DBI equations and holographic dc conductivity

DBI equations and holographic dc conductivity

ICTP South American Institute for Fundamental Research, Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, 01140-070 Sa˜o Paulo, Sa˜o Paulo, Brazil

Cristine N. Ferreira†

Nu´cleo de Estudos em Fı´sica, Instituto Federal de Educac¸a˜o, Cieˆncia e Tecnologia Fluminense, Rua Dr. Siqueira, 273, Campos dos Goytacazes, Rio de Janeiro CEP 28030-130, Brazil

and The Abdus Salam International Centre for Theoretical Physics Strada Costiera, 11 I-34151 Trieste Italy Victor J. Vasquez Otoya‡

Instituto Federal de Educac¸a˜o, Cieˆncia e Tecnologia do Sudeste de Minas Gerais, Rua Bernardo Mascarenhas 1283, 36080-001, Juiz de Fora, Minas Gerais, Brazil

(Received 11 March 2013; published 21 May 2013)

We provide a simple method for writing the Dirac-Born-Infeld equations of a Dp-brane in an arbitrary static background whose metric depends only on the holographic radial coordinatez. Using this method we revisit the Karch-O’Bannon procedure to calculate the dc conductivity in the presence of constant electric and magnetic fields for backgrounds where the boundary is four- or three-dimensional and satisfies homogeneity and isotropy. We find a frame-independent expression for the dc conductivity tensor. For particular backgrounds we recover previous results on holographic metals and strange metals.

DOI:10.1103/PhysRevD.87.106007 PACS numbers: 11.25.Tq, 11.25.Uv

I. INTRODUCTION

Holography provides a new approach for investigating strongly coupled field theories. The most important case is given by the AdS/CFT correspondence [1] that maps gauge theories with conformal symmetry to string/M theories compactified into anti-de Sitter spacetimes. The AdS/ CFT correspondence was conjectured by Maldacena after investigating N coincident branes in string/M theory in the large-N limit. At low energies some of these configurations describe conformal gauge theories at weak coupling, while at strong coupling they lead to supergravity solutions in-volving anti-de Sitter spacetimes.

For example, the holographic dual ofN _¼4U_ðN_Þsuper Yang-Mills theory in four dimensions is Type IIB string theory inAdS₅S5_{. This duality arises from a}

configura-tion of N coincident D3-branes in the large-N limit. This example is very interesting for many theoretical reasons (for instance integrability) and provides new insights into the old problem of nonperturbative QCD. Another interest-ing example is the N _¼6 k-level U_ðN_Þ U_ðN_Þ super Chern-Simons gauge theory in three dimensions, whose holographic dual is M theory in AdS₄S7_=Z

k. In this case the duality arises after considering a configuration of M2-branes in M theory. This example has inspired recent holographic models for condensed matter physics.

An interesting problem in condensed matter systems is to describe charge transport in strongly coupled theories.

In holography, this is achieved by the inclusion of fermionic charge carriers in the AdS/CFT correspondence. A very simple method of including fermions in the AdS/CFT correspondence consists of the inclusion ofNf Dp-branes in the probe limitNfN. This was originally proposed in the four-dimensional case [2], where N_f D7-branes are included on top ofND3-branes. The probe limitN_fN guarantees that the backreaction of the D7-branes on the bulk geometry can be ignored. The fermions are interpreted as quarks in the four-dimensional field theory, whose dual are strings stretching from the D3-branes to the D7-branes. In this paper we consider the problem of probe Dp-branes in an arbitrary static background, whose metric depends only on the holographic radial coordinate z. We provide a simple and elegant method to write the Dirac-Born-Infeld (DBI) equations that describe the dynamics of the Dp-brane. Using these results we calculate the electro-magnetic currents through holographic methods for a gen-eral gauge field configuration. We find a nice expression for the case when the gauge field strengths on the brane depend only on the radial coordinatez. As an application, we consider the case of constant electric and magnetic fields and calculate the dc conductivity tensor when the boundary is four and three dimensional and satisfies ho-mogeneity and isotropy. This is done following the Karch-O’Bannon procedure [3,4] (see also [5]), which is based on the reality of the brane action. We check our results for particular backgrounds, namely the D3-D7 model [2] and the Lifshitz background [6–10]. Our results for the D3-D7 model are consistent with those of [3–5] and are interpreted as holographic metals, while in the Lifshitz case the

*_{aballonb@ift.unesp.br}

†_{cferreir@ictp.it}

‡_{victor.vasquez@ifsudestemg.edu.br}

conductivity is consistent with [11] and can be interpreted in terms of strange metals. Our results are also consistent with a recent method proposed in [12,13], which is based on the properties of an effective metric, called open string metric, that arises from the DBI action.

Our paper ends with some conclusions and an Appendix describing the Drude model and the effect of a five-dimensional Chern-Simons term on the DBI equations for constant electric and magnetic fields.

II. DP-BRANE DYNAMICS AND HOLOGRAPHIC CURRENTS

Consider the dynamics of a probe Dp-brane in an arbitrary static background of the form

ds2_{¼aðzÞgmnðzÞdx}m_dxn_þbðzÞd2_{; (2.1)}

wheregmndescribes aDþ1spacetime that includes the radial coordinate z. We will be interested in the cases D_¼4 and D_¼3, although the results in this section are general.

We assume for simplicity thatis a compact space. We consider the case where the Dp-brane fills the spacetime of the internal coordinatesxm _{and wraps a submanifoldof} dimensionp Dinside. In a general class of Dp-brane models, the submanifold is specified by fixing some coordinates in to constant values while one of the coordinates, say, depends on the radial coordinatez.

The induced metric on the Dp-brane takes the form

ds2

Dp¼aðzÞgmnðzÞdxmdxnþbðzÞd2 ¼:GMNdxMdxN;

(2.2)

where gzz now includes a contribution arising from d2 _¼02_ðzÞdz2_.

A. The DBI action

The dynamics of a single Dp-brane is described by the Abelian DBI action

S_{DBI¼ p}Z dD_xdzdp D_e

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidet_ðG_MNþ20_F MNÞ

q

; (2.3)

whereG_MNis the induced metric on the probe brane,F_MN is the world volume gauge field strength and

_{p¼ ð}2_Þ p0 pþ1

2 (2.4)

is the brane tension.

