Exact microstate counting for dyonic black holes in AdS4

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Exact

microstate

counting

for

dyonic

black

holes

in

AdS

₄

Francesco Benini

a

,

b

,

Kiril Hristov

c

,

Alberto Zaffaroni

d

,

e

,

∗

a_{SISSA,INFN,SezionediTrieste,viaBonomea265,34136Trieste,Italy} b_{BlackettLaboratory,ImperialCollegeLondon,LondonSW72AZ,UnitedKingdom} c_{INRNE,BulgarianAcademyofSciences,TsarigradskoChaussee72,1784Sofia,Bulgaria} d_{DipartimentodiFisica,UniversitàdiMilano-Bicocca,I-20126Milano,Italy} e_{INFN,sezionediMilano-Bicocca,I-20126Milano,Italy}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received23April2017 Accepted25May2017 Availableonline7June2017 Editor: N.Lambert

We present a counting ofmicrostates ofa class ofdyonic BPS black holes in AdS4 which precisely reproduces their Bekenstein–Hawking entropy. The counting is performed in the dual boundary description, thatprovidesanon-perturbativedefinition ofquantumgravity,intermsofatwisted and mass-deformedABJMtheory.Weevaluate itstwistedindexandproposeanextremization principleto extracttheentropy,whichreproducestheattractormechanismingaugedsupergravity.

©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

Supersymmetricblackholes instringtheory constitute impor-tantmodelstotestfundamental questionsaboutquantumgravity in a relatively simple setting.The main question we would like toaddresshereistheoriginoftheblackhole(BH)entropy,which statisticallyisexpectedtocountthenumberofdegenerateBH con-figurations. String theory provides a microscopic explanation for theentropyofaclassofasymptoticallyflatblackholes [1].Much lessisknown aboutasymptotically AdSonesin fourormore di-mensions.

InprincipleAdS/CFT [2]provides anon-perturbativedefinition ofquantumgravityinasymptotically AdSspace,asadual bound-ary quantum field theory (QFT). The BH microstates appear as particular states in the boundary description. The difficulty with thisapproach is theneed to performcomputations ina strongly coupledQFT,butthedevelopment ofexactnon-perturbative tech-niques makes progress possible. We recently reported [3] on a particular example of magnetically charged BPS black holes in AdS4 [4] with a known field theory dual—topologically twisted

ABJM theory.Using the technique ofsupersymmetric localization wewereabletocalculateinanindependentwaythe(regularized) numberof groundstatesof thetheory and successfullymatchit withtheleadingmacroscopicentropyoftheblackholes.

In this Letter we discuss a function Z

(

ua

)

that encodes the

quantumentropiesofstaticdyonicBPSblackholesinAdS4,

com-*

Correspondingauthor.

E-mailaddresses:fbenini@sissa.it(F. Benini),khristov@inrne.bas.bg(K. Hristov), alberto.zaffaroni@mib.infn.it(A. Zaffaroni).

putednon-perturbativelyfromthedualQFTdescription,andshow that its leading behavior reproducesthe Bekenstein–Hawking en-tropy

[5,6]

.

In particular we show that, at leading order, the entropy of BPS blackholeswithmagneticcharges

p

a,electriccharges

q

aand

asymptotictoAdS4

×

S7 canbeobtainedbyextremizingthe

quan-tity

I

=

log Z(ua

)

−

i



aua

q

a (1)

with respect to a set of complexified chemical potentials ua for

the globalU

(

1

)

flavorsymmetries oftheboundary theory. Z

(

ua

)

isthetopologicallytwistedindex

[7]

oftheABJMtheory

[8]

which explicitlydependsonthemagneticcharges

p

a (see

[9,10]

forother

examples). Theentropyis givenby S

=

I(ˆ

u

)

evaluatedatthe ex-tremum,withaconstraintonthechargesthat S berealpositive.

As wewillsee,theextremizationof

I

isequivalent tothe at-tractormechanismforAdS4blackholesingaugedsupergravity.We

alsoargue,generalizing

[3]

,thattheextremizationof

I

selectsthe exactR-symmetryofthesuperconformalquantummechanicsdual totheAdS2 horizonregion.We noticestrongsimilaritiesbetween

our formalism and those based on Sen’s entropy functional [11]

andtheOSVconjecture

[12]

.

