Gauge invariance and holographic renormalization

Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Gauge invariance and holographic renormalization

Keun-Young Kim

^a

, Kyung Kiu Kim

^a

, Yunseok Seo

^b

, Sang-Jin Sin

^c,d,∗

aSchoolofPhysicsandChemistry,GwangjuInstituteofScienceandTechnology,Gwangju500-712,RepublicofKorea bResearchInstituteforNaturalScience,HanyangUniversity,Seoul133-791,RepublicofKorea

cDepartmentofPhysics,HanyangUniversity,Seoul133-791,RepublicofKorea dSchoolofPhysics,KoreaInstituteforAdvancedStudy,Seoul130-722,RepublicofKorea

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received5March2015

Receivedinrevisedform20July2015 Accepted22July2015

Availableonline28July2015 Editor:M.Cvetiˇc

Keywords:

Gauge/gravityduality Holographicrenormalization Gaugeinvariance

We study the gauge invariance of physical observables in holographic theories under the local diffeomorphism.Wefindthatgaugeinvarianceisintimatelyrelatedtotheholographicrenormalization:

thelocalcountertermsdefinedintheboundarycancelmostofgaugedependencesoftheon-shellaction aswellasthedivergences.Thereisamismatchinthedegreesoffreedombetweenthebulktheoryand theboundaryone.Weresolvethisproblembynoticingthatthereisaresidualgaugesymmetry(RGS).By extendingtheRGSsuchthatitsatisfiesinfallingboundaryconditionatthehorizon,wecanunderstand theprobleminthecontextofgeneralholographicembeddingofaglobalsymmetryattheboundaryinto thelocalgaugesymmetryinthebulk.

©^{2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

AccordingtoAdS/CFT correspondence, anyglobalsymmetry at theboundarytheoryisliftedtoalocalsymmetryinthebulk[1,2].

Thegaugesymmetry isessential toreducethedegreeoffreedom whichisenlargedbygoingintoonehigherdimension.Thephysical goalinholographyistheboundaryquantitieswhichdo notknow the presence of higher dimension or gauge degrees of freedom, while we use the tools in the bulk theory. Therefore the gauge invarianceofaphysicalquantityisacriticalissueforthevalidityof theAdS/CFT.Alsotracingthegaugeinvariancegivesmuchintuition onthewayhowholographyactuallyworks, especiallyhowglobal symmetryisencodedinthelocalgaugesymmetry.

One can find gauge invariant combinations of the fields, and expressthephysicalquantities intermsofsuch mastervariables, however,itisnotalwayseasytofindsuchgaugeinvariantcombi- nation.Eveninthecasetheyareavailable,itisnotveryconvenient to usesuch fields,especially if manyfields are coupled,because thephysical quantitiesare definedinterms ofthefield variables whichareformallygaugedependent.Forexample[2],energymo- mentum tensor and chemical potential are defined in terms of metric/gaugefieldwhichis notgaugeinvariant.Similarly,heatcur- rentscan be relatedtothe metricperturbationdefinedonly ina

*

Correspondingauthor.

E-mailaddresses:fortoe@gist.ac.kr(K.-Y. Kim),kimkyungkiu@gmail.com (K.K. Kim),yseo@hanyang.ac.kr(Y. Seo),sjsin@hanyang.ac.kr(S.-J. Sin).

specific gauge where time periodhas definite relationwithtem- perature.

In recentworks[3,4],based on[5,6], wedeveloped asystem- aticmethodtonumericallycalculatetheGreen’sfunctionsandall AC transportsquantities simultaneouslyforthe casewheremany fields are coupled and there are constraints due to gauge sym- metry. Althoughwe have testedthe validityof the procedureby showingtheagreementofzerofrequencylimitsofACconductivi- tieswiththeknownanalyticDCconductivities[7–9]westillthink thatwe needtoprovethegaugeinvarianceofourprocedureasa matterofprinciple.Wefoundthatthebulkgaugeinvarianceisin- timately relatedto theholographic renormalization. Althoughthe local counter termswere introduced to kill thedivergences, they alsokill mostofgaugedependence.

Furthermore,thereisaresidualgaugesymmetry(RGS)evenaf- terwefixtheaxialgauge grx=^{0.While equationsofmotioncan} be written in terms of the gauge invariant master fields Ph,Pχ (3.8),itturnsoutthatthequadraticon-shellaction,thegenerating functionfortwopointretardedGreen’sfunctions,cannot bewrit- tenassuch.However,weprovethattheGreen’sfunctionsarestill invariantundersuchasymmetry.