We assume for simplicity, that the gauge fields have nonzero components in the xm _{directions only and these} components do not depend on the coordinates of. Thus, after integrating over the compact submanifold , we obtain the D_þ1 dimensional effective action that can be written as

S_DBI¼ Z dD_xdz

ðz_ÞpffiffiffiffiffiffiffiffiE; (2.5)

whereEis the determinant of the tensor defined by E_mn:_¼g_mnþðz_ÞF_mn; (2.6)

and

ðzÞ ¼2_a0

ðz_Þ ; ðzÞ ¼pVe

_ðaðzÞÞDþ1 2 ðbðzÞÞ

p D 2 : (2.7)

Here we have used the notationVto refer to the volume of the compact submanifold . Note that F_mn denotes the components of the field strength in thexm _directions.

B. DBI equations and holographic currents

The relevant fields in the effective action (2.5) are the D_þ1 dimensional gauge field A_m and the scalar field _ðz_Þ. Therefore, the variation of the action (2.5) takes the form

S_DBI¼SBulk DBI þS

Bdy

DBI; (2.8)

where

SBulk DBI¼

Z

dD_xdz

@_m

2

ffiffiffiffiffiffiffiffi

E

p _~

Eð‘;mÞ_A ‘

þ

@z

2

ffiffiffiffiffiffiffiffi

E

p _~

Ezz@gzz @0

_{ffiffiffiffiffiffiffiffi}

E

p @

@

; (2.9)

SBdy_DBI¼ Z dD_xdz

@_m

2

ffiffiffiffiffiffiffiffi

E

p _~

Eð‘;mÞ_A ‘

þ@_z

2

ffiffiffiffiffiffiffiffi

E

p _~

Ezz@gzz @0

; (2.10)

andE~ð‘;mÞ_¼E~‘m _E~m‘_{, withE}~‘m_{being the inverse ofE} ‘m. Since the field variationsA‘ andare arbitrary, the bulk term (2.9) leads to the DBI equations

@_m

2

ffiffiffiffiffiffiffiffi

E

p _~

Eð‘;mÞ_{¼0; (2.11)}

@z

2

ffiffiffiffiffiffiffiffi

E

p _~

Ezz@gzz @0

_{ffiffiffiffiffiffiffiffi}

E

p @

@¼0: (2.12)

A natural prescription for the boundary electromagnetic currents in holography is the following:

j _¼ S A

z¼zBdy

: (2.13)

In our case the only contribution comes from the first term in the boundary term (2.10), so we find

j _{¼ lim} z!zBdy

2

ffiffiffiffiffiffiffiffi

E

p _~

Eð;zÞ_{: (2.14)}

@₀

2

ffiffiffiffiffiffiffiffi

E

p _~

Eðz;0Þ_þ@ i

2

ffiffiffiffiffiffiffiffi

E

p _~

Eðz;iÞ_{¼0; (2.15)}

@_z

2

ffiffiffiffiffiffiffiffi

E

p _~

Eðz;0Þ_þ@ i

2

ffiffiffiffiffiffiffiffi

E

p _~

Eð0;iÞ_{¼0; (2.16)}

@_z

2

ffiffiffiffiffiffiffiffi

E

p _~

Eðz;iÞ _@

0

2

ffiffiffiffiffiffiffiffi

E

p _~

Eð0;iÞ

þ@_j

2

ffiffiffiffiffiffiffiffi

E

p _~

Eði;jÞ_{¼0: (2.17)}

In the case where the gauge field strength tensor F_mn depends only on the radial coordinate z, the equation (2.15) is automatically satisfied, while the others can be integrated out leading to the equation

2

ffiffiffiffiffiffiffiffi

E

p _~

Eðz;Þ_{¼j; (2.18)}

where j _{is the holographic current associated with the} DBI action, as obtained in (2.14). This is a nontrivial equation involving the field strengths on the lhs and the holographic currents on the rhs.

1. The contribution from a-term

In the D_¼4 case, the boundary field theory is four dimensional and may include a-term of the form1

S_¼ Z _dtd3_xF

F; (2.19)

where _¼ðt; ~xÞ is a four-dimensional axion field and F_¼@A @A is the four- dimensional field strength.

Integrating by parts (2.19), we obtain S¼2

Z

dtd3_xf@½A

FŠ þAð@ÞFg:

(2.20)

This term provides a contribution to the holographic current,

j _¼S

A ¼

2 _ð@

ÞF: (2.21)

C. The method of-expansion

The method proposed in this paper consists of a series expansion in the parameter_ðz_Þfor the relevant quantities appearing on the DBI equations, namely the determinantE and the inverse tensorE~m;n_.

Consider the matrix representation of the metricg^ and the field strength F^. Then the tensor Emn has a matrix representation given by

^

E_¼g^_þF^ _¼g^_ð1_þX^_Þ; (2.22)

where we have definedX^ :_¼g^ 1_F^_{. Note that this matrix} is matrix linear in . Recalling some properties of the determinant, we can writeEin terms ofX^,

E_¼det_ðE^_{Þ ¼}det_ðg^_Þdet_ð1_þX^_Þ

¼det_ðg^_Þexp_fTr_½ln_ð1_þX^_ފg: (2.23)

On the other hand, the tensor Em;n _{corresponds to the} inverse matrix E^ 1 and can also be represented in terms ofX^,

^

E 1 _{¼ ð1þX^Þ} 1_g 1_{: (2.24)}

Now we consider a series expansion inX^ for the determi-nantEand the inverse tensorE~m;n_{. For the determinant we} first expand the matrixln_ð1_þX^Þand take the trace

Tr_½ln_ð1_þX^_{ފ ¼} 1 2TrðX^

2_Þ 1

4TrðX^

4_Þ

1 6TrðX^

6

Þ þ ; (2.25)

where we used the identityTr_ðX^n_{Þ ¼}0for odd values ofn. Substituting (2.25) into (2.23) we get

E_¼det_ðg^_Þ

1 1

2TrðX^

2_Þ 1

4TrðX^

4_Þ

þ1₈½Tr_ðX^2_ފ2 1

6TrðX^

6_{Þ þ}1

8½TrðX^

2_ފ½TrðX^4_ފ

1 48½TrðX^

2

ފ3_{þ : (2.26)}

The traces appearing in the expansion can be written in a tensorial form,

Tr_ðX^2_{Þ ¼} 2_Fmn_F mn;

