Logarithmic corrections to entropy of magnetically charged AdS 4 black holes

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Logarithmic corrections to entropy of magnetically charged AdS 4 black holes

Imtak Jeon, Shailesh Lal

To cite this version:

Imtak Jeon, Shailesh Lal. Logarithmic corrections to entropy of magnetically charged AdS 4 black

holes. Physics Letters B, Elsevier, 2017, �10.1016/j.physletb.2017.09.026�. �hal-01591923�

(2)

Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

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Logarithmic corrections to entropy of magnetically charged AdS ₄ black holes

Imtak Jeon

^a

, Shailesh Lal

^b

aHarish-ChandraResearchInstitute,ChhatnagRoad,Jhusi,Allahabad211019,India bLPTHE–UMR7589,UPMCParis06,SorbonneUniversités,Paris75005,France

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

Articlehistory:

Received25August2017 Accepted12September2017 Availableonlinexxxx Editor:M.Cvetiˇc

Keywords:

Stringtheory

AdS/CFTcorrespondence Blackholes

Quantumentropy AdSblackhole Logarithmiccorrections

Logarithmictermsarequantumcorrections toblackholeentropydeterminedcompletelyfromclassical data,thusprovidingastrongcheckforcandidatetheoriesofquantumgravitypurelyfromphysicsinthe infrared.We computethesetermsintheentropyassociatedtothe horizonofamagneticallycharged extremal black hole inAdS4×^S⁷ ûsing^the ^quantum êntropy^function ând ^discuss^the possibility of matchingagainstrecentlyderivedmicroscopicexpressions.

©²⁰¹⁷^TheÂuthor(s).^Published^byÊlsevier^B.V.^Thisîsânôpenâccessârticleûnder^the^CC^BY^license (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

ProvidingamicroscopicinterpretationtotheBekensteinHawk- ing formula in the context of certain classes of supersymmetric extremal black holes in ﬂat space has been a main success of stringtheoryasatheoryofquantumgravity[1–6].Theexpression forthemicroscopicentropyobtainedbyexplicitenumerationand countingofblackholemicrostatesinthesecasescontainsthearea lawastheleadingformula,butalsocontainshigher-derivativeand quantumcorrectionstoit.Wereferthereaderto [7]forareview ofthesedevelopmentsaswell asmoreexhaustivereferences.Im- portantly, since extremalblack holes expectedlypossess an AdS2 factorinthe nearhorizongeometry, onemay usetheAdS2/CFT1 correspondencetoprovideanalternative,butequivalent,deﬁnition ofthequantum entropy ofextremalblack holes instringtheory.

Thisproposalisknown asthequantum entropyfunction,andfor extremalblackholescarryingcharges

q

≡

^qi[8,9],

d_hor

q

≡

exp

i

qid

θ A

ⁱ_θ

f inite

AdS₂

,

(1)

where dhor is the full quantum degeneracy associated with the blackholehorizon,and

A

ⁱ_θ îs^the^componentôf^the^{ith gauge}^field

E-mailaddress:shailesh.hri@gmail.com(S. Lal).

alongthe boundaryofthe AdS2.Inthispicturetheentropyasso- ciatedtothehorizondegreesoffreedomofanextremalblackhole isessentially thefreeenergycorresponding tothepartition function (1). The superscript ‘finite’ reminds us that the quantity on the righthandside of(1)isnaively divergent duetothe infinite volume of AdS2 but this divergence may be regulated in accor- dancewithgeneralprinciplesoftheAdS/CFTcorrespondenceand a cutoff-insensitive finite partextracted, whichis then identified todhor [8–10].The pathintegraliscarriedout overallfieldsthat asymptote to the black hole near horizon geometry. In the con- textofsupersymmetricextremalblackholesinflatspace,thishas beenevaluatedusingsaddlepointtechniques[10–14,16,17]aswell as supersymmetriclocalization [18–24] and the answer matched withmicroscopic results whereveravailable. Importantly, even in the cases where thefull microscopic formulais unavailable, this quantitymaybeevaluated atleastusingsaddle-pointmethodsto gainsomeinsightintothefullmicroscopicformula[13,17].