Thereisamismatchinthedegreesoffreedominthebulkand thoseattheboundary:thereareonlytwoindependentbulksolu- tions satisfyingthe in-fallingboundaryconditions whilewe need threesolutionsattheboundarysincetherearethreeindependent sourcefields.TheRGSistheonethatresolvestheproblem:sinceit cannotsatisfyaproperboundarycondition,itisnotapropergauge http://dx.doi.org/10.1016/j.physletb.2015.07.058

0370-2693/©^{2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

symmetrybuta‘solutiongeneratingsymmetry’.Itgeneratethede- siredsolutionattheboundaryandthereforeweshouldaccept its bulkcounterpartasanewphysicaldegreeoffreedomaswellal- thoughit cannot satisfytheinfalling boundarycondition (BC).By extendingtheRGSsuch that itsatisfies infallingboundary condi- tionatthehorizon,wecanmakethebulksolutionmorenaturalin thesense thatit satisfiestheinfalling BC. Withsuchsolution we canalsounderstand theprobleminthe contextof generalstruc- tureofholography, namely thecorrespondence betweena global symmetry atthe boundaryandthe local gauge symmetryin the bulk.

2. Actionandbackgroundsolution

Letusfirstbrieflyreviewthesystemwewilldiscuss,whichhas beenanalysedin detailin[3,7,10].The holographicallyrenormal- ized action(Sren)isgivenby

Sren

=

SEM

+

Sψ

+

Sc

,

(2.1)

where S_EM

=

M

d⁴x

√

−

^g

R

−

²

−

¹ 4F²

−

²

∂M

d³x

− γ

K

,

(2.2)

is the usual action for charged black hole in AdS space (<0) withtheGibbons–Hawkingtermand

S_ψ

=

M

d⁴x

√

−

^g

−

¹ 2

2 I=¹

(∂ψ

I

)

²

,

(2.3)

istheactionfortwofreemasslessscalarsaddedforamomentum relaxationeffect.S_c isthecounterterm

S_c

= η

_c

∂M

dx³

− γ −

4

−

^R

[ γ ] +

¹

2

2 I=¹

γ

^μν

∂

_μ

ψ

_I

∂

_ν

ψ

_I

,

(2.4)

whichisincludedtocancelthedivergencein SEM+^Sψ.Here we introduced

η

c to keeptrackof theeffectof thecounter term. At theendofthecomputationwewillset

η

c=^1.

Theaction(2.1)yieldsgeneralequationsofmotion¹ RMN

=

¹

2gMN R

−

²

−

¹ 4F²

−

¹

2

2 I=1

(∂ψ

I

)

²

+

¹ 2

I

∂

M

ψ

I

∂

N

ψ

I

+

¹

2F_M^PF_NP

,

(2.5)

∇

MF^MN

=

⁰

, ∇

²

ψ

I

=

⁰

,

(2.6)

whichadmitthefollowingsolutions ds²

=

^GMNdx^Mdx^N

= −

^f

(

r

)

dt²

+

^dr2

f

(

r

) +

^r2

δ

i jdxⁱdx^j

,

(2.7) f

(

r

) =

^r2

− β

²

2

−

^m0 r

+ μ

²

4 r²₀ r²

,

m₀

=

^r3₀ ¹

+ μ

²

4r²₀

− β

² 2r₀²

,

(2.8)

A

= μ

1

−

^r0 r

dt

,

(2.9)

ψ

I

= β

Iixⁱ

= βδ

Iixⁱ

.

(2.10)

1 Indexconvention:M,N,· · · =⁰,1,2,r,andμ,ν,· · · =⁰,1,2,andi,j,· · · =¹,2.

These are reduced to AdS–Reissner–Nordstrom (AdS–RN) black branesolutionswhenβ=^{0.Herewehavetakenspecial}βIi,which satisfies ¹₂2

I=1βI· βI=β² forgeneralcases.

The solutions (2.7)–(2.10) are characterized by three parame- ters:r0,

μ

,andβ.r0istheblackbranehorizonposition(f(r0)=⁰⁾ andcanbereplacedbytemperatureT forthedualfieldtheory:

T

=

^f

(

r₀

)

4

π =

1 4

π

3r0

− μ

²

+

2

β

² 4r0

.

(2.11)

Non-vanishing components of energy–momentum tensor and chargedensityread

^Ttt

=

^2m0

,

^Txx

=

^{Ty y}

=

^m0

,

^Jt

= μ

r₀

.

(2.12) ^Ttt=^{2Txx implies that charge carriers are still of massless} character.Fromherewesetr0=^{1 nottoclutter.}

3. Gaugefixingandresidualgaugetransformation

Tostudyelectric,thermoelectric,andthermalconductivitieswe introducesmallfluctuationsaroundthebackground(2.7)–(2.10)

δ

A_x

(

t

,

r

) =

∞

−∞

d

ω

2

π

^e

−ⁱωta_x

( ω ,

r

) ,

(3.1)

δ

g_tx

(

t

,

r

) =

∞

−∞

d

ω

2

π

^e

−ⁱωtr²h_tx

( ω ,

r

) ,

(3.2)

δ

g_rx

(

t

,

r

) =

∞

−∞

d

ω

2

π

^e

−ⁱωtr²h_rx

( ω ,

r

) ,

(3.3)

δψ

1

(

t

,

r

) =

∞

−∞

d

ω

2

π

^e

−ⁱωt

χ ( ω ,

r

) .