Tr_ðX^4_{Þ ¼}4_Fmn_F

npFpqFqm; Tr_ðX^6_{Þ ¼}6_Fmn_F

npFpqFqrFrsFsm; :

(2.27)

Then we conclude that the determinant has the -expansion,

E_¼det_ðg^_Þ

1_þ

2

2 F mn_F

mnþ 4

4!F mnpq_F

mnpq

þ

6

6!F

mnpqrs_F

mnpqrsþ

; (2.28)

where

F_mnpq:_¼F_mnF_pq F_mpF_nqþF_mqF_np;

F_mnpqrs:_¼F_mnF_pqrs F_mpF_nqrsþF_mqF_nprs

F_mrF_npqsþF_msF_npqr; : (2.29)

It is easy to check that the covariant tensors F_mnpq and F_mnpqrsare antisymmetric in the index.2

1_{A-term may arise from instanton configurations in}

QCD-like theories and break parity symmetry; see for instance [14].

2_{The antisymmetry of these tensors implies that they admit a}

The expansion (2.28) involves only even powers of. It can also be truncated according to the dimensionD_þ1of the spacetime relevant to the effective action. This is due to the antisymmetry of the tensors appearing in the expan-sion. Consider for instance the cases D_¼4 or D_¼3, where the effective bulk theory is five or four dimensional, respectively. In those cases the antisymmetry of the tensor F_mnpqrsdefined in (2.29) implies that this tensor vanishes. This is also true for the tensors appearing at higher order in. Therefore, for D_¼4andD_¼3, the -expansion (2.28) reduces to a polynomial or order4_.

Now consider the matrix representation (2.24) of the inverse tensor Em;n_{. Expanding ð1þX^Þ} 1 _{in a geometric}

series and using (2.27), we get the following expansion: ~

Emn_¼gmn _Fmn_þ2_Fm

pFpn 3FmpFpqFqn þ4_Fm

pFpqFqrFrn 5FmpFpqFqrFrsFsn

þ : (2.30)

We are also interested in the antisymmetric tensorE~ðm;nÞ_¼ ~

Emn _E~nm_{. The terms in the expansion (2.30) that contain} even powers ofare symmetric inm_$n. Therefore, the tensorE~ðm;nÞ_{takes the form}

~

Eðm;nÞ_{¼ 2F}mn ₂3_Fmp_F pqFqn 25_Fmp_F

pqFqrFrsFsnþ : (2.31) If we multiply the expansion (2.31) with the determinant given in (2.28), we obtain the expansion

EE~ðm;nÞ_{¼detðg^Þ 2F}mn 3_F

pqFmnpq

5

4 FpqFrsF

mnpqrs_{þ : (2.32)}

1. The DBI effective action and DBI equations

The-expansion of the determinantEallows us to find a useful expansion for the effective DBI action given in (2.5). In (2.5) the Lagrangian density is proportional to pffiffiffiffiffiffiffiffiE. Using the-expansion in (2.28), we find that

ffiffiffiffiffiffiffiffi

E p

¼pffiffiffiffiffiffiffigQ_ðz; x_Þ; (2.33)

withQ_ðz; x_Þgiven by

Qðz; xÞ ¼

1þ

2

2 F mn_F

mnþ 4

4!F mnpq_F

mnpq

þ

6

6!F

mnpqrs_F

mnpqrsþ

₁₌₂

: (2.34)

As a consequence of (2.33) and (2.32), we find that

ffiffiffiffiffiffiffiffi

E

p _~

Eðm;nÞ_{¼ 2}

ffiffiffiffiffiffiffi_g

p

Qðz; xÞ

Fmn_þ2 2 FpqF

mnpq

þ

4

8 FpqFrsF

mnpqrs_þ

: (2.35)

This is a very useful expansion. In the general case, where the field strengths depend not only on zbut also on time and spatial coordinates, we can use (2.35) in (2.11) to rewrite the DBI equations as

@_m

2 ffiffiffiffiffiffiffi_g

p

Q_ðz; x_Þ

F‘m_þ2 2 FpqF

‘mpq

þ

4

8 FpqFrsF

‘mpqrs_{þ ¼0: (2.36)}

The DBI equations written in the form (2.36) show explicitly the expansion in , which is equivalent to an expansion in the string length squared 0_{. This is useful} if we want to truncate the DBI equations in the limit of small . In many brane models the product2 _is

inde-pendent of z, so that in the limit of small the DBI equations reduce to Maxwell equations in a curved space-time with metricgmnðzÞ.

The traditional way of solving DBI equations is to choose first a gauge field ansatz and then work with an effective determinant. The advantage of Eq. (2.36) is that they are valid for a general class of gauge field configura-tions, including time- and space-dependent field strengths. As a simple application of the method described above, we will consider the case the field strengths F_mn depend only on the radial coordinate z. This case is relevant to describe the physics of constant electric and magnetic fields in the boundary. In this case we can integrate out (2.36) and obtain

j_¼ 2 ffiffiffiffiffiffiffi_g

p

Q_ðz_Þ

Fz_þ

2

2 FF z

þ

4

8 FFF

z_{þ : (2.37)}

This equation is equivalent to (2.18) and will be used in the next section to calculate the conductivity in the presence of constant electric and magnetic fields.

III. DC CONDUCTIVITY IN3_þ1_DIMENSIONS

Now we consider the problem of charge transport in a 3_þ1 dimensional strongly coupled field theory in the presence of constant electric and magnetic fields. The way of describing this system in holography is by consid-ering a five-dimensional effective DBI action of the form (2.3) with a gauge field of the form

A_0ðz_{Þ ¼}f_0ðz_Þ; A~ _ðz;x_{Þ ¼}f~_ðz_Þ Et~ _þ1

2B~~x: (3.1) For simplicity, we also assume that the metric is homoge-neous and isotropy in the spatial directions ~xin the sense thatg11¼g22 ¼g33:¼gxxðzÞ.

