Thermodynamic properties of asymptotically anti-de Sitter black holes in d=4 Einstein-Yang-Mills theory

(1)

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Thermodynamic

properties

of

asymptotically

anti-de

Sitter

black

holes

in

d

=

4 Einstein–Yang–Mills

theory

Olga Kichakova

a

,

∗

,

Jutta Kunz

a

,

Eugen Radu

b

,

Yasha Shnir

a

,

c

,

d

a_Institut_für_Physik,_Universität_Oldenburg,_Postfach_{2503, D-26111}_Oldenburg,_Germany

b_Departamento_de_Fisica_da_Universidade_de_Aveiro_and_I3N_Campus_de_Santiago,_3810-183_Aveiro,_Portugal c_Department_of_Theoretical_Physics,_Tomsk_State_Pedagogical_University,_Russia

d_BLTP,_JINR,_Dubna,_Russia

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received12March2015

Receivedinrevisedform15May2015 Accepted19May2015

Availableonline1June2015 Editor:M.Cvetiˇc

Weinvestigatethethermodynamicsofsphericallysymmetricblackholesolutionsinafour-dimensional Einstein–Yang–Mills-SU(2) theory with a negative cosmological constant. Special attention is paid to configurations with aunit magnetic charge. We findthat aset of Reissner–Nordström–Anti-de Sitter blackholescanbecomeunstabletoformingnon-Abelianhair.However,thehairyblackholesarenever thermodynamicallyfavouredoverthefullsetofabelianmonopolesolutions.Thethermodynamicsofthe genericconfigurationspossessinganonintegermagneticchargeisalsodiscussed.

1. Introduction

Black holes are non-perturbative objects whose existence ap-pearsto beanunavoidableconsequenceofgeneralrelativity(and its variousextensions). Moreover, asis often statedin the litera-ture, theblack holes (BHs) arethe quantum gravity counterparts ofthehydrogenatominordinaryquantummechanics.Thus some basicresults derived atthe semiclassical level,like the existence ofHawkingradiationtogetherwithanintrinsicBHentropyare ex-pectedtobeverybasicfeaturesthatanyputativequantumtheory ofgravitywillhavetotakeintoaccount.

As a result, the subjectof BHthermodynamics hasenjoyed a constantinterestover thelast fourdecades.BHsolutions in anti-deSitter(AdS)spacetimebackgroundhavebeenconsideredalsoin thiscontext.Forexample,asshownbyHawkingandPage

[1]

,the presenceofanegative comologicalconstantmakes itpossiblefor aBHtoreachstablethermalequilibriumwithaheatbath. More-over, accordingto the AdS/CFT conjecture [2], BH solutions with AdSasymptoticswouldofferthepossibilityofprobingthe nonper-turbativestructureofsomeconformalﬁeldtheories.

ThestudyofthermodynamicsofBHsolutionsviolating theno hair conjecture is particularly interesting. This conjecture states thatthe onlyallowed characteristicsofa stationaryBHare those associatedwiththeGausslaw,such asmass,angularmomentum andU(1)charges

[3]

.Apartfromapuremathematicalinterest,the

*

Correspondingauthor.

E-mailaddress:[email protected](O. Kichakova).

BHswithhairmaybeusefulforprobingnotonlyquantumgravity, butalsomayplayanimportantroleinthecontextoftheAdS/CFT correspondence.

The ﬁrst (and still the best known) example of hairy BH so-lutionsisthat inEinstein–Yang–Mills(EYM)theory.Moreover,this examplecanberegardedascanonicalinthesensethatotherhairy solutions usuallyshare anumberofcommoncharacteristics with the EYM case(areview ofhairynon-Abelian BHs solutions with a cosmological constant

≥

0 can be found in [4]). BHs with non-Abelian (nA) hair in AdS background have also been exten-sively studied, starting withthe pioneering work [5]. TheseEYM solutions possess a variety ofinteresting features which strongly contrast withthose ofthe asymptotically ﬂat spacetime counter-parts in

[6]

.Forexample, stableBHs witha magneticcharge are knowntoexistevenintheabsenceofaHiggsﬁeld (see

[7]

fora reviewofthesesolutions).

Moreover, consideringsuch conﬁgurations isa legitimatetask, sincethegaugedsupergravitymodels(ofinterestinAdS/CFT con-text) generically contain the EYM action as the basic building block.