(3.4)

The fluctuationsare chosen to be independent of x and y. This isallowed since all thebackground fieldsappearingin theequa- tionsofmotionturnouttobeindependentofxand y.Thegauge fieldfluctuation(δA_x(t,r))sourcesmetric(δg_tx(t,r),δg_rx(t,r))and scalarfield(δψ1(t,r))fluctuationandviceversaandalltheother fluctuationsaredecoupled.Wewillworkinmomentumspaceand htx(

ω

,r) andhrx(

ω

,r)is definedso that it goesto constant asr goestoinfinity.

By linearizing the full equation of motion, we get four equa- tions. However oneof themcan be obtainedby theothers.Thus wemayconsiderfollowingthreeequations:

( χ

− β

hrx

) −

ⁱ

μω

ax

β

r²f

(

r

) −

^ir2

ω (

h_tx

+

ⁱ

ω

hrx

)

β

f

(

r

) =

0

,

(3.5) a_x

(

r

) +

^ax

(

r

)

f

(

r

)

f

(

r

) + ω

²ax

(

r

)

f

(

r

)

²

+ μ (

h_tx

+

ⁱ

ω

hrx

)

f

(

r

) =

⁰

,

(3.6) f

(

r

)

f

(

r

)( χ

(

r

) − β

hrx

) +

f

(

r

)

²

( χ

− β

hrx

)

+

^2f

(

r

)

²

( χ

− β

h_rx

)

r

+ ω

²

χ (

r

) −

ⁱ

β ω

htx

(

r

) =

⁰

.

(3.7) Ifwe differentiate thethird equation withrespect tor,all equa- tions can be written interms of three variables, Pχ,Ph, andax, where

P

χ

≡ χ

− β

h_rx

, P

h

≡

^h_tx

+

ⁱ

ω

h_rx

.

(3.8) Therefore, hrx is a non-dynamical degree of freedom. Indeed, Pχ,Ph,anda_x areinvariantunderadiffeomorphismgeneratedby

ξ^μ=(0,ζ (r)e⁻ⁱωt,0,0),underwhichthefieldsaretransformedas follows:

δ

h_rx

=

¹

r²

( ∇

r

ξ

_x

+ ∇

x

ξ

_r

) = ζ

(

r

)

e^−iωt

,

(3.9)

δ

h_tx

=

¹

r²

(∇

t

ξ

x

+ ∇

x

ξ

t

) = −

ⁱ

ω ζ (

r

)

e^−iωt

,

(3.10)

δ χ = βζ (

r

)

e^−iωt

,

(3.11)

δ

ax

=

0

.

(3.12)

Usingthisgaugedegreeoffreedom,onemaysethrx=^0,whichis so-called theaxialgauge.Thenumericalcalculationin[3]hasbeen performedin thisgauge. Aquestion arises whethertheresulting physicalquantitiesareindependentofsuchgaugefixingcondition.

Furthermore,evenafterwefixhrx=^{0,onecanstillfindaresid-} ualgauge transformation whichis givenby constant ζ [11]. This residualdiffeomorphismdoesn’tchangethegaugefixingcondition hrx=^{0 and generates constant shift on h}tx and

χ

, because the equationsofmotioncontainonlyderivativesofh_tx and

χ

andthe linear combinationof them,

ωχ

(r)−ⁱβhtx(r), which is invariant under

htx

→

htx

+

h0

,

and

χ → χ +

i

β

ω

^h0

,

^(3.13)

whereh0isaconstant.Thusthereisoneparameterconstantsolu- tiongivenby

ax

=

0

,

htx

=

h0

, χ =

i

β

ω

^h0

,

^(3.14)

whichdoesnotsatisfy in-fallingboundaryconditionsoitisnota physicaldegreeoffreedom.²Wecallittheresidualgaugesymme- try(RGS)becauseitisgeneratedbythezeromodeofadiffeomor- phismgenerator.Thiskindofsolutionwasfirstintroducedin[12].

Whyshouldtherebesucharesidualdegreeoffreedom?Itcan betracedtothedifferenceofthedifferentialequationnearhorizon andthosenearboundary.Neartheblackholehorizon(r→^1)the solutionsareexpandedas

htx

= (

r

−

1

)

^ν±+1

h_tx^(I)

+

h⁽_tx^{I I)}

(

r

−

1

) + · · · ,

ax

= (

r

−

¹

)

^ν±

a⁽_x^I)

+

^a(x^{I I)}

(

r

−

¹

) + · · · , χ = (

r

−

¹

)

^ν±

χ

^(I)

+ χ

^{(I I)}

(

r

−

¹

) + · · ·

,

(3.15)

where

ν

_±= ±ⁱ⁴

ω

/(−¹²+²β²+

μ

²)= ∓ⁱ

ω

/(4

π

T)andtheincom- ingboundarycondition correspondsto

ν

=

ν

₊.Byinsertingthese to theequationsof motion,one caneasily find a linearrelations betweenthezero-thmodes:

( ν +

1

)

h_tx^(I)

+ μ

a⁽_x^I)

+ β χ

^(I)

=

0

.

