Charged isotropic non-Abelian dyonic black branes

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Charged

isotropic

non-Abelian

dyonic

black

branes

Yves Brihaye

a

,

∗

,

Ruben Manvelyan

b

,

Eugen Radu

c

,

D.H. Tchrakian

d

,

e a_{Physique-Mathématique,UniversitedeMons-Hainaut,Mons,Belgium}

b_{YerevanPhysicsInstitute,AlikhanianBr.St.2,0036Yerevan,Armenia}

c_{DepartamentodeFísicadaUniversidadedeAveiroandCIDMACampusdeSantiago,3810-183Aveiro,Portugal} d_{SchoolofTheoreticalPhysics–DIAS,10BurlingtonRoad,Dublin4,Ireland}

e_{DepartmentofComputerScience,NationalUniversityofIrelandMaynooth,Maynooth,Ireland}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received12March2015 Accepted16April2015 Availableonline20April2015 Editor:M.Cvetiˇc

We construct black holes with a Ricci-flat horizon in Einstein–Yang–Mills theory with a negative cosmologicalconstant,whichapproachasymptoticallyanAdSdspacetimebackground(withd≥4).These solutionsareisotropic,i.e. allspacedirectionsinahypersurfaceofconstantradialandtimecoordinates are equivalent,and possessbothelectricandmagneticfields.We findthatthebasicpropertiesofthe non-Abelian solutions are similar to those ofthe dyonic isotropic branesin Einstein–Maxwelltheory (which, however, exist in even spacetime dimensions only). These black branes possess a nonzero magneticfieldstrengthontheflatboundarymetric,whichleadstoadivergentmassofthesesolutions, asdefinedintheusualway.However,adifferentpictureisfoundforoddspacetimedimensions,wherea non-AbelianChern–Simonstermcanbeincorporatedintheaction.Thisallowsforblackbranesolutions withamagneticfieldwhichvanishesasymptotically.

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction and motivation

In recent years there has been some interest in studying the AdS/CFT correspondence [1,2], in the presence of a background magnetic field. Onthe bulk side, this corresponds to solving the Einstein-gauge field system ofequations, with suitable boundary conditions such that the AdS background is approached asymp-totically,whilethemagneticfield doesnottrivialize. Severalnew classes of such solutions have been found in this way, most of them for the case of main interest of asymptotically AdS5 con-figurations with Abelian fields. For example, the results in [3,4]

revealedtheexistenceofavarietyofunexpectedfeaturesofthese solutions;herewementiononlythattheirstudyisrelevantforthe issueofthethirdlawofthermodynamicsintheAdS/CFTcontext.

The investigation of the non-Abelian (nA) generalizations of these solutions is only in its beginning stages. Considering such configurationsisalegitimatetask,sincethegauged supersymmet-ricmodels genericallycontain Yang–Millsfields(althoughusually onlyAbeliantruncationsareconsidered).Todate,theonlycase in-vestigatedsystematicallycorrespondstothatinfour(d

=

4) space-time dimensions (see [5] for a review of these solutions). The

*

Correspondingauthor.

E-mailaddress:[email protected](Y. Brihaye).

four-dimensional nA asymptotically-AdS (AAdS) solutions exhibit manynewfeatureswhichareabsentfor



≥

0.Forexample, sta-ble1 _{solitonsandblackholes,possessing aglobalmagneticcharge,}

areknowntoexistinagloballyAdS4 backgroundeveninthe ab-senceofaHiggsfield

[6,7]

.However,theresultsin

[8,9]

showthat these Einstein–Yang–Mills (EYM) black holes solutions have also generalizationswithanonsphericaleventhorizontopology,in par-ticular witha Ricci-flat horizonanda magnetic field whichdoes not vanishasymptotically.Theysharemanyofthefeatures ofthe sphericalconfigurations in

[6,7]

,inparticulartheexistence of so-lutions stableagainstlinearfluctuations. Theonlyd

>

4 nA AAdS solutions blackholes studiedmoresystematicallysofararethose possessingsphericaleventhorizontopology

[11–14]

,thoughsome solutionswithRicci-flat horizonhavebeenstudiedin

[15,16]

.