The main purposeof thiswork is to address theissue ofthe thermodynamical behaviour of the AdS solutions with a spheri-cal horizon topology in EYM-SU(2) theory, a problem which, to our knowledge,has not yet beenaddressed ina systematic way. Specialattentionispaidtoconfigurationssharingtheasymptotics withtheSchwarzschild-AdSandthe(embeddedAbelian)Reissner– Nordström-AdSBHs.Also,forsimplicity,weshallrestrictourstudy toconfigurationsfeaturingmagneticfieldsonly.

http://dx.doi.org/10.1016/j.physletb.2015.05.052

(2)

2 L2

=

8

π

G

F_μα(a)F_ν(a_β)gαβ

−

1 4gμνF (a) αβF (a)αβ

,

DμFμν

=

0

.

(1)

We are mainly interested in spherically symmetric solutions, withmagneticﬁeldsonly,acasewhichcanbestudiedwithinthe followingansatz ds2

=

dr 2 N

(

r

)

+

r 2

₍

_d

_θ

2

₊

_sin2

_θ

_d

_φ

2

₎

₋

_σ

2

₍

_r

₎

_N

₍

_r

₎

_dt2

_,

with N

(

r

)

=

1

−

2m

(

r

)

r

+

r2 L2

,

(2)

forthemetric,and

A

=

1 2g

ˆ

w

(

r

)

τ

1d

θ

+ (

cos

θ

τ

3

+

w

(

r

)

sin

θ

τ

2

)

d

φ

,

(3)

forthe gauge ﬁelds (with g the

ˆ

gauge coupling constant and

τ

a thePaulimatrices).Thentheﬁeldequations

(1)

reduceto

m

=

α

2

N w2

+

V 2

₍

_w

₎

2r2

,

σ

=

2

α

2

_σ

w2 r

,

w

+

N N

+

σ

w

+

w V

(

w

)

r2_N

=

0 (4)

(withV

(

w

)

= (

1

−

w2

₎

_),_where_we_have_deﬁned_the_coupling

con-stant

α

2

=

4

π

G

ˆ

g2

.

(5)

We wantthe metric (2)to describe a nonsingular, asymptoti-callyAdS spacetimeoutsidea horizonlocatedatr

=

rH

>

0 (here

N

(

rH

)

=

0 isonlyacoordinatesingularitywhereall curvature in-variantsareﬁnite).Therearetwoexplicitsolutionssatisfyingthese assumptions.If w2

₍

_r

₎

_≡

_1,_then N

(

r

)

=

1

−

rH r

1

+

r 2 L2

+

rrH L2

+

r2 H L2

,

σ

(

r

)

=

1

,

(6) whichisjusttheSchwarzschild-AdS(SAdS)metric,withvanishing YM curvature and mass M

=

rH

2

(

1

+

r2

H

L2

)

. A different solution is foundforw

(

r

)

≡

0,with

N

(

r

)

=

1

−

rH r

1

+

r 2 L2

+

rrH L2

+

r2_H L2

−

α

2 rrH

,

σ

(

r

)

=

1

.

(7) ThisdescribetheembeddedAbelianmagnetic Reissner–Nordström-AdS(RNAdS)metric,withmassM

=

_2r1

H

(

α

2

₊

_r2 H

(

1

+

r2 H L2

))

,andunit magneticcharge.

ingtemperatureandeventhorizonareaofthesolutions(withthe entropy S

=

AH

/

4G) TH

=

1 4

π

σ

H

3r4_H

+

r2_HL2

−

α

2_L2

₍

₁

₋

_w2 H

)

2

r3_HL2

,

AH

=

4

π

r 2 H

.

(9) A similar expression can also be written for r

→ ∞

as a power seriesin1

/

r,with m

(

r

)

=

M

−

α

2

₍

_{2 J}2

₊

_L2

₍

₁

₋

_w2 0

)

2

)

2L2_r

+ . . . ,

σ

(

r

)

=

1

−

1 2r4

α

2_J2

_{+ . . . ,}

_w

₍

_r

₎

₌

_w 0

−

J r

+ . . . ,

(10)

with three free parameters: M, which ﬁxes the ADM mass of

thesolutions, w0 whichgivesthe(SU(2)-valued)magneticcharge

Q

M

=

QM₂τ3_g_ˆ (where QM

=

1

−

w2₀)and J – an orderparameter whichprovidesameasureofnon-Abelianess.