(3.16) Noticethat allother modesaregeneratedby these. Thusthereis a well defined constraintequation which reducesthe degrees of freedom.

Ontheotherhand,byinsertingtheexpansionnearthebound- ary(r→ ∞⁾

htx

=

h_tx⁽⁰⁾

+

¹

r²h⁽_tx²⁾

+

¹

r³h⁽_tx³⁾

+ · · · ,

a_x

=

^a(x⁰⁾

+

¹

ra⁽_x¹⁾

+ · · · , χ = χ

⁽⁰⁾

+

¹

r²

χ

⁽²⁾

+

¹

r³

χ

⁽³⁾

+ · · · ,

(3.17)

2 Itisaregularsolutionatfuturehorizon.

to the equations of motion,we cannot get any relationbetween the zero-thmodesa⁽_x⁰⁾,h⁽_tx⁰⁾,and

χ

⁽⁰⁾,allofwhichare relatedto thehighermodes.Moreexplicitly,

ω ( ωχ

⁽⁰⁾

−

i

β

h⁽_tx⁰⁾

) −

²

χ

⁽²⁾

=

0

,

i

β( ωχ

⁽⁰⁾

−

i

β

h⁽_tx⁰⁾

) −

2h_tx⁽²⁾

=

0

,

(3.18) which are evolution equations in r-direction. Therefore, there is no constraintequation. Then thereis a crisis ofmismatchofde- greesoffreedomandthiscrisisisresolvedbytheeffectiveresidual degree of freedom described above. However, thisresidualgauge degree of freedom raises another issue of invariance of physics underthissymmetry.WewilladdressthisissueattheendofSec- tion5.

4. Holographicrenormalizationandgaugeinvariance

Nowwecomebacktothequestionwhetherphysicalquantities are independent of the choice of the gauge condition h_rx(r)=^0.

Wewillshowthisbyprovingthatthegeneratingfunctionofphys- ical quantities, the on-shell action, is invariant even in the case withh_rx(r)=^0.

Theon-shellrenormalized actiontoquadraticorderinfluctua- tionfields,S⁽_ren²⁾,is

S_ren⁽²⁾

=

^lim

r→∞

d³x

δψ

1

1

2

β

f

δ

g_rx

−

¹ 2f r²

δψ

₁

+

²

r

δ

g_tx²

−

¹

2f

δ

A_x

δ

A_x

− δ

gtx

1

2

δ

g

˙

rx

−

¹ 2r²

( δ

gtx

r²

)

+ μ

2r²

δ

Ax

+ η

c

δψ

1

r²

δψ ¨

1

2

f

− β δ ˙

g_tx 2

f

+ β δψ ˙

1

δ

gtx

2

f

−

²

f

δ

g²_tx

,

(4.1)

where f(r)=^r2−^β₂² −^m_r⁰ +_4r^μ22.Wedroppedtheboundarycon- tributionfromthehorizonasaprescriptionfortheretardedGreen function[13].³Nearboundaryr→ ∞^{,thefluctuationfieldsinmo-} mentumspace,(3.1)–(3.4),maybeexpandedas

h_tx

( ω ,

r

) =

∞ n=⁰

h⁽_txⁿ⁾

( ω )

rⁿ

,

h_rx

( ω ,

r

) =

∞ n=⁰

h⁽_rxⁿ⁾

( ω )

rⁿ

,

a_x

( ω ,

r

) =

∞ n=⁰

a⁽_xⁿ⁾

( ω )

rⁿ

, χ ( ω ,

r

) =

∞ n=⁰

χ

⁽ⁿ⁾

( ω )

rⁿ

,

(4.2)

andusingtheequationsofmotion,wecan obtainaquadraticac- tionasfollows

3 Infact,thecontributionoftheincomingsolutionatthehorizoniszeroin(4.1), whichisreal.However,forageneratingfunctionofretardedGreen’sfunctions,we willtakeonlypartof(4.1)asexplainedbelow(4.3),whichiscomplex.Inthiscase, it turnsoutthatthecontributionfromthehorizonispureimaginary.Fromthis perspective,weshoulddropthecontributionfromthehorizon.