QðzÞ 2j0 ¼

ffiffiffiffiffiffiffi_g

p _gzz_g00_f½1þ2_ðgxx_Þ2_jB~j2_Š@

zf0

þ2_ðgxx_Þ2_{ðE~B~Þ ð@}

zf~Þg; (3.2)

2

Q_ðz_Þ~j¼

ffiffiffiffiffiffiffi_g

p _gzz_gxx_f½1þ2_g00_gxx_jE~_j2_Š@

zf~

2_g00_gxx_½ðE~@

zf~ÞE~ ð@zf0ÞE~B~Š

þ2_ðgxx_Þ2_ðB~_@zf~_ÞB~_{g; (3.3)}

where

Q_ðz_{Þ f}Q_0ðz_{Þ þ}2_gzz_g00_½1þ2_ðgxx_Þ2_jB~j2_Šð@

zf0Þ2

þ2_gzz_gxx_½1þ2_g00_gxx_jE~_j2_Šj@

zf~j2

4_gzz_g00_ðgxx_Þ2_{½ 2ð@}

zf0ÞðE~B~Þ ð@zf~Þ

þ ðE~@_zf~_Þ2_{Š þ}4_gzz_ðgxx_Þ3_ðB~@

zf~Þ2g1=2; (3.4)

and

Q_0ðz_{Þ ¼}1_þ2_½g00_gxx_jE~j2_{þ ðg}xx_Þ2_jB~j2_Š

þ4_g00_ðgxx_Þ3_ðE~B~Þ2_{: (3.5)}

Note thatQ0ðzÞdoes not depend on the functionsf0ðzÞand

~ f_ðz_Þ.

Taking the scalar product of (3.3) withE~,B~ andE~B~, respectively, we get the following equations:

QðzÞ 2E~ ~j¼

ffiffiffiffiffiffiffi_g

p _gzz_gxx_fðE~@ zf~Þ

þ2ðgxx_Þ2_{ðE~B~ÞðB~@}

zf~Þg; (3.6)

Q_ðz_Þ 2B~ ~j¼

ffiffiffiffiffiffiffi_g

p _gzz_gxx_f½1þ2_g00_gxx_jE~_j2

þ2_ðgxx_Þ2_jB~j2_ŠðB~@

zf~Þ

2_g00_gxx_{ðE~B~ÞðE~@}

zf~Þg; (3.7)

Q_ðz_Þ

2ðE~B~Þ ~j¼ ffiffiffiffiffiffiffi_g

p _gzz_gxx_{f ½1þ}2_g00_gxx_jE~_j2_Š

ðE~B~_{Þ ð}@zf~Þ 2_g00_gxx_jE~_B~_j2_@

zf0g: (3.8)

A. Solutions of the DBI equations

We can solve the system (3.6) and (3.7) as a function of Q_ðz_Þ. The result is

ffiffiffiffiffiffiffi_g

p _gzz_gxx_B~@ zf~

¼ QðzÞ 2_Q

0ðzÞf

~

Bþ2_g00_gxx_ðE~_B~_ÞE~_{g ~j}

¼: QðzÞ 4_Q

0ðzÞ

j_B; (3.9)

ffiffiffiffiffiffiffi_g

p _gzz_gxx_E~_@zf~

¼ Q2_QðzÞ 0ðzÞ½

Q_0ðz_Þ~jE~ _ðgxx_Þ2_ðE~B~ÞjBŠ

¼: QðzÞ 4_Q

0ðzÞ

j_E: (3.10)

Similarly, we can solve (3.2) and (3.8) as a function of Q_ðz_Þ. The solutions are given by

ffiffiffiffiffiffiffi_g

p _gzz_gxx_{ðE~B~Þ @} zf~

¼ Q2_QðzÞ 0ðzÞf

2_ðgxx_Þ2_jE~_B~_j2_j0

þ ½1_þ2_ðgxx_Þ2_jB~_j2_ŠðE~_B~_{Þ ~jg}

¼: QðzÞ 5_Q

0ðzÞ

j_EB; (3.11)

ffiffiffiffiffiffiffi_g

p _gzz_g00_@

zf0

¼Q2_QðzÞ 0ðzÞf ½

1_þ2_g00_gxx_jE~j2_Šj0

þ2_g00_gxx_{ðE~B~Þ ~jg}

¼: QðzÞ 3_Q

0ðzÞ

~

j0: (3.12)

Using these results in (3.3), we get

ffiffiffiffiffiffiffi_g

p _gzz_gxx_½1þ2_g00_gxx_jE~j2_Š@

zf~

¼Q2_QðzÞ 0ðzÞf

Q0ðzÞ~jþg00gxxjEE~

ðgxx_Þ2~_j0E~_B~ _ðgxx_Þ2_j

BB~g: (3.13) Finally, using the solutions in the expansion for QðzÞ [Eq. (3.4)], we find a solution forQ_ðz_Þ,

Q_ðz_{Þ ¼}

2_Q 0ðzÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Q_0ðz_Þðz_{Þ þ} ðgxxÞ3

½1þ2_g00_gxx_jE~j2_Šj~20 g

00_ðgxx_Þ4 ½1þ2_g00_gxx_jE~j2_Šj2B

r ;

(3.14)

where

_ðz_Þ:_¼24_þ 2g00ðgxxÞ2

½1_þ2_g00_gxx_jE~j2_Š

½j~jj2_þ2_g00_gxx_ð~jE~_Þ2_{Š: (3.15)}

density j0_{, the electromagnetic current ~j and the electric}

and magnetic fieldsE~ andB~.

B. The Karch-O’Bannon procedure and the conductivity tensor in a general frame

The on-shell DBI action can be written as

So:s DBI¼

Z

d4_xdzðzÞpffiffiffiffiffiffiffi_gQ

ðz_Þ; (3.16)

withQ_ðz_Þgiven by (3.14).