Note also that the equations (4)possess two scaling symme-tries:

(

i

)

σ

→ λ

σ

and

(

ii

)

r

→ λ

r,m

→ λ

r,L

→ λ

L,

α

→ λ

α

(with

λ

apositivescalingparameter).Thesymmetry

(

i

)

wasusedtoset

σ

(

r

)

→

1 as r

→ ∞

. The symmetry

(

ii

)

can be used to ﬁx the valueoftheAdSradiusL orthevalueofthecouplingconstant

α

; inthisworkwesetL

=

1 andtreat

α

asaninputparameter(also, tosimplifytheexpressionofsomequantities,wesetG

=

1).

Restrictingtoacanonicalensemble(whichisthenaturalonein thepresenceofamagneticcharge),westudysolutionsholdingthe temperatureTH andthemagneticchargeQMﬁxed.Theassociated thermodynamicpotentialistheHelmholtzfreeenergy

F

=

M

−

THS

.

(11)

When different solutions exist at ﬁxed

(

TH

,

QM

)

, the conﬁgura-tionwithlowest F istheonethatisthermodynamicallyfavoured. Also, the condition for (local)thermodynamic stability of BHs in thecanonicalensembleis

C

=

∂

M

∂

TH

QM

=

TH

∂

S

∂

TH

QM

>

0

.

(12)

Asrecentlyfoundin

[8]

,thegenerichairysolutionssatisfytheﬁrst lawofthermodynamics dM

=

σ

H 2r2 HL2

3r4_H

+

r2_HL2

−

α

2L2

(

1

−

w2_H

)

2

drH

+

α

2_w 0J L2 dw0

=

THdS

+

d QM

,

(13) where

= −

α

2_J

_/(

_2w

0L2

)

isthemagnetostaticpotential.

The most remarkable feature ofthe AdS EYM BHs is perhaps that the value of the parameter w0 in

(10)

(i.e. thevalue of the

magneticcharge)isnot ﬁxedapriori.Thatis,forgiven

α

andany horizonsize(assetbytheinputparameterrH),solutionsarefound

(3)

Fig. 1. Left: ThefreeenergyisshownasafunctionoftemperatureforgenericembeddedAbelianReissner–Nordström-AdSblackholeswith0<α<L/6,inwhichcaseone noticestheexistenceofthreebranchesofsolutions.Thegenericpictureforα>L/6 isdisplayedintheinset,withtheexistenceofasinglebranchofsolutions.Right: The

mass,freeenergy,temperatureandareaoftheunstableReissner–Nordström-AdSblackholeswhereabranchofnon-Abeliansolutionsemergesisshownasafunctionofα.

forintervalsofw0 andnot only w2₀

=

1,asfor

=

0 (note

how-ever,that theallowed rangeof w0 decreases as

α

increases

[5,9,

10]).

Moreover, asfor the better known case of asymptotically ﬂat hairy BHs [6], the solutions here are also indexed by the node numberofthemagneticpotential.

Inthiswork we shallconsidernodeless (for w0

>

0) andone

node solutions (for w0

<

0) only. The conﬁgurations withhigher

numberofnodesrepresentexcitedstatesandarethereforeignored inwhatfollows.Moreover, the solutionswithnodesare unstable in linearized perturbation theory. A detailed discussion of these aspectstogetherwithalargesetofreferencescanbefoundin

[7]

. 3. Thethermodynamicsofnon-Abelianblackholes

3.1.Theunitmagneticchargesolutions

Among all sets ofnA BHs, of particularinterest are those so-lutionspossessingaunitmagneticcharge, i.e. with w0

=

0 inthe

farﬁeldexpansion

(10)

.Theexistenceofsuchconﬁgurations pro-vides an obvious violation of the no-hair conjecture, since two distinctsolutionsarefoundforthesamesetofglobalcharges.That is,apartfromthe gravitatingDiracmononopoleconﬁguration

(7)

withw

(

r

)

≡

0,thereisalsoan(intrinsicnA)conﬁguration possess-inganontrivialproﬁleofthemagneticpotential w

(

r

)

.

At thispoint it is instructive to brieﬂy review the thermody-namicsofthemagneticallychargedRNAdSsolutions1(7).The free-energyvs. temperaturebehaviourofthesesolutionsissummarized in Fig. 1 (left). For any

α

>

0, a branch of RNAdS BHs emerges at P0

0

,

₂√L 3p−

(

1

+

p 2 +

)

,a pointwhich corresponds toextremal BHs witha nonzero area AH

(

TH

=

0

)

=

2πL

2 3 p2− (where we note p_±

=

1

+ (

2

√

3 α_L

)

2

_±

_{1 and} _q_±

₌

1

±

1

− (

6_Lα

)

2_). _For ₀

_<

α

<

L

/

6, this branch continuesto the point P1

1 πL√3

(

q−

+

1 q₋

),

L 6√6q−

(

1

+

q 2 +

)

, wherea cusp occurs signalinga second branch ofsolutions extendingbackwardinTH,withanincreasing F (the correspondingvalueofthemassat P1 is L

√

2 3√3q

2

−).