2. Theblackholes

WeconsiderdyonicBPSBHsthatcanbeembeddedinM-theory andareasymptotictoAdS4

×

S7.Theyaremoreeasilydescribedas

solutions inthe STU model,a four-dimensional

N =

2 gauged su-pergravitywiththree vectormultiplets,whichisaconsistent

trun-http://dx.doi.org/10.1016/j.physletb.2017.05.076

0370-2693/©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

cationbothofM-theoryonS7_{,andofthe4dmaximal}

_{N =}

_{8 S O}

₍

₈

₎

gaugedsupergravity [13].The modelcontains fourAbelianvector fields(oneisthegraviphoton)correspondingtotheU

(

1

)

4

⊂

S O

(

8

)

isometriesofS7_.

In4d

N =

2 supergravities withnV vector multiplets, one can

usethestandard machineryofspecialgeometry

[14–16]

.The La-grangian

L

of the theory is completely specified by the prepo-tential

F(

X

₎

_{,whichis ahomogeneous holomorphic functionof}

sections X_{, and the vector of Fayet–Iliopoulos (FI) terms}

_{G =}

(

g

_,

_g



)

. The symplectic index



=

0

,

1

,

. . . ,

nV runs over the

graviphotonandthenV vectorsinvectormultiplets.Thescalars zi

invectormultiplets,withi

=

1

,

. . . ,

nV,parametrizeaspecial

Käh-lermanifold

M

andX _{aresectionsofasymplecticHodgevector}

bundle on

M

. The formalism is covariant withrespect to sym-plectic Sp

(

2nV

+

2

)

transformations. Indicating as

(

A

,

A

)

the

2nV

+

2 componentsofasymplecticvector A,thescalarproductis



A

,

B



=

AB

−

AB.One definesthecovariantly-holomorphic

sections

V

=

eK(z,z¯)/2



X

₍

_z

₎

F



(

z

)



(2)

on

M

, where

K

is the Kähler potential and

F



≡ ∂



F

. They

satisfy D_¯ı

V ≡



∂

¯ı

−

12

∂

¯ı

K



V =

0.The Kählerpotential isthen de-terminedby



V,

V

= −

i.

Theansatzfordyonicblackholesisoftheform

ds2

= −

e2U(r)dt2

+

e−2U(r)



dr2

+

V

(

r

)

2ds2_g



(3)

where



g isa Riemann surface ofgenus

g,

andthe scalarfields zi _{areassumedtoonlyhaveradialdependence.Wecanwritethe}

metricon



glocallyas ds_2_g

=

d

θ

2

+

f_κ2

(θ )

d

ϕ

2

,

fκ

(θ )

=

⎧

⎨

⎩

sin

θ

κ

=

1

θ

κ

=

0 sinh

θ

κ

= −

1 (4)

where

κ

=

1 forS2_,

_κ

₌

_{0 forT}2_,and

_κ

_{= −}

_{1 for}

_

g with

g

>

1. Thescalarcurvatureis2

κ

andthevolumeis

Vol(g

)

=

2

π η

,

η

=

2

|g −

1

|

for

g

=

1

1 for

g

=

1

.

(5)

Themagneticandelectricchargesoftheblackholeare



g F

=

Vol(g

)

p

,



g G

=

Vol(g

)

q

,

(6)

where G

=

8

π

GN

δ(

L

dvol4

)/δ

F and GN is the Newton

con-stant.ThisparticularnormalizationensuresthattheBPSequations areindependentof

g

(besidesalinearconstraint).Thechargesare collectedinthevector

Q

= (

p

_,

_q



)

.Thevector

G

ofFIterms

con-trolsthegauginganddeterminesthechargesofthegravitiniunder thegauge fields.In a frame withpurely electricgauging g,the

latticeofelectro-magneticchargesis

η

gp

∈ Z ,

η

4GNg

q

∈ Z

(7)

not summed over



.It turns out that theBPS equationsfix the morestringentcondition



G

,

Q

 = −

κ

,

(8)

thatwecallthelinearconstraint.