The Karch-O’Bannon procedure proposed in [3] and further developed in [4,5] consists of imposing the reality constraint on the on-shell DBI action. In a general class of backgrounds, the metric blows up at the boundary, while the component g_tt vanishes at the horizon. Under these circumstances, it is not difficult to see from (3.5) and (3.15) that the functionsQ_0ðz_Þand_ðz_Þshould have a zero at a pointz_¼z andz_¼z, respectively. Using the identity

½1_þ2_g00_gxx_jE~_j2_Š

1_þ2_ðgxx_Þ2ðE~B~Þ2

~ E2

¼Q0ðzÞ 2ðgxxÞ2j

~ EB~_j2

jE~_j2 ; (3.17)

we conclude that 1_þ2_g00_gxx_jE~j2_{<0 at z¼z}

. Therefore, in order to avoid an imaginary action, we find from (3.14) thatj~₀andj_Bmust vanish atz_¼z. Since1_þ 2_g00_gxx_E~2

is also positive at the boundary, it should have a zero atz_¼z_E2, which is closer to the boundary thanz. Analyzing all the possible localizations ofz with respect tozandz_E2leads to the conclusion thatz¼z(for more details see [5]).

In summary, we find from the reality constraint on the action that at some pointz_¼z, the following equations must be satisfied:

Q0 ¼ 1þ 2g00gxxjE~j2 þ2ðgxxÞ2jB~j2

þ 4_g00_ðgxx_Þ3_ðE~ _B~_Þ2

¼ 0; (3.18)

j_B¼ 2_½~jB~_þ2_g00_gxx_ðE~_B~_Þ~jE~_{Š ¼0; (3.19)}

~

j_{0¼f ½}1_þ2_g00_gxx_jE~_j2_Šj0_þ2_g00_gxx_{~j ðE}~_B~_Þg

¼0; (3.20)

_¼24_þ 2g00ðgxxÞ2

½1_þ2_g00_gxx_jE~j2_Š½j~jj

2_þ2_g00_gxx_ð~jE~Þ2_Š

¼0: (3.21)

From Eqs. (3.18), (3.19), and (3.20) we get

1_þ2_ðgxx_Þ2_jB~_j2 _¼ 2_g00_gxx_½jE~_j2_þ2_ðgxx_Þ2_ðE~_B~_Þ2_Š;

(3.22)

~j ðE~B~_{Þ ¼}

2_ðgxx_Þ2_jE~_B~_j2_j0

½1_þ2_ðgxx_Þ2_jB~j2_Š; (3.23)

~j_B~ _¼ 2_g00_gxx_ðE~_B~_Þ~jE:~ _(3.24)

The spatial current ~jcan be decomposed into projections onE~,B~ andE~B~ as

~j¼ 1

jE~B~j2f½ð~jE~ÞjB~j

2 _{ð~jB~ÞðE~B~ފE~}

þ ½ð~jB~_ÞjE~_j2 _{ð~jE~ÞðE~B~ފB~þ~j ðE~B~ÞE~B~g:}

(3.25)

Using (3.23), (3.24), and (3.25) we get ~j¼

2_gxx

½1_þ2_ðgxx_Þ2_jB~j2_Šfg

00_{ð~jE~Þ½E~þ}2_ðgxx_Þ2_ðE~B~ÞB~Š

gxx_j0_E~_B~_{g: (3.26)}

Using (3.21) and (3.26) we can solve ~jE~. Assuming ~j ~

E0, we get

~jE~ _¼ g00 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðg_xxÞ3_½1þ2_ðgxx_Þ2_jB~j2_Š24_þ2_ðj0_Þ2 q

:

(3.27)

Writing the current asj_i¼ijE_j, we get the conductivity tensor

_ij¼ g xx

½1_þ2_ðgxx_Þ2_jB~j2_Š

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðg_xxÞ3_½1þ2_ðgxx_Þ2_jB~j2_Š24_þ2_ðj0_Þ2

q

½ijþ2ðgxxÞ2BiBjŠþ2gxxj0 ijkBk

: (3.28)

This result is consistent with the one obtained in [3,5] for the D3-D7 brane model. In that case the induced metric can be written as

ds2

D7

R2 ¼

1 z2

ð1 z4 z4

HÞ 2

ð1_þz4 z4

HÞ

dt2_þ 1

z2

1_þz

4

z4

H

d ~x2

þ

₁

z2þ ð@zÞ2

dz2_þcos2_ðzÞd2

3: (3.29)

The black brane horizon is related to the field theory temperature byz_{H ¼}pffiffiffi2=_ðT_Þ. The dilaton is juste _¼ g 1

s , and the AdS radius is given byR¼ ð4Þ1=4

ffiffiffiffiffi

0 p

_ðz_{Þ ¼}20 R2 ¼

ffiffiffiffi

p

1=2_;

_ðz_{Þ ¼}7 g_sVS3R

8_cos3_{ðzÞ ¼}Nc

43cos3ðzÞ;

g_xxðz_{Þ ¼} 1 z2

1_þz

4

z4

H

:

(3.30)

Our result (3.28) is also consistent with that obtained recently in [12,13], using the method of open string metric.

C. The Drude limit and holographic metals

As observed in [3,5], the Drude limit is obtained by neglecting the effects of pair production (large mass limit) and higher powers in the electric field. In our framework, this corresponds to the limit _!0 and g_00!0. Then (3.28) reduces to

D ij ¼

j0

M_½1_þjB2~_Mj22Š

_ijþ BiBj 2_M2þ ijk

B_k M

; (3.31)

where

M_¼ 1

gxxðzÞ: (3.32)

As shown in Appendix A, this is exactly the Drude conductivity that describes a metallic behavior for particles with densityj0_{and massM. Note that in the case of a black}

brane the induced metric contains a horizon defined by g00¼0. Then in the Drude limit the pointz¼zcoincides with the horizon z_H. In particular, for the D3-D7 brane model we get from (3.30),

M_¼2

ffiffiffiffi

p

ffiffiffiffi

p _z2

H ¼ 2

ffiffiffiffi

p

T2_{; (3.33)}

where_¼4. As discussed in [3,5], this result can also be obtained by computing the drag force on the charge carriers in the holographic metals.

The temperature dependence in (3.33) is typical of metals. In the next section we will consider backgrounds with Lifshitz symmetry, where the conductivity has a different dependence on temperature.