In anunexpected development, thestudyofthed

=

4

,

5 EYM black brane solutionshas ledto thediscovery ofholographic su-perconductors andholographic superfluids, describing condensed phases of strongly coupled, planar, gauge theories [10]. Study-ing such solutions involves the construction of AAdS electrically chargedblackbranes, which,belowacriticaltemperaturebecome unstabletoformingYMhair.However,themagneticfieldofthese

1 _{Thestabilityisagainstlinearperturbations,andisnottopological.}

http://dx.doi.org/10.1016/j.physletb.2015.04.029

0370-2693/©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

configurations vanishes on the boundary, leading to a vanishing backgroundmagneticfieldforthedualtheory.

The main purpose ofthis work is to present an investigation ofd

≥

4 AAdSisotropicblackbranessupportingboth electricand magnetic nA fields. In contrast to previous studies in the litera-ture, themagnetic fieldsof thesesolutions donot vanishon the boundary,whichleadsto avarietyofinteresting features.For ex-ample,wefindthatthemassoftheseasymptoticallyAdSsolutions, asdefinedinthe usualway, always diverges,whilethe solutions donot possesaregularextremallimit.Inodd-dimensional space-times,whenaChern–Simonstermisaddedtothetotalaction,itis foundthataspecialclassofsolutionsexhibitanontrivialmagnetic fieldinthebulkwhilevanishingasymptotically.

2. The Einstein–Yang–Mills system

We consider the EYM theory in a d-dimensional spacetime, withacosmologicalconstant



= −(

d

−

2

)(

d

−

1

)/(

2L2

)

.The ac-tionis I

=



M ddx

√

−

g



1 16

π

G

(

R

−

2

)

−

1 4

∗

F

∧

F



+

Sbndy

.

(1)

TheboundarytermsSbndyincludetheGibbons–Hawkingterm

[17]

aswellasthecountertermsrequiredfortheon-shellactiontobe finite

[18]

.TheEinsteinandYang–Millsequationsderivedfromthe aboveactionare

Rμν

−

1

2R gμν

+ 

gμν

=

8

π

G Tμν

,

DμF

μν

₌

₀

_,

₍₂₎

whereDμ isthegaugederivativeandtheYang–Millsstress-energy tensor Tμν

=

1 2



F_μρI J F_νσI J gρσ

−

1 4gμνF I J ρσFI Jρσ



.

(3)

Weare interested instaticRicci-flatsolutions whichapproach asymptotically a (planar) AdSd background. Also, to simplify the picture,weshallrestrictourstudytothefollowingcase:denoting theradialandtimecoordinatebyr andt respectivelyand consid-eringthehypersurfacesparametrizedbyxi(i

=

1

,

. . . ,

d

−

2 andr

,

t

fixed),we assumethat allspacedirectionsinthesehypersurfaces areequivalent.Thusthefieldstrengthandthemetricaretakento be invariant underspacetranslations androtations in theplanes

(

xi

,

xj

)

;they arealsotime independent.Withoutanylossof gen-erality, a line element with this property can be written in the form

ds2

=

grr

(

r

)

dr2

+

g

(

r

)

d



d2−2

+

gtt

(

r

)

dt2

,

(4) with d



2

d−2

= (

dx1

)

2

+ . . . + (

dxd−2

)

2 themetriconthe

(

d

−

2

)

-flat space.

Theabove symmetry requirementsimplysome restrictions on thechoiceofthegauge group.RestrictingtoSO

(

n

)

YMfields,one findsthataYMansatz leadingtoanisotropicenergy–momentum tensorforboth evenandoddvaluesofd ispossibleforn

≥

d

+

1 only.2

Inthisworkweshallconsideran SO

(

d

+

1

)

gauge group,with

d

(

d

−

1

)/

2 SO

(

d

+

1

)

nA gauge fields represented by the 1-form potential AI J antisymmetric in I and J (with I

,

J

=

1

,

. . . ,

d

+

1)

2 _{Notethat,forevenvaluesofd,onecanconsiderinsteadagaugegroupSO(d−} 1),whichleadstoisotropicEYMbranes.Astudyofthiscasehasbeenproposedin [15](AnsatzIthere).However,thepropertiesofthosesolutionsareratherdifferent tothecaseofinteresthere.

and FI J

₌

_{d A}I J

₊

1 ˆ

gA

I K

_∧

_AK J_{, with g the}

_ˆ

_{Yang–Mills coupling.} Also,tosimplifytherelations,itisconvenienttodefine

α

2

₌

4

π

G

ˆ

g2

.