S⁽_ren²⁾

=

^V2 2

∞ 0

d

ω

2

π

− μ

a

¯

⁽_x⁰⁾h⁽_tx⁰⁾

− μ

h

¯

_tx⁽⁰⁾a⁽_x⁰⁾

−

2m0_h

¯

⁽_tx⁰⁾_h⁽_tx⁰⁾

+ ¯

a⁽_x⁰⁾a⁽_x¹⁾

+

¯ χ

⁽⁰⁾

+

ⁱ

β

ω

^h

¯

(0)

tx 3

χ

⁽³⁾

+ β

h_rx⁽⁴⁾

+ ( η

c

−

¹

)

−

³

4_h

¯

_tx⁽⁰⁾_h⁽_tx⁰⁾

−

²

4h

¯

⁽_tx¹⁾h_tx⁽⁰⁾

+

⁴ⁱh

¯

⁽_tx⁰⁾h⁽_rx²⁾

ω +

i

β

_h

¯

_tx⁽⁰⁾

χ

⁽⁰⁾

ω −

2i_h

¯

⁽_tx⁰⁾_h⁽_rx³⁾

ω + β

²_h

¯

⁽_tx⁰⁾_h⁽_tx⁰⁾

+

−

⁴ⁱ_h

¯

⁽_tx¹⁾_h⁽_rx²⁾

ω −

4_h

¯

⁽_tx²⁾_h⁽_tx⁰⁾

+

ⁱ

β χ ¯

⁽⁰⁾h⁽_tx⁰⁾

ω

− ¯ χ

⁽⁰⁾

χ

⁽⁰⁾

ω

²

−

^2m0h

¯

⁽_tx⁰⁾h_tx⁽⁰⁾

−

⁴h

¯

⁽_tx⁰⁾h⁽_tx³⁾

−

²ⁱ

ω

h

¯

_tx⁽¹⁾h⁽_rx³⁾

+ β

²_h

¯

⁽_tx¹⁾_h⁽_tx⁰⁾

+

ⁱ

β ω

h

¯

⁽_tx¹⁾

χ

⁽⁰⁾

−

4i

ω

h

¯

⁽_tx²⁾h⁽_rx²⁾

−

⁴_h

¯

⁽_tx³⁾_htx⁽⁰⁾

+

ⁱ

β ω χ ¯

⁽¹⁾h⁽_tx⁰⁾

− ω

²

χ ¯

⁽¹⁾

χ

⁽⁰⁾

+ [

^c.c

],

^(4.3)

wheretheargumentofthefields⁴is

ω

.V2 denotesvolumeinx–y spaceand[c.c]means the complexconjugated terms.From here, we will drop the [c.c] term since we want to compute retarded Green’sfunctions[13].

The second line is proportional to a gauge invariant combi- nation under (3.13). Furthermore,one ofthe equation ofmotion includingh⁽_rx⁴⁾is

h⁽_rx⁴⁾

−

¹

β

²

− ω

²

3i

ω

h_tx⁽³⁾

−

ⁱ

μω

a⁽_x⁰⁾

−

³

β χ

⁽³⁾

=

⁰

.

(4.4) Onecanshowthat(4.4) isequivalenttoaWardidentity

∇

μ

^Tμν

+

^Fλν

^Jλ

− O

^I

∂

^ν

ψ

I

=

⁰

,

(4.5) by using the boundary metric andthe other fields in the linear approximationgivenasfollows:

ds²

= η

μνdx^μdx^ν

+

^2h(tx⁰⁾e^−iωtdtdx

,

T^μν

=

T^(0)μν

+

T^(1)μν

F

= −

ⁱ

ω

a⁽_x⁰⁾e^−iωtdt

∧

^dx

,

J^μ

=

J^(0)μ

+

J^(1)μ

= ( μ ,

0

,

0

) +

0

,

a⁽_x¹⁾

− μ

h⁽_tx⁰⁾

,

0

e^−iωt

ψ

I

= (β

x

, β

y

) ,

O

^I

= O

^(1)I

=

3

χ

⁽³⁾

+ β

h⁽_rx⁴⁾

,

0

e^−iωt

,

(4.6) where

T^(0)μν

=

^m0

⎛

⎝

^{2 0 0}0 1 0 0 0 1

⎞

⎠ ,

T^(1)μν

=

−

^2m0h⁽_tx⁰⁾

−

^3htx⁽³⁾

+

ⁱ

ω

h⁽_rx⁴⁾

⎛

⎝

^{0 1 0}1 0 0 0 0 0

⎞

⎠

e^−iωt

.