It hasbeen noticed in [17] that the BPS equations of gauged supergravityforthenear-horizongeometry canbeputintheform

of“attractorequations”.Onedefinesthecentralchargeoftheblack hole

Z

andthesuperpotential

L

:

Z

= 

Q

,

V

 =

eK/2



qX

−

p

F





L

= 

G

,

V

 =

eK/2



gX

−

g

F





.

(9)

TheBPSequationsforthenear-horizongeometry

ds2_nh

= −

r 2 R2_Adt 2

₊

R2A r2 dr 2

₊

_R2 Sds2g (10)

with constant scalar fields zi _{imply the following two equations} [17]:

Z −

iR2 S

L

=

0 andDj



Z −

iR2 S

L



=

0,whereDj

= ∂

j

+

12

∂

j

K

,

besides



G,

Q

= −

κ

.Theseequationscanberewrittenas

∂

j

Z

L

=

0

,

−

i

Z

L

=

R 2 S

.

(11)

Inotherwords,thescalarszi _{atthehorizontakeavaluesuchthat}

thequantity

−

i

Z/L

hasacriticalpointon

M

andthenitsvalue isproportionaltotheBekenstein–Hawkingblackholeentropy.

NoticethataconditiontohaveBHswithsmoothhorizonisthat

−

i

Z/L

berealpositiveatthecriticalpoint.Sincethecritical-point equationsalreadyfix the valuesofthe scalars,thiscondition be-comes a second (non-linear)constrainton thecharges. Therefore the domain ofallowed electro-magneticcharges has real dimen-sion2nV (beforeimposingquantization).Thereareother

inequali-tiestobesatisfiedbythecharges,forinstancetoensurethatalso

R2_Abepositive.

InthecaseofveryspecialKählergeometry,i.e. thatthe prepo-tentialtakestheform

F =

di jkXiXjXk

/

X0 orsymplectic

transfor-mations thereof,generalsolutions to the near-horizon BPS equa-tionsaswellasfullBHsolutionshavebeenfoundin

[18–20]

.That analysis guarantees that all near-horizon solutions can be com-pletedintofullBHsolutions.

Ourfocusis ontheSTU model,which hasnV

=

3 and

prepo-tential

F

= −

2i



X0_X1_X2_X3

_,

₍₁₂₎

withpurely electricgauging g

≡

g, g

=

0. Thenthe AdS4

vac-uumhasradiusL2

₌

₁

_/

_2g2_{.NotethatalldyonicBHsolutionswith}

electriccharges havecomplex profilesfor thescalars, i.e. the ax-ionsareturnedon.

3. Thedualfieldtheory

M-theoryonAdS4

×

S7 hasadualholographicdescriptionasa

three-dimensionalsupersymmetricgaugetheory,theABJMtheory

[8],whichprovidesanon-perturbativedefinitionthereof.In

N =

2 notation, the ABJM theory is a U

(

N

)

1

×

U

(

N

)

−1 Chern–Simons

theory (the subscripts are the levels) with bi-fundamental chiral multiplets Ai and Bj, i

,

j

=

1

,

2, transforming in the

(

N

,

N

)

and

(

N

,

N

)

representationsofthe gauge group,respectively, andwith superpotential W

=

ε

ik

_ε

jl_{Tr A}

iBjAkBl. The theory has

N =

8

su-perconformalsymmetryandS O

(

8

)

R-symmetry.Theidentification betweengravitationalandQFTparametersis

L2 GN

=

1 2g2_G N

=

2

√

2 3 N 3/2

_.

₍₁₃₎

The“topologicallytwistedindex”ofan

N =

2 three-dimensional theoryisitssupersymmetricEuclideanpartition functiononS2

×

S1 with a topological twist on S2 [7]. Its higher-genus general-ization, namely the twisted partition function on



g

×

S1, has

been constructed aswell [21,22]. They depend on a set of inte-germagneticfluxes

p

aandcomplexfugacitiesya,alongtheCartan

generatorsoftheflavorsymmetrygroup.