Incontrast,thesituationforentropycomputationsforextremal blackholesinAdSspaceisrelativelyinitsinfancy.WhiletheWald entropy[25]associatedtothehorizonmaybecomputedforsuch blackholesusingforexampletheentropyfunctionformalism,the quantum correctionstoit are still unknown.¹ Inthissituation, it

1 RecentlythequantumentropyofthetopologicalblackholeinAdS4hasbeen computedusingsupersymmetriclocalization[26],buildingontheanalysisof[27],

https://doi.org/10.1016/j.physletb.2017.09.026

0370-2693/©²⁰¹⁷^TheÂuthor(s).^Published^byÊlsevier^B.V.^Thisîsânôpenâccessârticleûnder^the^CC^BY^license(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

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JID:PLB AID:33173 /SCO Doctopic: Theory [m5Gv1.3; v1.222; Prn:20/09/2017; 14:35] P.2 (1-5)

2 I. Jeon, S. Lal / Physics Letters B•••⁽••••⁾•••^–•••

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wouldclearlybeofinteresttoevaluate(1)forsuchblackholesto obtainthe setofquantum correctionsto theWaldformula. Such computations are further motivatedby the recent proposal fora microscopic computation of a CFT₃ index argued to capture the quantumentropyofanAdS₄ extremalblackhole[29].

In this note we will focus on the computation of (1) in the semi-classicalapproximationwheretheeventhorizonoftheblack holehas a large length scale a associated withit. In thecase of extremal black holes in ﬂat space, this large length scale arises whenthe chargesof theblack holeare takento be large. Inthe presentcaseofAdSblackholes,thiscorrespondstotakingthe rank N ofthegaugegroupinthedualCFTtobelarge,whilethecharges themselvesarenotscaled.

One saddle-point of the path integral (1) is the black hole’s near-horizongeometryitself.Byevaluatingtheon-shellactionon thisﬁeldconﬁguration,itispossibletoshowthat[8,9]

d_hor

^e^S^Wald

⇒

^SB H

=

^ln^dhor

^SWald

.

(2) Hencethequantumentropyfunctionproducesastheleadingcon- tribution, Wald’s formula for the entropy of the horizon. In this note, we will considersubleading correctionsof the formlna to thisformula,i.e.

SB H

=

SWald

+

clna

+ . . . ,

(3)

where c is a coeﬃcient which depends on the details of the quantum gravity that the black hole is embedded in. Forexam- ple,thesame four-dimensionalblackholewhichis aquarter-BPS black hole in

N =

4 supergravity may be embedded as a one- eighthBPS blackholein

N =

8 supergravity andwhilethelead- ing Bekenstein–Hawking answer is the same for the black hole, the log terms computed in both theories are different [12], and matchwiththemicroscopiccomputationscarriedoutrespectively in

N =

^{4 and}

N =

^{8 string}compactiﬁcations.Thismatchingisan importanttestoftheconsistencyofthequantumentropyfunction proposal.

The main reasonwhy the logterm is an important contribu- tiontothe microscopicformulaisthatit isagenuinelyquantum correction tothe BekensteinHawking formula, butisdetermined completely fromone-loop ﬂuctuationsof massless ﬁelds of two- derivate supergravity, which essentially constitute the IR data of theblackhole.² Toseethis, letusrecallsome elementsofascal- ing argument presented in[15]. Consider the

-loop free energy foratheorydeﬁnedonaD dimensionalbackgroundwithalength scaleaassociatedwithit.AtypicalFeynmangraphcontributingto thisquantitywouldscaleas(see[15]fordetails)