1 _The_{thermodynamics} _of_RNAdS _solution _has_been _discussed _from _a _slightly differentperspectivein[11], forelectricallychargedBHs;however,duetothe ex-istenceofelectric–magneticdualityind=4 Einstein–Maxwelltheory,theresults thereapplytomagneticallychargedRNAdSsolutionsaswell.

This secondary branch ends at P2

1 πL√3

(

q+

+

1 q+

),

L 6√6q+

(

1

+

q 2 −

)

, where another cusp is found, with the occur-rence of a third branch of solutions (the corresponding value of the massat P2 is L

√

2 3√3q

2

+). Thisbranchintersects the ﬁrstone at

some T(_Hc) (withtheoccurrenceofa“swallowtail” shape)and ex-tends to arbitrarily large TH (note that the free energyvanishes for TH

=

√ 2 πL

(

p+

+

1 p+

)

(apoint where M

=

L 3√2p+

(

1

+

p 2 +

)

) and

becomesnegativeforlargervaluesofTH.

The ﬁrst branch solutions dominate the thermodynamic en-semble for TH

<

TH(c);however, forhighertemperatures, thefree energy isminimized by the solutions on thethird branch. How-ever,therelative sizeofthesecondbranchdecreases with

α

,the points P1andP2 coincidingfor

α

=

L

/

6,wherethesecondbranch

ofsolutions disappears.Then,for

α

>

L

/

6 onlyone single branch ofsolutionsisfound,seetheinsetin

Fig. 1

(left)(fromthere,itis alsoclear that,for any

α

,at lowenough temperatures therecan onlybeonesolution).

The picture valid for the nA solutions is different. First, unit magnetic charge solutions are foundfor 0

<

α

<

α

max only, with

α

max

1

.

264.Anotherstrikingfeatureistheexistenceofasoliton limitoftheBHs,2 whichisapproachedasrH

→

0,whereTH

→ ∞

,

AH

→

0 and F

→

M

>

0.

Moreover,we noticethat thenAsolutions existabove a mini-malvalue ofTH

>

0 only.In particular,noextremalBHs withnA hairisfound,whichagreeswiththerecentresultsin

[13]

.

Also,asseenin

Fig. 2

,foragiventemperature,thefreeenergy of all solutions is minimized by a RNAdS solution. For

α

<

L

/

6, thisRNAdSsolutionislocatedontheﬁrstbranch(forT

<

T(_Hc))or on thethird branch(for highertemperatures). Thus we conclude thatall unitmagneticcharge nAsolutionsare globally

thermody-2 _An_exact_{particle-like}_solution_with_w

0=0 isknownforα=0 (i.e. theprobe limit–YMﬁeldsinaﬁxedAdSbackground),andhasamagneticpotentialw(r)=

1/

1+r_L22 [12].Moreover,onecanwriteaperturbativesolitonicsolution(withunit

magneticcharge)oftheEYMeqs.(4)byconsideringaperturbativeexpansioninα2 aroundthe globalAdSbackground.Forexample,the ﬁrstordercorrectedmetric functionsarem(r)=3α42 ₁ Larctan( r L)− r L2₊r2 ,σ(r)=1−α2 L2 2(L2₊r2₎2.

Thissolutioncapturessomeofthebasicfeaturesofthegeneralnonperturbative conﬁgurationandcanalsobeextendedtohigherordersinα;however,sofarwe couldnotidentifyageneral pattern;moreover,theexpressionsfor thefunctions becomeincreasinglycomplicated.

(4)

Fig. 2. ThefreeenergyandthehorizonareaareshownasafunctionoftemperatureforembeddedAbelian(redcurves)andnon-Abelian(bluecurves)solutionswithunit magneticchargeandseveralvaluesofα.(Forinterpretationofthereferencestocolorinthisﬁgurelegend,thereaderisreferredtothewebversionofthisarticle.)

namicallyunstable(despitethefactthattheirspeciﬁcheatcanbe positiveforsomerangeofrH).

