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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,
daInstitutfürPhysik,UniversitätOldenburg,Postfach2503, D-26111Oldenburg,Germany
bDepartamentodeFisicadaUniversidadedeAveiroandI3NCampusdeSantiago,3810-183Aveiro,Portugal cDepartmentofTheoreticalPhysics,TomskStatePedagogicalUniversity,Russia
dBLTP,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.
©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.
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:olga.kichakova@uni-oldenburg.de(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
0370-2693/©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3.
2 L2
=
8π
G Fμα(a)Fν(aβ)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(
dθ
2+
sin2θ
dφ
2)
−
σ
2(
r)
N(
r)
dt2,
with N(
r)
=
1−
2m(
r)
r+
r2 L2,
(2)forthemetric,and
A
=
1 2gˆ
w(
r)
τ
1dθ
+ (
cosθ
τ
3+
w(
r)
sinθ
τ
2)
dφ
,
(3)forthe gauge fields (with g the
ˆ
gauge coupling constant andτ
a thePaulimatrices).Thenthefieldequations(1)
reducetom
=
α
2 N w2+
V 2(
w)
2r2,
σ
=
2α
2σ
w2 r,
w+
N N+
σ
σ
w+
w V(
w)
r2N=
0 (4)(withV
(
w)
= (
1−
w2)
),wherewehavedefinedthecouplingcon-stant
α
2=
4π
Gˆ
g2
.
(5)We wantthe metric (2)to describe a nonsingular, asymptoti-callyAdS spacetimeoutsidea horizonlocatedatr
=
rH>
0 (hereN
(
rH)
=
0 isonlyacoordinatesingularitywhereall curvature in-variantsarefinite).Therearetwoexplicitsolutionssatisfyingthese assumptions.If w2(
r)
≡
1,then N(
r)
=
1−
rH r1
+
r 2 L2+
rrH L2+
r2 H L2,
σ
(
r)
=
1,
(6) whichisjusttheSchwarzschild-AdS(SAdS)metric,withvanishing YM curvature and mass M=
rH2
(
1+
r2
H
L2
)
. A different solution is foundforw(
r)
≡
0,withN
(
r)
=
1−
rH r1
+
r 2 L2+
rrH L2+
r2H L2−
α
2 rrH,
σ
(
r)
=
1.
(7) ThisdescribetheembeddedAbelianmagnetic Reissner–Nordström-AdS(RNAdS)metric,withmassM=
2r1H
(
α
2+
r2 H(
1+
r2 H L2))
,andunit magneticcharge.ingtemperatureandeventhorizonareaofthesolutions(withthe entropy S
=
AH/
4G) TH=
1 4π
σ
H 3r4H+
r2HL2−
α
2L2(
1−
w2 H)
2 r3HL2,
AH=
4π
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 J2+
L2(
1−
w2 0)
2)
2L2r+ . . . ,
σ
(
r)
=
1−
1 2r4α
2J2+ . . . ,
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=
QM2τ3gˆ (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
(withλ
apositivescalingparameter).Thesymmetry(
i)
wasusedtosetσ
(
r)
→
1 as r→ ∞
. The symmetry(
ii)
can be used to fix the valueoftheAdSradiusL orthevalueofthecouplingconstantα
; inthisworkwesetL=
1 andtreatα
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 thecanonicalensembleisC
=
∂
M∂
TH QM=
TH∂
S∂
TH QM>
0.
(12)Asrecentlyfoundin
[8]
,thegenerichairysolutionssatisfythefirst lawofthermodynamics dM=
σ
H 2r2 HL2 3r4H+
r2HL2−
α
2L2(
1−
w2H)
2 drH+
α
2w 0J L2 dw0=
THdS+
d QM,
(13) where= −
α
2J/(
2w0L2
)
isthemagnetostaticpotential.The most remarkable feature ofthe AdS EYM BHs is perhaps that the value of the parameter w0 in
(10)
(i.e. thevalue of themagneticcharge)isnot fixedapriori.Thatis,forgiven
α
andany horizonsize(assetbytheinputparameterrH),solutionsarefoundFig. 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 (notehow-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) andonenode solutions (for w0
<
0) only. The configurations withhighernumberofnodesrepresentexcitedstatesandarethereforeignored inwhatfollows.Moreover, the solutionswithnodesare unstable in linearized perturbation theory. A detailed discussion of these aspectstogetherwithalargesetofreferencescanbefoundin
[7]
. 3. Thethermodynamicsofnon-Abelianblackholes3.1.Theunitmagneticchargesolutions
Among all sets ofnA BHs, of particularinterest are those so-lutionspossessingaunitmagneticcharge, i.e. with w0
=
0 inthefarfieldexpansion
(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)
=
2πL2 3 p2− (where we note p±
=
1
+ (
2√
3 αL)
2±
1 and q±=
1±
1
− (
6Lα)
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 Thethermodynamics ofRNAdS solution hasbeen 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 +)
) andbecomesnegativeforlargervaluesofTH.