IV. DC CONDUCTIVITY IN2_þ1DIMENSIONS

In2_þ1dimensions the magnetic field is a scalar, so the gauge field ansatz takes the form

A_0ðzÞ ¼f0ðzÞ; A~ ðz;xÞ ¼f~ðzÞ Et~ þ

1

2B ijxix^j; (4.1)

where x^_i is the unit vector in the spatial directions i_{¼ ð}1;2_Þ, and _ijis the Levi-Civita symbol. Since the field strengths depend only on the z coordinate, we can use (2.37) to get the DBI equations

j0 _¼2 ffiffiffiffiffiffiffi_g

p

QðzÞ g

zz_g00_f½1þ2_ðgxx_Þ2_B2_Š@

zf0

þ2_ðgxx_Þ2_ðE~_~@

zf~ÞBg (4.2)

~j¼

2pffiffiffiffiffiffiffi_g

Q_ðz_Þ g

zz_gxx_f½1þ2_g00_gxx_jE~j2_Š@

zf~

2_g00_gxx_½ðE~@

zf~ÞE~ Bð@zf0ÞE~~Šg; (4.3)

whereE~~ _¼"_ijE_jx^_iandQ_ðz_Þis given by

Q_ðz_{Þ ¼ f}Q_0ðz_{Þ þ}2_gzz_g00_½1þ2_ðgxx_Þ2_B2_Šð@

zf0Þ2

þ2gzz_gxx_½1þ2_g00_gxx_jE~_j2_Šj@

zf~j2

4_gzz_g00_ðgxx_Þ2_{½ 2@}

zf0ðE~~@zf~ÞB

þ ðE~@_zf~_Þ2_Šg1=2_{; (4.4)}

with

Q_0ðz_{Þ ¼}1_þ2_g00_gxx_jE~j2_þ2_ðgxx_Þ2_B2_{: (4.5)}

Following a procedure similar to that described in the previous section, we solve (4.2) and (4.3) in terms ofQðzÞ. We find in this case

ffiffiffiffiffiffiffi_g

p _gzz_g00_@

zf0 ¼

QðzÞ 2_Q

0ðzÞf ½

1_þ2_g00_gxx_jE~j2_Šj0

þ2_g00_gxx_{B ~jE}~_~g

¼: QðzÞ 3_Q

0ðzÞ

~

j0; (4.6)

ffiffiffiffiffiffiffi_g

p _gzz_gxx_E~@ zf~¼

QðzÞ

2 ~jE;~ (4.7)

ffiffiffiffiffiffiffi_g

p _gzz_gxx_E~_~@ zf~¼

Q_ðz_Þ 2_Q

0ðzÞf½

1_þ2_gzz_gxx_B2_Š~jE~_~

2_ðgxx_Þ2_BjE~j2_j0_{g; (4.8)}

ffiffiffiffiffiffiffi_g

p _gzz_gxx_½1þ2_g00_gxx_jE~_j2_Š@

zf~

¼ Q2_QðzÞ 0ðzÞ½

Q_0ðz_Þ~j_þ2_g00_gxx_Q

0ðzÞð~jE~ÞE~

þ_ðgxx_Þ2_j~

0B ~E~Š: (4.9)

Finally, using the results (4.6), (4.7), (4.8), (4.9), and (4.4) we find a solution for Q_ðz_Þ in terms of the current and electromagnetic fields,

QðzÞ ¼

2_Q 0ðzÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Q0ðzÞðzÞ þ ðg xx_Þ2 ½1þ2_g00_gxx_jE~_j2_Š~j

2 0

r ; (4.10)

_ðz_Þ:_¼24_þ 2g00gxx

½1_þ2_g00_gxx_jE~j2_Š

½j~jj2_þ2_g00_gxx_ð~jE~Þ2_{Š: (4.11)}

A. The conductivity tensor

Following the Karch-O’Bannon procedure, described in Sec.III, we find that the reality constraint of the on-shell DBI action leads to the following conditions atz_¼z:

Q_{0 ¼}1_þ2_g00_gxx_jE~j2_þ2_ðgxx_Þ2_B2 _{¼0; (4.12)}

~

j0¼f ½1þ2g00gxxjE~j2Šj0þ2g00gxxB ~jE~~g ¼0;

(4.13)

24_þ 2g00gxx

½1_þ2_g00_gxx_jE~_j2_Š½j~jj

2_þ2_g00_gxx_ð~jE~_Þ2_{Š ¼0:}

(4.14)

From Eqs. (4.12) and (4.13), we get

1_þ2_ðgxx_Þ2_B2 _¼ 2_g00_gxx_jE~j2_{; (4.15)}

~j_E~~_¼ 2ðgxxÞ2jE~j2Bj0

½1_þ2_ðgxx_Þ2_B2_Š: (4.16)

The current can be decomposed as

~j¼½ð~j ~ ~

E_ÞE~~_{þ ð}~jE~_ÞE~_Š

jE~_j2 : (4.17)

Using (4.15), (4.16), and (4.17), we obtain ~j¼

2_gxx

½1_þ2_ðgxx_Þ2_B2_Šfg00ð~jE~ÞE~ gxxBj0

~ ~

Eg: (4.18)

Using (4.14) and (4.18) we solve ~jE~. Assuming ~jE~0, we get

~jE~ ¼ g00 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðgxxÞ2_½1þ2_ðgxxÞ2_B2_Š24_þ2_ðj0_Þ2 q

:

(4.19)

Writing the current as j_i¼ijE_j, we get the conduc-tivity tensor

_{ij ¼} ðg xx_Þ

½1_þ2_ðgxx_Þ2_B2_Š

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqðg_xxÞ2_½1þ2_ðgxx_Þ2_B2_Š24_þ2_ðj0_Þ2

ij

þ2_gxx_Bj0

ij

: (4.20)

In this case the conductivity has two identical diagonal components _11¼22 and two opposite nondiagonal components _{12¼ 21}. Remember that in (4.20) the parametersandas well as the metricg_xxare evaluated at the special pointz satisfyingQ_0ðz_{Þ ¼}0.

Our result (4.20) is also consistent with that obtained recently in [12,13], using the method of open string metric.

B. Lifshitz symmetry and strange metals

Here we consider the finite temperature background considered in [11] in a holographic approach to strange metals. The induced metric of a Dp-brane in the massless case can be written as

ds2 _¼R2 fðvÞdt2

v2z þ dv2

fðvÞv2þ

dx2_þdy2

v2

þd2_;

(4.21)

where is a compact submanifold and v is the radial direction that extends from the boundary v_¼0 to the horizonv_þ.