(5)

3. Embedded Abelian solutions

Before proceeding to the non-Abelian case, it is instructive to consider the dyonic black branes in Einstein–Maxwell theory, (i.e. thegauge fields takingtheir valuesin the U

(

1

)

subgroup of

SO

(

d

+

1

)

). A gauge field ansatz compatiblewiththe symmetries oftheline-element

(4)

canbeconstructedforanevennumberof spacetimedimensionsonly,d

=

2n

+

2 andreads3

AI J₁

=

w 2 0

ˆ

g x 2

_δ

I [d

δ

J d+1]

,

A I J 2

= −

w2₀

ˆ

g x 1

_δ

I [d

δ

J d+1]

,

. . . ,

AI J_2n−1

=

w 2 0

ˆ

g x 2n

_δ

I [d

δ

J d+1]

,

A I J 2n

= −

w2₀

ˆ

g x 2n−1

_δ

I [d

δ

J d+1]

,

AI J_r

=

0

,

A_tI J

=

V

(

r

)

ˆ

g

δ

I [d

δ

J d+1]

,

(6)

with w0 an arbitraryparameterwhichfixesthemagneticfield in a two plane, F₂₁I J

= . . . =

F_2n2nI J ₋₁

=

2w 2 0 ˆ g

δ

I [d

δ

J d+1]. Choosinga met-ric gauge with g

=

r2,one finds4 a black brane solution with

1

/

grr

= −

gtt

=

N

(

r

)

,where N

(

r

)

=

r 2 L2

−

M0 rd−3

+

2

(

d

−

3

)(

d

−

2

)

α

2_Q2 r2(d−3)

−

4

(

d

−

5

)

α

2_w4 0 r2

,

(7) and V

(

r

)

=

V0

−

Q

(

d

−

3

)

rd−3

,

(8)

with V0 a constant which is fixed by requiring that the electric potentialvanishatthehorizon.Apartfromw0,thissolutions pos-sesses twomoreparameters:M0andQ ,whichfixesthemassand theelectricchargedensities,respectively.

This black brane possesses an horizon at r

=

rH

>

0, where

N

(

rH

)

=

0 (and N

(

rH

)

≥

0). The Hawking temperature TH, the eventhorizonareadensity AH,the chemicalpotential

andthe electricchargedensityQe ofthissolutionare

TH

=

1 4

π



(

d

−

1

)

rH L2

−

2

α

2 rH



_2w4 0 r2_H

+

1

(

d

−

2

)

Q2 r2_H(d−3)



,

AH

=

rdH−2

,

=

1 d

−

3 Q rd_H−3

,

Qe

=

α

2 4

π

Q

.

(9)

One can easily verify that the total mass of the solutions, as definedaccordingtothecountertermprescriptionin

[18]

,diverges forany(even)d

>

4 due totheslowdecayofthemagneticfields, despitethefactthatthespacetimeisstillAAdS.Afinitemass den-sityresultswhenaboundaryterm

3 _{Theansatz(6),(4)canbeextendedtothecaseofoddd byaddinganumberof} codimensionsyμ,withAμI J=0;however,thisleadstoanisotropicconfigurations.

4 _{Aversionofthissolutionhasbeenconsideredinamoregeneralcontextin[24].} Also,itspurelymagneticlimit,Q=0,hasbeendiscussedin[3].

Fig. 1. ThereducedareaaH andmassμareshownasafunctionofreducedtemperaturetHford=6 isotropicblackbranesinEinstein–Maxwelltheory.HereandinFig. 3

thequantitiesarescaledwithrespecttothemagneticfieldontheboundary.