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 withα
,the points P1andP2 coincidingforα
=
L/
6,wherethesecondbranchofsolutions disappears.Then,for
α
>
L/
6 onlyone single branch ofsolutionsisfound,seetheinsetinFig. 1
(left)(fromthere,itis alsoclear that,for anyα
,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, withα
max1
.
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. Forα
<
L/
6, thisRNAdSsolutionislocatedonthefirstbranch(forT<
T(Hc))or on thethird branch(for highertemperatures). Thus we conclude thatall unitmagneticcharge nAsolutionsare globallythermody-2 Anexactparticle-likesolutionwithw
0=0 isknownforα=0 (i.e. theprobe limit–YMfieldsinafixedAdSbackground),andhasamagneticpotentialw(r)=
1/
1+rL22 [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.
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 areshowninFig. 1
(right). For0<
α
<
L/
6, theunstableRNAdSsolutionsare locatedonthesecond andthird branchesina smallregion aroundthepoint P2 (seeFig. 1
(left)).Furtherincreasing
α
,theunstablesolutionsmovetosmaller tem-peratures, the point P0 being approached forα
→
α
max, with W(
r)
≡
0 inthat limit. Also, an analytic estimate for the criticalhorizonradiusoftheRNAdSBHs, 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.3Thepicturewehavefoundisratherdifferentinthiscase.First, thesehairyBHs donotemerge asperturbations4 oftheSAdS
so-3 These arethe naturalAdScounterpartsofthewell-knownasympoticallyflat solutionsin[6],whichnecessarilyhavew2
0=1.Also,notethattheseareunstable configurationswithrespecttosmallperturbations,themagneticpotential possess-ingatleastonenode.
4 Thatis,onecanshowthatthelinearizedYMequationin(4)withw(r)= −1+
W1(r)inaSAdSbackgrounddoesnotpossesssolutionswhicharefiniteatr=rH
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 withα
(the well-known F(
TH)
diagram fortheSAdS blackholes isalsoshownthere;notethatthevacuumsolutionsdonotpossess adependenceonα
). 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, possessinga nonvanishing (and noninteger) magnetic charge.5 As shown in
Fig. 4,thethermodynamicsofthegenericsolutionsresemblesthat ofthe SAdSBHs,withtheexistenceoftwo branchesofsolutions.
5 Strictlyspeaking,theseEYMBHsdonotviolatethenohairconjecture(werecall 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 apartfromtheonein
Fig. 4
. 4. FurtherremarksItiswellknownthattheRNAdSBHsolutionspossessavariety 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 Arelatedexampleisthecaseofasymptoticallyflat,magneticallychargedRN BHs embeddedinEinstein–Yang–Mills–Higgs theory, whichdevelopnA hair for some rangeoftheparameters[15].For <0,it isthe asymptoticstructureof spacetimewhicheffectivelyreplacestheHiggsfield.
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(whichisageneralizationoftheconstantw0enter-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 andn
=
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(seee.g.[18]forvacuumsolutions),thecorrespondingmetricansatzbeing
ds2
=
dr 2 N(
r)
+
r 2(
dθ
2+
f2 k(θ )
dφ
2)
−
σ
2(
r)
N(
r)
dt2,
where N
(
r)
=
k−
2mr(r)+
rL22, with k adiscrete parameter which fixes the topology of the horizon. For k=
1, f1=
sinθ
and themetricansatz
(2)
isrecovered;thesolutionswithk=
0 have f0=
θ
andareplanarBHs,approachingasymptoticallyaPoincarépatch oftheAdSspacetime;finally,thesolutionswithk= −
1 possessa hyperbolichorizon,with f−1=
sinhθ
.Thecorrespondinggeneral-kYMansatzhasbeendisplayedin
[17]
andlooksverysimilarto(3)
, with7 However,suchconfigurationspossessthe sameamountofsymmetryfor any topologyofthehorizon.
A
=
1 2gˆ
w(
r)
τ
1dθ
+ (
dfk(θ )
dθ
τ
3+
fk(θ )
w(
r)
τ
2)
dφ
.
The equations of the model are still given by (4), with V
(
w)
=
k
−
w2(
r)
inthegeneralcase.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 impliesm
(
r)
=
M−
α
2k2/(
2r)
,σ
(
r)
=
1,whichisthe(planar)SAdSBHfor 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.8Moreover,thek=
0,
−
1 solutionsare(locally)thermallystable, since C
>
0 (similarresultshavebeenfoundfor othervaluesofthemagneticcharge,asfixedbytheparameterw0intheasymptoticexpansion
(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 dimensionalspacetimewithalineelementds2
=
L2(
dθ
2+
f2k
(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 ForthedatainFig. 5,wesettoonetheareaofthe
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
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