The function f_ðv_Þ is model dependent but satisfies the universal condition f_ðv_{þÞ ¼}0. As a consequence of this condition, we get

T_¼jf0ðvþÞj 4vz_þ 1

1

vz_þ : (4.22)

The metric (4.21) realizes Lifshitz symmetry through the following scaling transformations:

t_!z_{t; ~x! ~x; v!v: (4.23)}

The parameter z is then interpreted as the dynamical critical exponent of the quantum critical sector in the dual field theory.

The Dp-brane action corresponding to (4.21) can be written as in (2.5) with

_ðv_{Þ ¼}20_{; ðvÞ ¼}

pVe ¼:eff;

g_xxðv_{Þ ¼}R

2

v2: (4.24)

Therefore, from (4.20) we get for the Lifshitz background the conductivity tensor

ij¼

1

h

1_þ 20

R2

₂

v4

B2

i 8 <

:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1_þ

₂0

R2 ₂

v4

B2

2

effð20Þ4þ ₂0

R2 ₂

v4

ðj0Þ2

s

ijþ

₂0

R2 ₂

v4

Bj0 ij

9 =

This result is consistent with the one obtained in [11]. In the limit of large densities, the conductivity tensor reduces to

_{ij ¼}

20

R2

v2

j0

h

1_þ 20

R2

₂

v4

B2

i

_ijþ

₂0

R2

v2

B ij

¼:_h 1 1þ B2

2_M2

i

_ijþ B M ij

: (4.26)

The parameteris interpreted as the dc resistivity, which has the following temperature dependence:

_¼ R

2

20_v2

j0 T2=z

j0 : (4.27)

For the casez¼2, the dc resistivity is linear in the tem-perature, as expected for strange metals. Note that the second line in (4.26) is the same obtained in a 2_þ1-dimensional Drude model withMgiven by

M_¼ R

2

20_v2

T2=z_{: (4.28)}

V. CONCLUSIONS

We have proposed a simple expansion method to inves-tigate DBI equations and define holographic currents in a general class of backgrounds (whose metric depends on the radial coordinate only) and Dp-brane configurations. Our method is alternative to the traditional approach where an effective determinant is calculated for each field configu-ration. In our method, we write the DBI equations for arbitrary gauge field configurations before choosing any ansatz. We expect that our results would be useful in several holographic models that are derived from string theory. A future direction in our approach would be to extend the expansion method for the case of backgrounds with conical deficits (see for instance [15]).

As an application of our method we have discussed thoroughly the situation where constant electric and mag-netic fields are turned on. Following the Karch-O’Bannon method, we found for a general background, satisfying homogeneity and isotropy, a frame-independent expression for the conductivity tensor in3_þ1and2_þ1dimensions. For particular backgrounds we have recovered previous results on holographic metals and strange metals. An in-teresting point in that analysis was that the presence of a horizon guarantees the existence of a solution that satisfies the real constraint.

It would be interesting also to investigate physical systems, where the electric or magnetic fields have a time or spatial dependence. We leave this problem for future work.

ACKNOWLEDGMENTS

The authors would like to thank the ICTP in Trieste (Italy) and the ICTP-SAIFR in Sao Paulo (Brazil) for their

hospitality. The work of A. B. B. is supported by a Capes fellowship. The work of C. N. F is partially supported by a CNPq fellowship.

APPENDIX A: THE DRUDE MODEL

The Drude equation for an electromagnetic current can be written as

j0_{E~ M ~jþ~jB~ ¼0; (A1)}

where is the drag coefficient andM is the mass of the charge carrier. We can write this equation in a matrix form,

M_ijj_j¼ j0_E

i; (A2)

where

M_ij¼ M_{ijþ ijk}B_k: (A3)

The inverse of this object is

M 1

ij ¼

1

Mh1_þjB2~_Mj22

i

_ijþ BiBj 2_M2þ ijk

Bk M

:

(A4) Using this result we get the Drude conductivity

D ij¼

j0

Mh1_þjB2~_Mj22

i

_ijþ BiBj 2_M2þ ijk

B_k M

: (A5)

APPENDIX B: THE EFFECT OF A FIVE-DIMENSIONAL CHERN-SIMONS TERM

In some brane models that reduce to five dimensions, we obtain an effective Chern-Simons (CS) action,

SCS ¼

4

‘mnpqZ _d4_xdzA

‘FmnFpq: (B1)

In the special case of (3.1), where we have constant electric and magnetic fields, the DBI-CS equations reduce to

0_¼ ~EB;~ (B2)

j0 _¼2

Q_ðz_Þ

ffiffiffiffiffiffiffi_g

p _gzz_g00_f½1þ2_ðgxx_Þ2_B~2

Š@_zf₀

þ2_ðgxx_Þ2_ðE~_B~_{Þ ð@zf}~_{Þg 6 ~Bf;}~ _(B3)

~j¼

2

Q_ðz_Þ

ffiffiffiffiffiffiffi_g

p _gzz_gxx_f½1þ2_g00_gxx_jE~_j2_Š@

zf~

2_g00_gxx_½ðE~@

zf~ÞE~þ ð@zf0ÞE~B~Š

þ2_ðgxx_Þ2_ðB~@

zf~ÞB~g þ6½f0B~þf~E~Š: (B4)

Here the quantities j0 _{and ~j are integration constants, not}

necessarily the electromagnetic current.3

3_{The definition of holographic currents in the presence of a}

According to Eq. (B2) the electric and magnetic field must be orthogonal.4Note that we can rewrite the last term in (B4) as

6_½f0B~þf~E~Š

¼6

½f_0jB~_j2_{þ ðE~B~Þ f}~_Š B~ jB~_j2

ðB~f~_Þ jB~_j2 E~B~

:

(B5)

Here we used the vectorial decomposition

~

f_¼ 1

jE~B~_j2f½ðf~E~ÞjB~j 2

ðf~B~_ÞðE~B~_ފE~_{þ ½ð}f~B~_ÞjE~_j2

ðf~E~_ÞðE~B~_ފB~_þf~ _ðE~B~_ÞE~B~_g (B6)

and the constraintE~B~ _¼0.