⁽_P^D⁻²⁾⁽⁻¹⁾a⁻⁽^D⁻²⁾⁽⁻¹⁾

a/√

d^Dk

˜

k

˜

²⁻²F

k

˜

,

(4)

wherek

˜ =

^ka^and

{

^k

}

âre ^the^loop^momenta,ând^F îsâ^function whichapproaches1 atlargevalues ofits arguments.Byfocusing on the regime where all loop momenta are of the same order, andworking at large k,

˜

we see that a lna termarises fromthe k

˜

²⁻²⁻^D terminthe ¹_˜

k expansionof F atlargek,

˜

andthefull a dependenceofthistermis

toﬁndacompletematchwiththecorrespondingmicroscopicexpression.Wethank JunNianforhelpfulcorrespondenceregardingthis.Wealsonotethatthequantum entropyofAdS4hyperbolicblackholeswasstudiedin[28].

2 Theterm‘massless’hastobecarefullydeﬁnedoncurvedmanifolds.Themore precisestatementisthattheeigenvaluesofthekineticoperatorshouldscaleas_a¹2, whichishowtheeigenvaluesofthekineticoperatoroveramasslessscalar,− wouldscale.

1 a

(D−2)(−1)

lna

,

(5)

which ishighlysuppressed inthelargea limit unless

=

^1.^This verifies the above claim that only one-loop fluctuationsgive rise to the logterm. Further, by considering differentscaling regimes where various subsetsof loop momenta scale to be much larger thantherest,onemayverifythefactthatonlythetwoderivative sector ofmasslessfieldscontributestothelogterm.However,we do not do sohere andinstead referthe readerto [15] for those details.

Therefore,asarguedearlier,thelogtermmayberegardedasan IRprobeofthemicroscopictheory,inthesensethatanyputative microscopicdescriptionoftheblackholemustcorrectlyreproduce notonlytheleadingBekensteinHawkingarealaw,butalsothelog correctiontoit.

In this note we shall compute the log term for a class of magnetically charged extremal black holes which asymptote to AdS4

×

^S⁷ ^for ^which ^a ^complete ^expression ^for ^the microscopic entropy has recently been computed via the computation of a topologically twisted indexin ABJMtheory [29].We omitdetails of the microscopic formula, referring the reader to [29] as well asthecompanionpapers[30,31].Further,theonlyfeaturesofthe nearhorizongeometrywhichshallbe relevanttousarethatitis AdS2

×

^S²

×

^S⁷ ^where^the ^S⁷îs^bundledôver^the ^S² ând^that^the AdS₂,S² andS⁷haveacommonlengthscaleaassociatedtothem.

Thatis,the metricover thefull 11dimensionalnearhorizonge- ometrycanbebroughttotheformgμν

=

^a²^g⁽μν⁰⁾wherethemetric g⁽⁰⁾andthecoordinatesareaindependent.

Further the S² has S O

(

3

)

isometry, while the S⁷ has U

(

1

)

⁴ isometry.Theseinputssuﬃceforthemacroscopiccomputationof the logtermasa prediction forthemicroscopic formulaof [29].

Weshallﬁnallydiscussafewaspectsoftheproposedmatch.³ De- tailsofthefullblackholesolutionarereviewedin[29].

2. Thelogtermfromthequantumentropyfunction

Wewillnowdescribehowthelogtermmaybeextractedfrom the pathintegral(1).Todoso, itisusefultophrasetheproblem more generally. In particular, we consider a theory in D dimensions,withadynamicalﬁeld

,admittingasaddle-pointwhichis abackgroundwithlengthscalea.

Z

[

]

= D

e⁻¹^h¯S[]

.

