ε2)= ε2 N

Tpω0ρ

p−10

∫dτ dpξ

√gξ

(26)

×[

1

2

1

ω20(1− ρ2

0)(δρ)2 − 1

2

1

(1− ρ20)

∂δρ

∂ξi

∂δρ

∂ξjgξi ξj + 1

2

1− ρ20

ω20ρ

20

(δρ)2 − 1

2

(1− ρ2

0

)∂δφ∂ξi

∂δφ

∂ξjgξiξj

+ p − 1

ω0ρ0δρδρ + 1

2

6−p∑k=1

(δxk)2 − 1

2

1

f0r20

6−p∑k=1

∂δxk

∂ξi

∂δxk

∂ξjgξi ξj

].

J.Y. Kim, Y.S. Myung / Physics Letters B 509 (2001) 157–162 161

The equations of motion are

(27)1

ω20(1− ρ2

0)

∂2δρ

∂τ2 − 1

1− ρ20

∂

∂ξi

(∂δρ

∂ξjgξi ξj

)− p − 1

ω0ρ0

∂δφ

∂τ= 0,

(28)1− ρ2

0

ω20ρ

20

∂2δφ

∂τ2 − (1− ρ2

0

) ∂

∂ξi

(∂δφ

∂ξjgξi ξj

)+ p − 1

ω0ρ0

∂δρ

∂τ= 0,

(29)∂2δxk

∂τ2− 1

f 20 r

20

∂

∂ξi

(∂δxk

∂ξjgξi ξj

)= 0.

We observe thatδxk perturbations are decoupled fromδρ andδφ. Let us introduce the spherical harmonicsYl onSp ,

(30)gξiξj∂

∂ξi

∂

∂ξjYl(ξ1, . . . , ξp)= −QlYl(ξ1, . . . , ξp),

whereQl is the eigenvalue of the Laplace operator on unitp sphere. Possible values ofQl arel(l + p − 1) withl = 0,1,2, . . . . Choosing the harmonic oscillation, perturbations can be expressed as

(31)δρ(τ, ξ1, . . . , ξp)= δρ e−iωτ Yl(ξ1, . . . , ξp),

(32)δφ(τ, ξ1, . . . , ξp)= δφ e−iωτ Yl(ξ1, . . . , ξp),

(33)δxk(τ, ξ1, . . . , ξp)= δxk e−iωτ Yl(ξ1, . . . , ξp).

From Eq. (29), we find the natural frequency forδxk perturbations as

(34)ω2x = Ql

(f0r0)2=Qlω

20 −→ ωx = ±√

Q2 ω0.

And δρ andδφ perturbations are coupled and their normal frequencies are determined by the matrix equation

(35)

1(1−ρ2

0)

(−ω2

ω20

+Ql

)iω(p − 1) 1

ω0ρ0

−iω(p − 1) 1ω0ρ0

(1−ρ20)

ρ20

(−ω2

ω20

+ ρ20Ql

)(

δρ

δφ

)= 0.

This gives the frequencies of two modes (±)

(36)ω± = 1

2(f0r0)2

[(1+ ρ2

0

)Ql + (p− 1)2 ±

√(p − 1)4 + 2(p− 1)2

(1+ ρ2

0

)Ql +Q2

l

(1− ρ2

0

)2].

These modes oscillate with real and positiveω±, so their vibration is stable for any size of the probe brane. Tomake a connection with AdSm × Sn, we can choose the possible maximum value ofρ0 asρ0 = 1. Then the naturalfrequencies for this case are

(37)ω± = 1

(f0r0)2

[Ql + (p − 1)2

2± (p− 1)

√Ql + (p − 1)2

4

].

This agrees with the result of the case whose background geometry is of the form AdSm ×Sn (see, Eq. (4.17) of [9]and Eq. (73) of [13]). InsertingQl = l(l +p− 1), we obtainω+ = (1/f0r0)(l+p− 1) andω− = (1/f0r0)l. Sincel is integer, their motion is periodic. Here we would like to emphasize that Eq. (15) is the crucial condition in ourcalculation. It has been discussed in [8] that we can draw the giant graviton picture even in non-supersymmetrypreserving backgrounds whenever this condition is met.

It is worthwhile commenting that the case considered in this Letter is somewhat different from that in AdSm×Sn

space. Extended brane configurations in AdSm × Sn space, relevant for the giant graviton picture, sit at definite

162 J.Y. Kim, Y.S. Myung / Physics Letters B 509 (2001) 157–162

value ofr both for branes extended in AdS subspace and for branes extended on the sphere in which case the branesits at the origin of the AdS space. In our calculation we did not consider the brane motion along the transversedirection (r) in the background D(6 − p)-branes. Interestingly it is known that the brane motion in this directioninduces a cosmological evolution on the universe brane, called the mirage cosmology [12,14]. Recently it has beenstudied that the motion of giant graviton is related to the closed universe of mirage cosmology [14]. Hence themotion of a probep-brane along ther-direction is expected to induce an interesting cosmological evolution in thebackground whose geometry is not AdSm × Sn space. If we remove the restrictionr = 0, then the only changemight be the existence of the additional vibrational modes along ther direction which do not mix with the modesconsidered here. It would be interesting to study the vibration modes without the restrictionr = 0.

In summary, we found the normal modes of small vibrations of giant gravitons in the background of dilatonicD-branes whose geometry is not of the form AdSm × Sn and these vibrations are stable.

Acknowledgements

Research by J.Y. Kim was supported in part by the Korea Science and Engineering Foundation, No. 2001-1-11200-003-2. Research by Y.S. Myung was supported in part by the Brain Korea 21 Program, Project No. D-0025.

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