Remarkably,one ﬁnds that,astheminimal valueof TH is ap-proached, the branch of nA solutions joins smoothly a critical RNAdS solution with the same value of

α

, such that the mag-netic gauge function w

(

r

)

becomes identicallyzero. This bifurca-tion can alsobe viewed asindicating that the embedded RNAdS BHpresentsaninstability withrespecttostaticnAperturbations, for a critical value of the horizon radius, a feature that, to our bestknowledge,hasnotbeennoticedbeforeintheliteratureThis instability canbe studied withinthe sameAnsatz (3),by consid-ering values of the magnetic gauge potential w

(

r

)

close to zero everywhere, w

(

r

)

=

W

(

r

)

, and a ﬁxed RNAdS background. The perturbation W

(

r

)

startsfromsomenonzerovalueatthehorizon andvanishesatinﬁnity,beingasolutionofthelinearequation

(

N

(

r

)

W

(

r

))

+

W

(

r

)

r2

=

0

,

(14)

withN

(

r

)

givenby

(7)

.Althoughtheabove equationdoesnot ap-pear to be solvable in terms of known functions, one can write againanapproximatesolutionnearthehorizonandatinﬁnity.The general solution is constructed numerically. Given 0

<

α

<

α

max, thisreducestoﬁndingthecriticalvalue ofrH suchthatthe func-tionW

(

r

)

hastheproperbehaviour.Someparametersofthe crit-icalRNAdSsolutions areshownin

Fig. 1

(right). For0

<

α

<

L

/

6, theunstableRNAdSsolutionsare locatedonthesecond andthird branchesina smallregion aroundthepoint P2 (see

Fig. 1

(left)).

Furtherincreasing

α

,theunstablesolutionsmovetosmaller tem-peratures, the point P0 being approached for

α

→

α

max, with W

(

r

)

≡

0 inthat limit. Also, an analytic estimate for the critical

horizonradiusoftheRNAdSBHs, rH

(

α

)

,canbe foundby match-ing at some intermediate point the expansion of W

(

r

)

(and its ﬁrst derivative) atthe horizon, to that at inﬁnity. Surprisingly, it turns out that the simple expression rH

=

√

αL

31/4 provides a good approximationformostofthenumericaldata(typicallywith sev-eralpercenterror).

Let us also remark that the nA configurations cannot be the end points of this instability, since, as noticed above, they are not thermodynamically favoured overall sets ofAbelian configu-rations.Althoughfurtherworkisnecessary,weconjecturethatthe branchofhairyBHswouldrepresentintermediatestatesonly,with aphasetransitionto(final)RNAdSconfigurations.

3.2. Solutionswithavanishingmagneticcharge

Anothercaseofinterestcorresponds tosolutions with w2₀

=

1 in the far ﬁeld asymptotics (10) (i.e. with a vanishing magnetic charge)whichpossess,however,anon-vanishingmagneticﬁeldin thebulk.3

Thepicturewehavefoundisratherdifferentinthiscase.First, thesehairyBHs donotemerge asperturbations4 oftheSAdS

so-3 _These _are_the _natural_AdS_counterparts_of_the_well-known_{asympotically}_ﬂat solutionsin[6],whichnecessarilyhavew2

0=1.Also,notethattheseareunstable conﬁgurationswithrespecttosmallperturbations,themagneticpotential possess-ingatleastonenode.

4 _That_is,_one_can_show_that_the_linearized_YM_equation_in₍₄₎_with_w₍_r₎_{= −}₁₊

W1(r)inaSAdSbackgrounddoesnotpossesssolutionswhichareﬁniteatr=rH

(5)

Fig. 3. ThefreeenergyandthehorizonareaareshownasafunctionoftemperatureforthevacuumSchwarzschild-AdSblackholes(redcurves)andnon-Abelian(bluecurves) solutionswithvanishingmagneticchargeandtwovaluesofα.(Forinterpretationofthereferencestocolorinthisﬁgurelegend,thereaderisreferredtothewebversionof thisarticle.)

Fig. 4. Thefreeenergyandthehorizonareaareshownasafunctionoftemperatureforgenericnon-Abelianblacksolutions withanonintegermagneticchargeandseveral valuesofα.

lutions (6). Instead of that, one ﬁnds two branches of solutions, which,inan F

(

TH

)

diagram forma cuspforsomeminimal value of TH

>

0.As seen inFig. 3, thisminimal value of TH decreases with

α

(the well-known F

(

TH

)

diagram fortheSAdS blackholes isalsoshownthere;notethatthevacuumsolutionsdonotpossess adependenceon

α

). Moreover,the largeSAdS solutionis always thermodynamicallyfavoured,minimizingthefreeenergy.