(4.7) One may ask why Ward identity of the boundary theory is in- cluded in the bulk equation of motion. It is not accidental: The

4 ^a¯⁽x⁰⁾(ω)≡^a(x⁰⁾(−ω)=^a(x⁰⁾(ω)^∗bytherealityconditionofδAx.Thesamenota- tionandrealityconditionapplytoalltheotherfields.

translation,x→^x+ξ0attheboundarytheoryisimbeddedintothe bulk diffeomorphismx→^x+ξ(x),whichinduces thefield trans- formation→+δξ,whichinturnisaspecialcaseofgeneral variation, →+δ. Now the equation of motion is coming fromtheinvarianceofbulkactionδSB=^{0 underthegeneralvari-} ation,whiletheWardidentityistherequirementoftheboundary action under the translation δξ0Sb=^{0. Because AdS/CFT request} S_B=^Sb attheonshell,thelatteriscontainedinthehugetowerof equationofmotionasatinypiece.

Thetermsproportionalto(

η

c−¹)in(4.3)includethedivergent terms with,a regularization parameter, andfinite termswith- out .Aremarkablefactisthat withthecounter termofweight

η

c=^{1,notonlythedivergenttermsarecanceled,butalsoallthe} hrxdependentfinitetermsdisappearsfromtheon-shellaction,as weclaimedinthebeginningofthissection.

5. Gaugeinvarianceundertheresidualgaugetransformation Ourstartingpointistheaction⁵

S⁽_ren²⁾

=

^V2 2

∞ 0

d

ω

2

π

− μ

a

¯

⁽_x⁰⁾h⁽_tx⁰⁾

−

^2m0_h

¯

_tx⁽⁰⁾_h⁽_tx⁰⁾

+ ¯

^a(x⁰⁾a⁽_x¹⁾

−

³h

¯

_tx⁽⁰⁾h⁽_tx³⁾

+

³

χ ¯

⁽⁰⁾

χ

⁽³⁾

+

β χ ¯

⁽⁰⁾

+

ⁱ

ω

h

¯

⁽_tx⁰⁾

h_rx⁽⁴⁾

+

^c.c

,

(5.1)

which is still dependent on residual gauge (3.13) even after we set hrx=^{0. Since it is just a constant shift of the solution} , its effects are only shifts of zero-th modesand (r) andall of its modes,especially (a⁽_x¹⁾,h⁽_tx³⁾,

χ

⁽³⁾):=^a are intact. Noticethat the recurrence relations derived fromequations of motionrelate highermodeswiththezero-thmodes J^a=(a⁽_x⁰⁾,h⁽_tx⁰⁾,

χ

⁽⁰⁾).How- ever,alldependencesofhighermodesonzerothmodesisthrough thegaugeinvariant combination

ωχ

⁽⁰⁾−ⁱβh⁽_tx⁰⁾.See,forexample, (3.18).Thusallhighermodesaregaugeinvariant,whichmakesthe gaugeinvarianceofthe(r)intactinspiteofthecomplicatedde- pendenceofhighermodesonthezerothmodes.

The residual gauge dependence of(5.1) can be understood as follows. The full on shell action should be invariant under the residualgaugetransformation.However,whatwearelookingatis thequadraticpartoftheaction S⁽_ren²⁾,whichgeneratesthe2-point function,intheexpansionof

Sren

[δ] =

S⁽_ren⁰⁾

+

S⁽_ren¹⁾

[δ] +

S⁽_ren²⁾

[δ] + · · ·,

(5.2) whereδ=(δ_μν,δ_μ,δI) collectivelydenotes thesources of the dualfield theory,whichare boundaryvalues of _r¹2δgμν,δAμ andδψI.S⁽_ren¹⁾[δ]^andS(ren²⁾[δ]^{aregivenasfollows:}

S⁽_ren¹⁾

[δ] =

d³x

1

2

δ

μν

T^(0)μν

+ δ

μ

J^(0)μ

+ δ

I

O

^(0)I

,

(5.3)

S⁽_ren²⁾

[δ] =

d³x

1

2

δ

μν

T^(1)μν

+ δ

μ

J^(1)μ

+ δ

I

O

^(1)I

.

(5.4)

5 Itcomesfrom(4.1)beforewegetEq.(4.3),forwhichwehavetousetheequa- tionsofmotion.

Under the residual gauge transformation⁶ with h0= −ⁱ

ω

ζ0, the variationsoftheseactionsare

δ

S⁽_ren¹⁾

[δ] =

^V2

d

ω

2

π ζ ¯

0

i

ωμ

a⁽_x⁰⁾

+

²ⁱ

ω

m₀h⁽_tx⁰⁾

+

^c.c

,

(5.5)

δ

S⁽_ren²⁾

[δ] = −δ

^S(ren¹⁾

[δ]

+

V2

d

ω

2

π ζ ¯

0

3

β χ

⁽³⁾

−

3i

ω

h⁽_tx³⁾

+

i

ωμ

a⁽_x⁰⁾

+

β

²

− ω

²

h_rx⁽⁴⁾

+

^c.c

.

(5.6)

ThusthetotalvariationisproportionaltotheWardidentity(4.4).