In the present case, to make the enhanced symmetry more manifest,weintroduce anindexa

=

1

,

2

,

3

,

4 thatsimultaneously runsover the four ABJM chiral fields andthe four Abelian sym-metriesU

(

1

)

4

⊂

S O

(

8

)

.Thisisdonebyintroducingabasisoffour R-symmetries Ra,each actingwithcharge 2 onone ofthechiral

fieldsandzeroontheothers.Then themagneticfluxesidentifya

U

(

1

)

subgroupofS O

(

8

)

usedtotwist,andarerequiredby super-symmetrytosatisfy

_a

p

a

=

2g

−

2.Thecomplexfugacities ya

=

exp iua must satisfy



aya

=

1 (

aua

∈

2

π

Z

) andencode

back-groundvaluesforthe flavor symmetries.Writing ua

= 

a

+

i

β

σ

a

(where

β

isthelengthof S1_{),wecanidentify}



a withflavorflat

connectionsand

σ

a withrealmasses.

TheHamiltoniandefinitionoftheindexis

[7]

Z

(

ua

, p

a

)

=

Tr

(

−

1)Fei 3

a=1aJa_e−βH

_,

₍₁₄₎ where Ja

=

1₂

(

Ra

−

R4

)

are thethreeindependent flavor

symme-triesandH isthetwistedHamiltonianonS2,explicitlydependent on the magnetic charges

p

a and the real masses

σ

a. Due to the

supersymmetryalgebra Q2

₌

_H

₋

3

a=1

σ

aJa,theindex Z

(

ua

,

p

a

)

is a meromorphic function of ya. For simplicity, we will keep

the dependence on

p

a implicit and use the shorthand notation

σ

J

=

_a3₌₁

σ

aJa.We stress that, in general, (14) iswell-defined

onlyforcomplexua while theindexfor

σ

a

=

0 is definedby

an-alyticcontinuation.Wewouldliketoseehowwe canextractthe BHentropiesfromZ .

4. Statisticalinterpretation

ThepartitionfunctionZ

(

u

)

describesasupersymmetric ensem-blewhichiscanonicalwithrespecttothemagneticcharges(i.e. all stateshavethesame,fixed,magneticcharges)butgrandcanonical withrespecttothe electriccharges(i.e. itisa sumover all elec-tric charge sectors, with fixed chemical potentials ua). A similar

viewpointinBHphysics isadvocatedin

[23]

.Wecan decompose

Z asa sumover sectorswithfixed charges

q

a under Ra

/

2 (then

thelatticeofchargesissuchthatboth Ja

,

Ra

∈ Z

,uptoapossible

zero-pointshiftinthevacuum): Z

(

u

)

=



qe

i 3a=1ua(qa−q4)_Z

q

.

(15)

WewouldliketoidentifySq

≡ R

elog Zqwiththeleadingentropy ofa BH offixed electriccharges

q

a.We take thereal partto

re-movetheeffectofapossibleoverallsign.Animportantassumption isthat

(

−

1

)

F inthetrace

(14)

doesnotcausedangerous cancela-tionsatleadingorder.

WecanFouriertransformthepreviousexpressionwithrespect tothethreeindependent



a toobtain



q4 Zq

=



d3



a

(2

π

)

3e− i 3_b=1b(qb−q4)_Z

₍

_u

_{) ,}

₍₁₆₎

whereprimemeansthatthesumistakenatfixedinteger

q

a

− q

4.

As we willsee,for supergravityBHs both the electriccharges

q

a

andlog Z areoforderN3/2,thereforetheprevious expressioncan beevaluatedatlargeN usingasaddlepointapproximation:



q4 Zq

=

exp



log Z(u

ˆ

)

−

i



3 a=1u

ˆ

a

(q

a

− q

4

)



(17)

atleadingorder,whereu

ˆ

a isasolutionforua to

∂

∂

ua



log Z

(

u

)

−

i



3 b=1ub

(q

b

− q

4

)



=

0 (18)

witha

=

1

,

2

,

3.Thissaddlepointingeneralgivescomplexvalues for u

ˆ

a.The sumon the LHSof (17)will alsobe dominated by a

specific value of

q

4,corresponding totheelectricR-charge ofthe

blackhole.Forthatvalue: Sq

= R

e



log Z(u

ˆ

)

−

i



3 a=1u

ˆ

a

(q

a

− q

4

)



.