Hereitisinterestingtoconsiderthe

→

0 limitoftheseEYM BHsolutions.Then,byusingthedatain

[6]

,onecaneasily verify thatthedifferencebetweenthefreeenergyofahairyBHandthe Schwarzschildsolutionwiththesametemperatureisalways pos-itive.The asymptotically ﬂat EYM solutions areknown to be un-stableinperturbationtheory

[4]

.Then theirthermodynamicsalso suggeststhattheyshoulddecaytoaSchwarzschildvacuum BH.

3.3.Thegeneralcase

The generic solutions have w2₀

=

1 and w0

=

0, possessing

a nonvanishing (and noninteger) magnetic charge.5 As shown in

Fig. 4,thethermodynamicsofthegenericsolutionsresemblesthat ofthe SAdSBHs,withtheexistenceoftwo branchesofsolutions.

5 _Strictly_speaking,_these_EYM_BHs_do_not_violate_the_no_hair_conjecture_(we_recall thattheRNAdS solutionpossessesaunitmagneticchargeonly,whenviewedas solutionoftheEYMtheory).

However, inthiscase, thesmallBHs branchemergesfroma soli-tonicsolutionwitha vanishinghorizonarea anda nonzeromass, extending backwards in TH. Then a cusp occurs for a minimal temperature, with the emergence of a branch of large BHs. The freeenergyisminimizedbythesecondarybranchsolutions,which possessalsoapositivespeciﬁc heat,C

>

0.Also,note thata sim-ilarpicturehasbeenfoundforothervaluesofw0 apartfromthe

onein

Fig. 4

. 4. Furtherremarks

ItiswellknownthattheRNAdSBHsolutionspossessavariety ofinterestingthermodynamicalfeatures,seee.g.[11].Asexpected, wehavefoundthatenlargingthegaugegroupofthematterﬁelds toSU(2)leadstoan evenmorecomplicatedpicture.Forexample, as seen already in other cases when the Einstein–Maxwell sys-tem is considered as part of a larger theory (see e.g. [14]), we have found that the (unit magnetic charge) RNAdS BH can be-come unstable to forming hair atlow temperatures. This results innewbranchesofhairysolutionsofthelargertheory.6However,

6 _A_related_example_is_the_case_of_{asymptotically}_ﬂat,_magnetically_charged_RN BHs embeddedinEinstein–Yang–Mills–Higgs theory, whichdevelopnA hair for some rangeoftheparameters[15].For <0,it isthe asymptoticstructureof spacetimewhicheffectivelyreplacestheHiggsﬁeld.

(6)

Fig. 5. ThefreeenergydensityisshownasafunctionoftemperatureforEYMblackholessolutionswithplanar(k=0)andhyperbolic(k= −1)horizontopologyandseveral valuesofα.

ourresultsshowthatthehairyBHsareneverthermodynamically favouredoverthefullsetofRNAdSsolutions.

There are various possible extensions of theresults discussed above.First,ourpreliminaryresultsindicatetheexistenceofstatic axially symmetricBH solutions withnA hair. Thesesolutions are thecounterpartsoftheparticle-likeEYMconﬁgurationsdiscussed in [16] being constructed by using a similar approach, and pos-sessaneventhorizonofsphericaltopology,which,however,isnot aroundsphere.Theyare foundforanaxially symmetric general-izationof themagnetic YMansatz (3)withtwo positive discrete parametersn andm (which aretheazimuthal andpolarwinding numbers,respectively),thesphericallysymmetricansatz

(3)

being recoveredforn

=

m

=

1.These BHspossessaswell a continuous parameterw0(whichisageneralizationoftheconstantw0

enter-ingthe asymptotics

(10)

), whichﬁxesthe magneticchargeofthe solutions

[16]

.

Again,ofparticularinterestare

(

i

)

theconfigurationswhosefar field asymptotics of the YM fields describe a gauge transformed charge-n Abelianmultimonopole, and

(

ii

)

theconﬁgurations with avanishingnetmagneticﬂux.Theaxiallysymmetricsolutionswe haveconstructedsofarinamoresystematicway,havem

=

1 and

n

=

2

,

3

,

4 and thus represent deformationsofthe conﬁgurations in Section 2, sharing their basic properties. In particular, as ex-pected,wehavefoundthatthecharge-n RNAdSsolutionisalways thermodynamicallyfavouredoverthenAones. Wehopetoreport elsewhereontheseaspects.