(6) Then,asarguedfromEquation(4),toextracttheterminthefree energy proportional to lna, it is suﬃcient to concentrate on the one-loop partition function ofthetheory.The techniquesforthis analysisare well-known andwe referthe readerto[14,38,16,17]

foraccountsofhowthesecomputationsarecarriedout.Firstly,the one-looppartitionfunctionisthengivenby

Z

1−

=

^det

O

⁻¹²

· ( Z

zero

)

ⁿ⁰

,

(7) wheredet

O

^is^thedeterminantof

O

êvaluatedôverîts^non-zero modes, n0 is the number of zero modes of

O

, and

Z

zero is the residualzero-modeintegral.Therefore

ln

Z

1−

= −

¹

2ln det

O +

ⁿ0ln

Z

zero

.

(8) The ﬁnal result is that the coeﬃcient of the lna term in the freeenergycomputedaboutthissaddle-pointdependson K

(

t

;

⁰

)

,

3 Whilethisdraftwasbeingreadiedforsubmission,welearnedof[32]where thiscomparisonhasbeencarriedoutinanumericalschemetoﬁndamismatch.

(4)

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which is the t⁰ coeﬃcient of the heat kernel expansion of the kineticoperator

O

^about^thissaddle-point, and

β

,whichdeter- mineshowthe zeromodecontribution

Z

zero tothe pathintegral scaleswitha.

Z

zero

=

^a^β

_Z ˆ

_zero

,

(9) where

Z ˆ

zero doesnotscale witha.We eventuallyobtain thefor- mula

ln

Z

1−

=

^K

(

t

;

⁰

)

lna

+ (β

−

¹

)

n₀lna

.

(10) Itiswell known thatin odd-dimensionalspacetimes, K

(

0

;

^t

) =

⁰ andhencewehavetheformula

ln

Z

1−

= (β

−

1

)

n0lna

.

(11)

The extension to the case of multiple ﬁelds

{ }

^having ^zero modesisapparent.

ln

Z

1−

=

φ∈{}

β

φ

−

1

n^φ₀lna

,

(12)

wheren^φ₀ isthenumberofzeromodesofthekineticoperatorover theﬁeld

φ

.The speciﬁcvalues for

β

that willbe relevantto us areforthe vector,the graviton, thethree-formandthegravitino.

Thesearegivenby

β

v

=

^D

−

²

2

, β

m

=

^D

2

, β

_f

=

^D

−

¹

, β

C

=

^D

2

−

³

.

(13) Herevdenotesthevectorﬁeld,mthemetricorthegraviton, f the gravitino,andthecorresponding

β

valueshavebeenlistedinEqua- tion(2.37)of[14].C denotes thethree-formﬁeldandits

β

value maybe determined exactly in the same manneras the previous ﬁelds[14,38].Westart withtheexpressionforthenormalization oftheﬁeld CM N P inD-dimensions.

D

^CM N Pe −

d^dx√_{g g}MUg^{N V}g^{P W}CM N PCU V W

=

¹

.

(14)

HerethemetricscalesasgM N

=

^a²^g⁽_{M N}⁰⁾ ^where ^g⁽⁰⁾^does^not^scale witha.Thereforewehave

D

^CM N Pe −^a^D⁻⁶

d^dx

g⁽⁰⁾g⁽⁰⁾^MUg⁽⁰⁾^{N V}g⁽⁰⁾^{P W}C_{M N P}C_{U V W}

=

¹

.

(15) Hencethecorrectlynormalizedintegrationmeasureis

x,(M N P) d

a^D²⁻³C_{M N P}

.

(16)

FromthiswecanobtainthatthezeromodeofCM N P corresponds to

β

C

=

^D

2

−

³

.