Noticethat S_renisgaugeinvariantbutS⁽_ren²⁾,whichisstartingpoint toderivetheGreenfunction,isnotinvariantbyitself.Nevertheless physical observablesderived from S⁽ren²⁾ are invariant becausethe Greenfunctionsare second derivatives ofthefull on shellaction atthezerosourcelimit.

At thispoint one can discussa puzzlein countingdegreesof freedom.Thereareonlytwoindependentbulksolutionssatisfying thein-fallingboundaryconditions,⁷whileweneedthreesolutions atthe boundary since there are three independent source fields.

Therefore,thereisa crisisofmismatchofdegreesoffreedombe- tweenthebulkandboundary.WhatsolvestheproblemistheRGS (3.14). We call it RGS because it is generated by the zero mode ofadiffeomorphismgenerator.Ontheotherhand,tobeaproper gaugedegreeoffreedominthebulk,thediffeomorphismgenerator shouldsatisfytheproperboundaryconditions:infallingathorizon andDirichlet at boundary. The residual gauge symmetry genera- torisaglobalshiftandthereforeitcansatisfyneitherofthem.So suchashiftbythediffeomorphismzeromodeisnotatruegauge symmetry,whileitisasymmetryofthebulkequationsofmotion.

Inotherwords,theRGSisa“solutiongeneratingsymmetry”rather thanagaugesymmetry.Therefore,thegaugeorbitofRGScanpro- videusthenecessarydegree offreedom (d.o.f)nearboundary. To matchthed.o.f,weneedtoacceptitsbulkorbitasphysicalconfig- urationinspiteofthefactthattheresultingbulksolutiondoesnot satisfytheinfallingBC.⁸Onecangiveamorenaturalbulksolution byextendingRGStoadiffeomorphismwhichsatisfiestheinfalling boundaryconditionanditisreducedtoourpreviousRGSnearthe boundary.Itisgeneratedbyξ^μ=(0,ζ (r)e⁻ⁱωt,0,0),with⁹

ζ (

r

) = (

f

(

r

)/

r²

)

^{−iω/(4πT)}

,

(5.7) where f isthemetricfactorgiveninEq.(2.8)and isaconstant parameter.NoticethattheRGSisthecasewhereζ (r)isconstant.

We willcall this“boundaryshifting diffeomorphism”(BSD). Now we can understand the degree of freedom mismatchas follows:

Since it is not satisfyingthe Dirichlet bc, it is still not a proper gaugetransformation.Noticealsothatunder(5.7),thegaugeslice is shiftedand some ofthe gauge fieldsbecome singular. Forthe discussiononthetreatingtheseissues,wereferthereadertop. 24 ofRef. [9].¹⁰This isthereasonwhytheBSD cangeneratea new

6 Thistransformation changesthe sourcesoftheaction, δμν,δμ,δI. One shouldnotethattherearenon-vanishingtransformationsforδ00andδ0.

7 Wehavetwosecondorderdifferentialequationsand onefirstorderonein threevariables:ax,htx,χ.Therefore,thereare5boundaryconditionstofix.Ifwe fixthein-fallingboundaryconditionsforallthreevariables,weareleftwithtwo degreesoffreedom.Werecallequations(3.15)and(3.16).

8 SofarwediscussedthedegreeoffreedommismatchusingtheRGS,sinceour formalismin[3]tocalculatetheconductivityisbasedonit.

9 Wethanktheanonymousrefereeforsuggestingtoconsiderthis.

10 ItisverytemptingtoconsiderBSDasagaugetransformationatleastfrombulk pointofview.Ifwedoit,wegettotheproblem:Itsorbitintheboundarygenerate physicalconfigurationwhileitdoesnotinthebulk,sothatcrisisofd.o.fbecomes real!

solution in theboundary. It is preciselythe same logic whyRGS generate new solution.¹¹ Since RGS and BSD shift the boundary values of fields,they generate the Ward identity forthe transla- tion invariance.Thisisatypical examplehowa globalsymmetry isencodedinalocalgauge transformationandhowtheapparent paradox ofthedegree offreedomcan beresolved becauseofthe holographiccorrespondence.¹²

6. Basisindependence

In[3],weconstructed aformalism toperformtheACconduc- tivities for the case where multiple fields are coupled together.

We had to choose a basis of initial conditions and one can ask whetherdifferentchoicesofbasisgivethesameresult.Answering this question will also provide an alternative reasoning ofgauge invariance.Toprovidethesetup,let usconsider N fields^a(x,r), (a=¹,2,· · ·,N),

^a

(

x

,

r

) =

d^dk

(

2

π )

^de

−^ikxr^p

^a

(

k

,

r

) ,

(6.1)

where the index a may include components of higher spin fields. For convenience, r^p is multiplied such that the solution ^a(k,r)goestoconstantatboundary.Inourcase, (¹,²,³)= (a_x,h_tx,

χ

)andp=^{0 for1},³ andp=^{2 for2}.