(19)

We can restore the permutation symmetry betweenthe charges, partoftheWeylgroupofS O

(

8

)

,byintroducing

I

(

u

)

≡

log Z(u

)

−

i



4

a=1ua

q

a

.

(20)

Eqn. (18) is equivalent to extremization of

I

andthe entropy is givenby Sq

= R

e

I(ˆ

u

)

.

ThisargumentdoesnotdeterminetheR-chargeoftheBH, es-sentiallybecausetheindex Z

(

u

)

lacks achemical potentialforit. Howeverfromtheattractorequations

(11)

itfollowsthat,forgiven magneticcharges

p

aandflavorelectriccharges

q

a

− q

4,thereisat

mostonevalue of

q

4 leadingtoalargesmoothBH.Ourargument

then gives an unambiguousprediction forthe leading entropyof thatBH.

5. RGflowinterpretation

We can extract more information fromthe index ifwe inter-prettheBHasanholographicRGflow.Thenear-horizongeometry of BPS black holes contains an AdS2 factor permeated by

con-stant electric flux,where thesuper-isometry algebra isenhanced to

su(1

,

1

|

1

)

. Thus we can think of the BH solution as a holo-graphic RG flow from the 3d theory on S2 _{to an ensemble of}

su(1

,

1

|

1

)

-invariantstates ina1dsystem. The bosonicsubalgebra is

sl(2

,

R)

×u(

1

)

c wherethesecondfactoristheIRsuperconformal

R-symmetry, whichissome linearcombinationofU

(

1

)

4

⊂

S O

(

8

)

. In the near-horizon canonical ensemble this implies that all BH stateshave zeroU

(

1

)

c charge (byan argumentsimilar tothat in [24,25]).Wewillassumethattherearenoothercontributions out-sidethehorizon.

The asymptotic behavior of electrically charged BH solutions withaxionsturnedonsuggeststhatthedualABJMtheory isalso deformedbyrealmasses

σ

a.Ingeneral,theyliftapossiblevacuum

degeneracyoftheHamiltonianH .ThepresenceofAdS2with

con-stant electricflux, though, indicates that there shouldbe alarge vacuumdegeneracyforamodified Hamiltonian Hnh inwhichthe

energy of states gets an extra contribution linear in the charge:

Hnh

(

σ

)

=

H

(

σ

)

−

σ

. From thesupersymmetry algebra Q2

=

Hnh

we concludethat Hnh

≥

0, and the index gets contribution only

fromitsgroundstates.Wecanrewritetheindexin

(14)

as Z

(,

σ

)

=

TreiπRtrial()_e−βσJ

_,

₍₂₁₎ where Tr

=

TrHnh=0. We introduced a trial R-current Rtrial

()

≡

R0

+ 

J

/

π

thatparametrizesthemixingoftheR-symmetrywith

theflavor symmetries,with R0 areferenceR-symmetrysuch that eiπR0

= (−

₁

₎

F_.

We want to argue, generalizing [3], that the superconfor-mal R-symmetry Rc of the Hamiltonian Hnh can be found by

extremizing Z

(,

σ

)

for fixed values of

σ

a. Let

ˆ

a be the

value such that Rtrial

( ˆ

)

=

Rc. One computes

∂

log Z

/∂

a



_ˆa

=

i



Jae−βσJ

/

e−βσJ



, using that at zero temperature the density

matrixisuniformlydistributedoverthegroundstatesof Hnh,and

that Rc

=

0 inthosestatesasarguedabove.Theexpressiononthe

rightisimaginary, implyingthat

ˆ

a aredetermined by

extremiz-ingtheindexwithrespectto



a atfixed

σ

a:

∂

R

e log Z

(,

σ

)

∂

a



This is the generalization of the

I

-extremization principle pro-posed in [3]. Assuming the large N factorization



J e−βσJ



=



J



e−βσJ



,wealsohave

∂

I

m log Z

(,

σ

)

∂

a





_ˆ

=

i



Ja

 ≡

i

(q

a

− q

4

) ,

(23)

where



Ja



isthe charge ofthe vacuumdensitymatrix.This

de-terminesthe relationbetweenthe flavorcharges

q

a

− q

4 and

σ

a.