(17)

3. Countingthenumberofzeromodes

Fromthe above discussionit isclear that ifwe areto extract thecontributiontothe(logarithmofthe) quantumentropyfunction which scales as lna, where a is the length scale associated withthe radii of AdS₂, S² and S⁷, it is enough to compute the zeromodesof the fieldsappearing in thepath integral (1). Zero modescanin principle appearinthe massless spectrum ofAdS2 fieldsobtainedfromfieldsofthe11dimensionalsupergravityre- ducedonto S²

×

^S⁷^.^In^this^section^we^will^enumerate^these^zero modes and compute their contribution to the quantum entropy ofmagneticallycharged AdS4 black holes.In isolating thesezero modes, aspecial role isplayed by theso-calleddiscretemodesof

theLaplacian forspin-1 andspin-2fields,andtheDiracoperator forspin-³₂ fieldsonAdS2 explicitlyenumeratedin[33,34]respec- tively,andcountingthetotalnumberofzeromodesisessentially equivalenttocountingthetotalnumberofdiscretemodesofthese fields.Thesemodeshavealsobeenlistedin[12,14,17].Animpor- tantobservationforusisthatitturnsoutthatnaivelythenumber ofzeromodesforeachofthesefieldsturnsouttobeinfinite.How- ever,thisdivergenceturns out tobe essentiallyequivalent to the volumedivergenceofthefreeenergyandmayberegulatedinthe same way. Two slightly different, but equivalent, procedures for doingthisareavailablein[11,12,14]and[16,17]andweshalluse thoseresultsfortheregularizednumberofzeromodesinthecom- putationsthatfollow.

3.1. Bosoniczeromodes

ThebosonicﬁeldsarethegravitonhM N andthe3-formCM N P. Quantization ofthegraviton givesriseto aghost vectorﬁeld but this hasno zero modes. Quantization of the 3-form C gives rise to aghost 2-form B withGrassmann oddstatistics, andaghost- for-ghost 1-form A with Grassmann even statistics. We use the following conventions: M is an 11-d vector index,

μ

is an AdS2 index,a isan S² indexandi isan S⁷ index. Finally

α

iseithera ori.

Considerthemetriczeromodesﬁrst.ThegravitonhM N decom- posesintotheAdS2gravitonhμν,3masslessAdS2vectorshμa,and 4 masslessAdS₂ vectorshμi, alongwiththe AdS₂ scalars g_ia

,

g_{i j} andgab.Incountingthenumberofmasslessvectors,weusedthe factthatthesearegivenby

h_μα

=

^vμk_α

,

(18)

wherekα isaKillingvector alongan internaldirection.Since the internal spacehas SU(2)

⊗

Û(1)^⊗⁴ îsometry, ^there âre ³

+

⁴

=

⁷ Killingvectors.EachmasslessvectorﬁeldonAdS₂ contributes

n₀^v

= −

1

,

(19)

hencethereare

−

^{7 zero} ^modes^from^the^hμα.Alsothereare

−

³ zeromodesfromthe AdS2 metrichμν.Thereforetotalnumberof metriczeromodesis

n^m₀

= −

⁷

−

³

= −

¹⁰

.

(20)

Then the contribution to the log termfromthe 11d metric zero modesis

δ

Z

|

metric

= (β

m

−

1

)

n^m₀lna

= −

45 lna

.

(21) Wenextconsiderthe3-formﬁeldCM N P.Itsquantizationwascar- riedoutin[35–37]andisreviewedinSection3.1of[38].Wewill justrequirethefollowingresult.Thequantizationofa p-formre- quirespgeneralizedghostﬁeldswhichare

(

p

−

^j

)

-forms,where j runsfrom1 top.Furtherthelogarithmiccontributionofallthese ﬁeldstothefreeenergymaybepackagedintotheexpression

F

=

p

j=0

(−

1

)

^j

β

p−^j

−

j

−

1

n⁰_O

p−jlna

,

(22)

where

O

is the kinetic operator. For j

=

¹

, . . . ,

p which are the generalizedghostﬁelds,thisisjusttheHodgeLaplacian.For j

=

^0, whichisthephysicalfield,thiscanhavecouplingstobackground fluxesaswell.WenowconsiderthereductionofCM N P ontoAdS2. ThisfirstlyleadstoCμνaCμνiwhicharetwo-formsonAdS2 which are Hodge duals ofscalars. These haveno zero modes. Next,we considera 3-formwhichisthe wedgeproduct ofa1-formalong AdS2 withaharmonic2-formalong S²

×

^S⁷^.