Aratherdifferentpicture isfound whenconsidering

‘topologi-calblackhole’generalizationsofthesolutionsinSection 2.Inthis case,thetwo-sphereinthemetricansatz

(2)

isreplacedbya two-dimensionalspaceofnegativeorvanishingcurvature7_(see_e.g._[18]

forvacuumsolutions),thecorrespondingmetricansatzbeing

ds2

=

dr 2 N

(

r

)

+

r 2

₍

_d

_θ

2

₊

_f2 k

(θ )

d

φ

2

₎

₋

_σ

2

₍

_r

₎

_N

₍

_r

₎

_dt2

_,

where N

(

r

)

=

k

−

2m_r(r)

+

r_L22, with k adiscrete parameter which ﬁxes the topology of the horizon. For k

=

1, f1

=

sin

θ

and the

metricansatz

(2)

isrecovered;thesolutionswithk

=

0 have f0

=

θ

andareplanarBHs,approachingasymptoticallyaPoincarépatch oftheAdSspacetime;ﬁnally,thesolutionswithk

= −

1 possessa hyperbolichorizon,with f₋1

=

sinh

θ

.Thecorrespondinggeneral-k

YMansatzhasbeendisplayedin

[17]

andlooksverysimilarto

(3)

, with

7 _However,_such_{conﬁgurations}_possess_the _same_amount_of_symmetry_for _any topologyofthehorizon.

A

=

1 2g

ˆ

w

(

r

)

τ

1d

θ

+ (

dfk

(θ )

d

θ

τ

3

+

fk

(θ )

w

(

r

)

τ

2

)

d

φ

.

The equations of the model are still given by (4), with V

(

w

)

=

k

−

w2

₍

_r

₎

_in_the_general_case.

Astudyofthegeneral-k caseshowsthatthe EYMBHs witha sphericaleventhorizontopology(k

=

1)arespecial.First,asoliton limit oftheBHsexists onlyinthiscase

[17]

;also,theembedded Abelian solution can possess an instability for k

=

1 only. More-over, the EYM BHs with a planar orhyperbolic horizon topology are intrinsicnA,sincethey cannotapproachasymptotically a vac-uumSAdSoraRNAdSconﬁguration(notethatthemagneticgauge potential w

(

r

)

isanodelessfunctionfork

=

1).Infact,an embed-dedAbelian solutionexists fork

= ±

1 only,i.e. w

(

r

)

≡

0 implies

m

(

r

)

=

M

−

α

2_k2

_/(

_2r

₎

_,

_σ

₍

_r

₎

₌

_1,_which_is_the_(planar)_SAdS_BH_for k

=

0.Also,noreasonableYMvacuumstateexistsfork

= −

1.

As shown in Fig. 5, in strong contrast to the k

=

1 case, the EYM BHs with a non-spherical horizontopology exhibit a single branchofsolutions.8_Moreover,_the_k

₌

₀

_,

₋

_{1 solutions}_are_(locally)

thermallystable, since C

>

0 (similarresultshavebeenfoundfor othervaluesofthemagneticcharge,asﬁxedbytheparameterw0

intheasymptoticexpansion

(10)

).

Asanavenueforfurtherresearch,itwouldbeinterestingto ex-tendtheanalysisinthisworktoBHdyons,i.e. solutionsfeaturing bothmagneticandelectricnAﬁelds.Thestudyofaspecialclassof such EYMblackbranesolutions(i.e. withaplanarhorizon, k

=

0) has ledto thediscovery ofholographic superconductors, describ-ing condensed phasesof stronglycoupled, planar, gauge theories in d

=

3 dimensions [19,20] (notethat in thiscasethe magnetic ﬁeldvanishesontheboundary,suchthatthehairysolutionsshare theasymptoticswiththeelectricallychargedRNAdSblackbranes). WeexpecttheEYMsolutionswithasphericalorhyperbolicevent horizontopologytoexhibitarathersimilarpattern.Moreover,itis likely that theinterpretation ofEYM black branesasholographic superconductors

[19,20]

remainsvalidforanonsphericaltopology of the horizon (Note that in the general case the dual theory is deﬁnedinad

=

2

+

1 dimensionalspacetimewithalineelement

ds2

₌

_L2

₍

_d

_θ

2

₊

_f2

k

(θ ))

−

dt2). Also, asnoticed above, the generic EYM solutions present a nonvanishing magnetic flux at infinity. Therefore,the studyoftheseconfigurationsshouldgive some in-formationaboutthebehaviourofaholographicsuperconductorin an externalnA magneticfield.Wehope toreturnelsewherewith adiscussionofsomeoftheseaspects.