SinceZ

(,

σ

)

isaholomorphicfunctionofua

= 

a

+

i

β

σ

a,wecan

summarizetheresultinthecomplexequation

∂

log Z(u

)

∂

ua





ˆ u

=

i

(q

a

− q

4

) ,

(24)

whichdeterminesboth

ˆ

aand

σ

a asfunctionsof

q

a.

Fromeqn.

(21)

,atthecriticalpoint

Z

( ˆ

,

σ

)

=

e−βσJTr1

=

e− 4a=1βσaqa_eSq

_.

₍₂₅₎

The real partof the logarithm of this expression reproduces the resultofthestatisticalargument,namely

Sq

= R

e



log Z(u

ˆ

)

−

i



4 a=1u

ˆ

a

q

a



.

(26)

An advantage ofthis derivation isthat we can argue,at leastat leading order, that eSq _{is the number of groundstates, without} dangeroussignsthatcouldcausecancelations.

Wecanalsowrite theentropyinaslightlydifferentformand makeaconjectureforthevalueofthefourthcharge.Sinceu only

ˆ

dependson thedifferences

q

a

− q

4 and

aua

∈

2

π

Z

,we can

al-ways shift the integer charges

q

a and write the entropy in the

permutationallysymmetricandholomorphicform

Sq

=

log Z(u

ˆ

)

−

i



4

a=1u

ˆ

a

q

a

=

I

(

u

ˆ

) ,

(27)

up to

O(

N0

)

termswhich are invisible inthe large N limit. The determinationofthelogarithmissuchthatlog Z isrealfor

σ

a

=

0

and extended by continuity. The requirement that (27) be real positivefixes thefourthcharge. Interestingly,thisis preciselythe constraint

(11)

thatcomesfromsupergravity.

6. ExplicitmatchforABJM

ThelargeN expressionfortheindexofABJMwasfoundin

[3,

21]forthe caseofrealua, andwe can extenditto thecomplex

planeusingholomorphy:

log Z

=

N 3/2 3



2u1u2u3u4



4 a=1

p

a ua

.

(28)

Thisisvalidfor

_aua

=

2

π

and0

<

R

eua

<

2

π

.The

I

-extremi-zationprinciple

(24)

isequivalenttotheextremizationof

I

QFT

=



4 a=1



N3/2 3



2u1u2u3u4

p

a ua

−

i

q

aua



.

(29)

Thentheentropyisgivenby Sq

=

I

QFT

(

u

ˆ

)

,withtheconstrainton

thechargesthat

I

QFT

(

u

ˆ

)

bepositive.

Insupergravity,theBHentropyisdeterminedby

SBH

=

Area 4GN

= −

i

Z

L

2

π η

4GN

≡

I

SUGRA (30)

using(12),and

I

SUGRA shouldbe extremizedwithrespectto X.

We canidentifythe index



= {

0

,

1

,

2

,

3

}

witha

= {

1

,

2

,

3

,

4

}

, as

well as2

π

Xa

_/

bXb

=

ua sincethey havethesamedomainand

constraint:

I

SUGRA

=

η

4gGN



4 a=1



_√

u1u2u3u4 pa ua

−

iqaua



.

(31)

Identifyingtheintegersin

(7)

withthecharges

p

a,

q

a,respectively,

andusing

(13)

weobtainaperfectmatch

I

QFT

=

I

SUGRA.Thefield

theoryextremizationprinciplecorrespondstothesupergravity at-tractormechanism:theyleadtothesameentropyandnon-linear constraintonthecharges.

Acknowledgements

WethankJ.deBoer,A.Gnecchi,N.HalmagyiandS.Murthyfor instructive clarifications. FB is supported by the MIUR-SIR grant RBSI1471GJ.AZissupportedbytheMIUR-FIRBgrantRBFR10QS5J.

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