C⁽³⁾

=

^C₍⁽₁^AdS₎ ²⁾

∧

^C₍⁽₂^S₎²^×^S⁷⁾

.

(23)

(5)

JID:PLB AID:33173 /SCO Doctopic: Theory [m5Gv1.3; v1.222; Prn:20/09/2017; 14:35] P.4 (1-5)

4 I. Jeon, S. Lal / Physics Letters B•••⁽••••⁾•••^–•••

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Thenumberofsuch harmonicformsisgivenby thesecond Betti number b2 of the manifold, which is 1 in this case, as may be readilyseenfrommultiplyingthePoincaré polynomialsof S² and S⁷. Hence there is a single massless vector ﬁelds along AdS₂ fromthe11dimensional3-formﬁeld.Finally wehavethescalars Cabi

,

Cai j

,

Ci jk which don’t have zero modes. Hence the 3-form contributes

δ

Z

|

C

= (β

C

−

¹

)

n^C₀lna

= −

³

2lna (24)

to the log term. We next consider the ghost BM N which arises from the quantization of C. This decomposes into the following massless ﬁeldsonAdS₂.Firstwe have Bμν whichcontributesno zeromodes.Nextwemayobtainamassless1-formonAdS2 from thisﬁeld bydecomposing BM N intoa wedge productofan AdS2 1-formwithaharmonic1-formon S²

×

^S⁷^.^The ^number^of^such harmonic 1-forms is the ﬁrst Betti number of S²

×

^S⁷ ^which ^is zero.FinallywehavethescalarsB_ab

,

B_aiandB_{i j},whichcontribute nozeromodes.Thereforethecontributiontothelogtermis

δ

Z

|

B

= − (β

B

−

2

)

n₀^Blna

=

0

.

(25) Theoverall minus signison accountofGrassmann oddstatistics ofthisfield.Finallywehavetheghost-for-ghostfield AM fromthe quantizationof B.Thisleadstoone masslessvectorfieldonAdS₂ andtherefore

δ

Z

|

A

= (β

A

−

³

)

n₀^Alna

= −

³

2lna

.

(26)

Wethereforeadd(21),(24),(25)and(26)toobtain

δ

Z

=

−

45

−

³ 2

−

³

2

lna

= −

48 lna

.

(27)

Theresidualscalargeneralizedghosthasnozeromodes.

3.2. Fermioniczeromodes

Tocount fermion zeromodes, we need to compute the regu- larizednumberofdiscretemodes

ξ

_μ⁽^k⁾⁺ and

ξ ˆ

_μ⁽^k⁾⁺,k

=

¹

,

2

, . . .

,on AdS2.Therelevantcomputationsareavailablein[12,14,17]andwe onlymentionﬁnal results.Firstly,itmaybeshownthat theregu- larizednumberofmodesofboth

ξ

and

ξ ˆ

isgivenby

−

^1.^Further, thesemodesshouldbe tensoredwiththespinors associatedwith directionstransversetoAdS2.Thiswillgiverisetoadditionalmul- tiplicity factors. Todetermine the multiplicity, we note ﬁrst that thenearhorizongeometryoftheblackholehasasuperconformal symmetry su

(

1

,

1

|

¹

)

withfermionic generators Gα

n where

α = ±

andn

∈ Z +

¹₂^.^The^gravitino^zero^modes^we^consider^are^associated withthegeneratorsGα

n where

|

ⁿ

| ≥

³₂^.^Inparticular,weidentifyGα n withn

≥

³₂ ^with

ξ

^kwheren

=

^k

+

¹₂ ^and^G^αn withn

≤ −

³₂ ^with

ξ

^k wheren

= −

^k

−

¹₂^.^Hence,^thereîsânôverallmultiplicityfactorof 2comingfrom

α

takingvalues

+

^and

−

^.^Therefore^the^number^of fermionzeromodesis

n₀^f

= (−

1

−

1

) ×

2

= −

4

.