Nearhorizon(r=^{1),solutionscanbeexpandedas}

^a_i

(

k

,

r

) = (

r

−

¹

)

^νa±

ϕ

_i^a

+ ˜ ϕ

^a_i

(

r

−

¹

) + · · ·

,

(6.2)

whereanewsubscripti isintroducedtodenotethesolutionscor- responding toa specificindependent setofinitial conditions.For example,

ϕ

^a_i maybechosenas

ϕ

₁^a

=

1

, −( μ ˜ + ˜ β)/(

1

+ ν ),

1

, ϕ

₂^a

=

1

, −( μ ˜ − ˜ β)/(

1

+ ν ), −

¹

,

(6.3)

whereweused(3.16)and

ν

= −ⁱ

ω

/(4

π

T)asshownbelow(3.15) forincomingboundaryconditiontocomputetheretardedGreen’s function[13].Duetoincomingboundarycondition,

ϕ

^a_i determines

˜

ϕ

^a_i throughhorizon-regularityconditionsothatwecandetermine thesolutioncompletely.Eachinitialvalue vector

ϕ

i yieldsasolu- tion,denotedbyi(r),whichisexpandedas

^a_i

(

k

,

r

) → S

^a_i

+ · · · + O

^a_i

r^δa

+ · · · (

near boundary

) ,

(6.4) where S^a_i ^{are the sources (leading terms) of i-th solution and} O^a_i ^{are the operator}expectation values corresponding to sources (δa≥^1).

Notice thatwehaveonlytwosolutions whilewehaveathree dimensional vector spaceJ^{of boundaryvalues Ja,a}=¹,2,3.To

11 Thisargumentisfurtherjustifiedifweconsiderthenumericalcalculationstart- ingfromtheboundaryinsteadfromhorizon.Afterchoosing3fields’svalues,wecan adjusttwo“expectationvalues”suchthatwecangetinfallingboundaryvaluesat thehorizon.Itiseasytoshowthatonlywhenwestartfromasubspaceofcodimen- sion 1,wegetthreeinfallingsolutionnearthehorizon.Ifwestartfromapointoff thisplane,wegetoneinfallingandtwofieldswhicharemixtureofinfallinganda constant.Inthiscalculationthegaugeconditionhrx=0 isintact.Thisdemonstrates thatwecannot imposeinfallingbcforallfieldsathands.Ifwedothesamenumer- icalexperimentforBSD,thepictureisfollowing.TheBSDgeneratetheorbitandit alsomovethegaugeslice.Nowinthiscaseeveninthecasewestartfromtheoff theplane,wecangetthreeinfallingfieldsatthehorizon.Weneedtocalculatethe r-evolutionateach‘gaugefixing’planewhichpassthroughtheinitialdata.

12 Theapparent‘mismatch’isduetothedifferenceinviewingthegaugeorbitof BSD(orRGS)betweenthebulkandboundary.Inthebulk,onecouldviewitas gaugeorbit.Ontheotherhand,fromtheboundarytheorypointofview,thereisno gaugestructureandtheorbitoftranslationsymmetryisphysicaldegreeoffreedom.

fixsuchmismatchofdegreeoffreedom, weintroduce aconstant solution0(r)= S0=(0,1,iβ/

ω

)alongthegauge-orbitdirection oftheresidualgaugetransformationsothatS^a₁,S^a₂,S^a₀ ^formabasis ofJ^.NowS^andO^{aregenericregularmatricesoforder 3.}

Thegeneralsolutionisalinearcombinationofthem:let

^a

(

k

,

r

) =

^a_i

(

k

,

r

)

cⁱ

,

(6.5) withrealconstantscⁱ’s.Wecanchoosecⁱ suchthatthecombined sourcetermmatchestheboundaryvalue J^a:

J^a

= S

^a_i^ci

,

(6.6)

whichyields

^a

(

k

,

r

) =

^a_i

(

k

,

r

)

cⁱ

→

J^a

+ · · · +

^a r^δa

+ · · · ,

(

near boundary

)

(6.7)

where,with(6.4)and(6.6),

^a

= O

^a_i^ci

= O

^a_i

(S

⁻¹

)

ⁱ_bJ^b

=:

^Ca_b^Jb

.

(6.8) Noticethatboth^aandC^a_bareinvariantunderthetransformation J^b→^Jb+

S^b₀becauseC_b^aS^b₀=O^a_i(S⁻¹)ⁱ_bS^b₀=O^a0=^0,whereO^a0= 0 sinceitisthesub-leadingtermoftheconstantsolutions.

Ageneralon-shellquadraticactioninmomentumspacehasthe formof

S⁽_ren²⁾

=

¹ 2

d^dk

(

2

π )

^d

¯

J^a

A

ab

(

k

)

J^b

+ ¯

^Ja

B

ab

(

k

)

^b

,

(6.9)

whereA^andB^{areregularmatricesoforderN.} ¯J^a means