8 _For_the_data_in_{Fig. 5,}_we_set_to_one_the_area_of_the

(7)

Acknowledgements

E.R.wouldliketothankCristianSteleaforusefulcommentson anearlyversionofthispaper.WeacknowledgesupportbytheDFG Research Training Group 1620 “Models of Gravity”. The work of E.R.issupportedbytheFCT-IFprogrammeandtheCIDMA strate-gicprojectUID/MAT/04106/2013.Y.S. acknowledges supportfrom A. von Humboldt Foundation in the framework of the Institute LinkageProgrammandtheJINRHeisenberg–LandauProgramm. References

[1]S.W.Hawking,D.N.Page,Commun.Math.Phys.87(1983)577.

[2]J.M.Maldacena,Int.J.Theor.Phys.38(1999)1113,Adv.Theor.Math.Phys.2 (1998)231,arXiv:hep-th/9711200;

E.Witten,Adv.Theor.Math.Phys.2(1998)253,arXiv:hep-th/9802150. [3]R.Ruﬃni,J.A.Wheeler,Phys.Today24 (1)(1971)30.

[4]M.S.Volkov,D.V.Gal’tsov,Phys.Rep.319(1999)1,arXiv:hep-th/9810070. [5]E.Winstanley,Class.QuantumGravity16(1999)1963,arXiv:gr-qc/9812064. [6]M.S.Volkov,D.V.Galtsov,JETPLett.50(1989)346,Pis’maZh.Eksp.Teor.Fiz.

50(1989)312;

H.P.Kuenzle,A.K.M.Masood-ul-Alam,J.Math.Phys.31(1990)928; P.Bizon,Phys.Rev.Lett.64(1990)2844.

[7]E.Winstanley,Lect.NotesPhys.769(2009)49,arXiv:0801.0527[gr-qc]. [8]Z.Y.Fan,H.Lü,J.HighEnergyPhys.1502(2015)013,arXiv:1411.5372[hep-th]. [9]J.Bjoraker,Y.Hosotani,Phys.Rev.D62(2000)043513,arXiv:hep-th/0002098. [10]Y.Hosotani,J.Math.Phys.43(2002)597,arXiv:gr-qc/0103069.

[11]A. Chamblin,R.Emparan,C.V.Johnson,R.C.Myers, Phys. Rev.D60 (1999) 064018,arXiv:hep-th/9902170;

A. Chamblin,R.Emparan,C.V.Johnson,R.C.Myers, Phys. Rev.D60 (1999) 104026,arXiv:hep-th/9904197.

[12]H.Boutaleb-Joutei,A.Chakrabarti,A.Comtet,Phys.Rev.D20(1979)1884. [13]C. Li,J.Lucietti,Class. QuantumGravity30(2013) 095017,arXiv:1302.4616

[hep-th].

[14]S.S.Gubser,Phys.Rev.D78(2008)065034,arXiv:0801.2977[hep-th]. [15]K.M.Lee,V.P.Nair,E.J.Weinberg,Phys.Rev.Lett.68(1992)1100,

arXiv:hep-th/9111045.

[16]O.Kichakova,J.Kunz,E.Radu,Y.Shnir,Phys.Rev.D90 (12)(2014)124012, arXiv:1409.1894[gr-qc].

[17]J.J.VanderBij,E.Radu,Phys.Lett.B536(2002)107,arXiv:gr-qc/0107065. [18]J.P.S.Lemos,Phys.Lett.B353(1995)46,arXiv:gr-qc/9404041;

S. Aminneborg,I. Bengtsson,S.Holst, P.Peldan,Class. QuantumGravity 13 (1996)2707,arXiv:gr-qc/9604005;

R.G.Cai,Y.Z.Zhang,Phys.Rev.D54(1996)4891,arXiv:gr-qc/9609065; D.R.Brill,J.Louko,P.Peldan,Phys.Rev.D56(1997)3600,arXiv:gr-qc/9705012. [19]S.S.Gubser,Phys.Rev.Lett.101(2008)191601,arXiv:0803.3483[hep-th]. [20]S.S.Gubser,S.S.Pufu,J.HighEnergyPhys.0811(2008)033,arXiv:0805.2960