(28)

Then the contribution to the log term from the fermionic zero modesis

δ

Z

|

fermions

= − β

_f

−

¹

n₀^flna

= +

^{36 ln}^a

.

(29) The overall minus is on account of Grassmann odd statistics.

Adding(27)and(29)weseethatthelogtermisgivenby

F

= (−

⁴⁸

+

³⁶

)

lna

= −

^{12 ln}^a

.

(30)

4. Onthecomparisonwithmicroscopics

Wehavesofarcomputedthelogtermformagneticallycharged AdS₄extremalblackholesusingthequantumentropyfunction.In thissectionweshallverybrieﬂydiscusshowinprincipleamatch withtheproposedmicroscopicanswerof[29]maybecarriedout.

At ﬁrstglanceone mayexpectthat thelarge N expansion ofthe logarithm ofthetopologicallytwistedindexintheCFTcomputed in[29]maybematchedtermby termwiththelargea expansion ofthelogarithm ofthe quantumentropyfunction.Wenote however, that severalsteps should in principle be necessary tocarry outbeforeamatchcanbemeaningfullyproposed.

Firstly,theindexcomputedby[29]measurestheblackholeen- tropyinthegrandcanonicalensembleasitemploystheAdS4/CFT3 correspondence. In contrast, the natural boundary conditions for thequantumentropyfunction (1)pickout themicrocanonicalen- semble[9].Whilethechoiceofensembleisirrelevantinthestrict large N limit, it is typically important when ﬁnite N effects are takenintoaccount.Indeedexamplesexistwherethechoiceofen- sembleisexplicitlyshowntoaffectthevalueofthelogterm[15].

Therefore,asaﬁrststep,weexpectthatone wouldneedtogoto the microcanonicalensemblewhencomputingtheCFTanswer to matchwiththequantumentropyfunction.

Second,itisnotobvioushowthenaivelarge N scalingsofthe log of the index are to be reproduced by the quantum entropy function. In particular, it appears from the analysis of [29] that lnZ scalesas

lnZ

∼

^N³^/²

+ O (

NlnN

) .

(31)

TheN³^/²termpreciselymatcheswiththeBekensteinHawkingen- tropy oftheblackhole,butthepossiblesubleadingtermoforder NlnN isparticularlysurprisingfromthepointofviewofthegen- eralscalingsofcontributionstothequantumentropyasexpected fromthequantumentropyfunction.Inparticular,ongoingthrough thescaling analysisofEquation(4),itisapparent thatwe donot expect atermoftheformNlnN,andthatthistermshoulddrop fromtheCFT3 answerifamatchwiththequantumentropyfunc- tionistobepossible.

Remarkably, itseemsthat anumericalestimate ofthe large N behavior of the index produces the pattern one would naturally expect from thequantum entropyfunction scalings, includingan absence of the NlnN term, and the presence of the lnN term, albeitwithamismatchingcoeﬃcient[32].Thisremarkablefeature shouldcertainlybebetterunderstoodbycarryingoutasystematic largeN expansionoftheCFTindexwhileaccountingforthechoice ofensembleaswehaveindicatedabove.Itwouldbeinterestingto returntothesequestionsinthefuture.

Acknowledgements

We would like to thank Rajesh Gupta, Nick Halmagyi, Eui- hunJoung,TomokiNosaka,ChiungHwang,Leopoldo PandoZayas, AshokeSen,PiljinYiandAlbertoZaffaroniforhelpfuldiscussions.

SL thanks theKorea Institute of AdvancedStudy and Kyung Hee University for hospitality while this work was carried out. SL’s work issupported by a Marie Sklodowska Curie 2014Individual Fellowship.

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