Taking the scalar product of (B4) withE~,B~ andE~B~, respectively, we get the equations:

ffiffiffiffiffiffiffi_g

p _gzz_gxx_E~_@zf~_¼ QðzÞ

2 ~jE;~ (B7)

ffiffiffiffiffiffiffi_g

p _gzz_gxx_B~_@zf~

¼ Q2_QðzÞ 0ðzÞf

~jB~_þ6_½f_0jB~_j2_{þ ðE~B~Þ f}~_Šg;

(B8)

ffiffiffiffiffiffiffi_g

p _gzz_gxx_f½1þ2_g00_gxx_jE~j2_{ŠðE~B~Þ @}

zf~

2_g00_gxx_jE~B~j2_@

zf0g

¼QðzÞ

2½~j ðE~B~Þ 6ðB~f~ÞjE~j2Š: (B9)

Here we usedE~B~¼0. From (B3) and (B9) we get

ffiffiffiffiffiffiffi_g

p _gzz_gxx_{ðE~B~Þ @} zf~

¼ Q2_QðzÞ 0ðzÞf

2_ðgxx_Þ2_jE~_B~_j2_j0

þ ½1_þ2_ðgxx_Þ2_jB~j2_{Š~j ðE~B~Þ 6ðf}~_B~_ÞjE~_j2_g

¼: QðzÞ 5_Q

0ðzÞf

^

jEB 63ðf~B~ÞjE~j2g and (B10)

ffiffiffiffiffiffiffi_g

p _gzz_g00_@

zf0 ¼

QðzÞ 2_Q

0ðzÞf ½

1_þ2_g00_gxx_jE~j2_Šj0

þ2_g00_gxx_{~j ðE}~_B~_{Þ þ6 ~fB}~_g

¼: QðzÞ 3_Q

0ðzÞf

^

j_0þ6 ~fB~_g: (B11)

Using (B5), (B7), (B8), and (B11) in (B4), we get

ffiffiffiffiffiffiffi_g

p _gzz_gxx_½1þ2_g00_gxx_jE~_j2_Š@

zf~ ðB~@zf~Þ ~ B jB~2_j

¼Q4_QðzÞ 0ðzÞ

Q_0ðz_Þ

2_~jþ2_ð~jB~Þ B~

jB~_j2

4_Q

0ðzÞg00gxxð~jE~ÞE~

2_ðgxx_Þ2_j^_{0 6}

jB~_j2½1þ

2_g00_gxx_jE~_j2_ŠðB~_f~_ÞE~_B~_{: (B12)}

We can take the derivative of (B8) and use (B10) and (B11) to get a second-order differential equation forB~f~,

ffiffiffiffiffiffiffi_g

p _gzz_gxx5Q0ðzÞ

QðzÞ

@z

2_Q 0ðzÞ

QðzÞ

ffiffiffiffiffiffiffi_g

p _gzz_gxx_@zðB~_f~_Þ

¼ 6_fg₀₀gxx2_jB~j2_j^

0 j^EBþ6½g00gxx2jB~j2þ2jE~j2Š ~Bf~g: (B13)

Finally, using (B7) and (B10)–(B12) in the definition ofQ_ðz_Þ, we get

4_{If we want to consider parallel electric and magnetic fields, we would need to introduce a time dependence on theA}

0component

Q_ðz_Þ 2_Q

0ðzÞ¼

ffiffiffiffiffiffiffiffiffiffiffiffi

Q0ðzÞ

q

1_þ2_gzz_gxxðB~@zf~Þ2 jB~_j2

₁₌₂

f24_Q2

0ðzÞ þ ðgxxÞ3½1þ2ðgxxÞ2jB~j2Š½j^0þ6 ~Bf~Š2

2g00_ðgxx_Þ4_½j^

0þ6 ~Bf~Š½j^EB 63ðB~f~ÞjE~j2Š ðg00Þ2ðgxxÞ34Q20ðzÞð~jE~Þ2

þ g

00_ðgxx_Þ2

2_½1þ2_gzz_g00_jE~_j2_Š Q0ðzÞ

2_~jþ2_ð~jB~_Þ B~ jB~j2

4_Q

0ðzÞg00gxxð~jE~ÞE~

2_½ðgxx_Þ2_j^_{0 6}

jB~_j2½1þ

2_g00_gxx_jE~_j2_ŠðB~_f~_ފE~_B~_j2_g 1=2_{: (B14)}

[1] J. M. Maldacena, Adv. Theor. Math. Phys.2, 231 (1998). [2] A. Karch and E. Katz,J. High Energy Phys. 06 (2002) 043. [3] A. Karch and A. O’Bannon, J. High Energy Phys. 09

(2007) 024.

[4] A. O’Bannon,Phys. Rev. D76, 086007 (2007).

[5] M. Ammon, T. H. Ngo, and A. O’Bannon,J. High Energy Phys. 10 (2009) 027.

[6] S. Kachru, X. Liu, and M. Mulligan, Phys. Rev. D 78, 106005 (2008).

[7] U. H. Danielsson and L. Thorlacius,J. High Energy Phys. 03 (2009) 070.

[8] R. B. Mann,J. High Energy Phys. 06 (2009) 075. [9] G. Bertoldi, B. A. Burrington, and A. Peet,Phys. Rev. D

80, 126003 (2009).

[10] K. Balasubramanian and J. McGreevy, Phys. Rev. D80, 104039 (2009).

[11] S. A. Hartnoll, J. Polchinski, E. Silverstein, and D. Tong,

J. High Energy Phys. 04 (2010) 120.

[12] K.-Y. Kim, J. P. Shock, and J. Tarrio,J. High Energy Phys. 06 (2011) 017.

[13] K.-Y. Kim and D.-W. Pang, J. High Energy Phys. 09 (2011) 051.

[14] D. E. Kharzeev, Ann. Phys. (Amsterdam) 325, 205 (2010).

[15] C. A. B. Bayona, C. N. Ferreira, and V. J. V. Otoya,

Classical Quantum Gravity28, 015011 (2011).

[16] A. Rebhan, A. Schmitt, and S. A. Stricker,J. High Energy Phys. 01 (2010) 026.

[17] A. Gynther, K. Landsteiner, F. Pena-Benitez, and A. Rebhan,J. High Energy Phys. 02 (2011) 110.