I_ct(YM)

= −

1 d

−

5



∂M dd−1x



−

hL 4 F I J abF I J ab

_,

₍₁₀₎

is included in (1), with hab the boundary metric and F_abI J the gauge field on the boundary. Then the boundary stress tensor

Tab

=

√2_−hδhδIab acquires a supplementary contribution from (10),

whichleadstofinitemassdensity5

M

=

(

d

−

2

)

16

π

G M0

.

(11)

Note that thisrelation holds alsofor the simplestcased

=

4, in whichcasenomattercountertermisrequired.

Onecanseethat thequantities

(9)

,

(11)

verifythefirstlawof thermodynamics(withaconstantbackgroundmagneticfield) dM

=

1

4GTHd AH

+

1

G

d Qe

.

(12)

Indiscussingthe propertiesofthesesolutions(andtheir non-Abeliangeneralizations), it isconvenient to work withquantities scaled withrespect tothe magneticfield ina twoplane asfixed bytheparameter w0: aH

=

AH wd₀−2

,

tH

=

TH w0

,

μ

=

G M wd₀−1

,

q

=

Qe wd₀−2

.

(13) As seen in Fig. 1, the properties of the solutions with a back-groundmagnetic field are not really sensitive to the presence of an electriccharge sincethe constant-q curves preserve theq

=

0 shape,whichisapproachedasymptoticallyforlargetH.These dy-onicblackbranespossessaregularextremallimitTH

=

0,withan

AdS2

×

Rd−2 nearhorizongeometry.Aninterestingfeatureisthat thetotal mass ofthe d

>

4 solutionsis allowed to take negative values.Thiscaneasilybeseenintheextremalcase, alimitwhich isapproachedfor Q

=

r d−2 H √ 2αL

√

(

d

−

2

)(

d

−

1

)



1

−

4α2L2w 4 0 (d−1)r4_H .The ex-tremalsolutionshaveamass

M

=

(

d

−

2

)

2

₍

_d

₋

₁

₎

8

(

d

−

3

)

π

L2



1

−

4

(

d

−

4

)

α

2_L2_w4 0

(

d

−

5

)(

d

−

2

)(

d

−

1

)

r4_H



,

(14)

whichbecomesnegativefor ₄(d−5)(d−2)

(d−4)α2_L2

<

w4 0 r4 H

≤

(d−1) 4α2_L2.

5 _{Asusual inthis context, the mass is the chargeassociated with the} time-translationsymmetryoftheboundarymetric.

4. Non-Abelian solutions

AsimplenAansatzleading toan isotropiclineelementcanbe constructed foranyd

≥

4,interms ofamagnetic potential, w

(

r

)

andanelectricone V

(

r

)

A_iI J

=

w

(

r

)

ˆ

g

δ

I [i

δ

J d−1]

,

ArI J

=

0

,

AtI J

=

V

(

r

)

ˆ

g

δ

I [d

δ

J d+1]

.

(15)

Unfortunately, no AAdS exact solutions of the EYM equations seemstoexistinthiscase.However,thesystempossessesasimple globallyregularLifshitz-typeconfigurationwith

ds2

=

c1 dr2 r2

+

c2r 2_d

_

2 d−2

−

r2zdt2

,

w

(

r

)

=

u0r

,

V

(

r

)

=

0

,

(16) where c1

=

4

α

2

(

d

−

2

)

p2

,

c2

=

2

α

2

₂

₍

_d

₋

₃

₎

_{− (}

_d

₋

₂

₎

_p2

₎

(

d

−

2

)

2_p2 u 2 0

,

z

=

(

d

−

3

)((

d

−

2

)

p 2

₊

₂

₎

2

(

d

−

3

)

− (

d

−

2

)

p2

>

1

,

(17)

hereu0

=

0 isanarbitraryconstantwhile p isaparameterrelated tothecosmologicalconstantby



= −

(

d

−

2

)

p 2 2

α

2

₍₍

_d

₋

₂

₎

_q2

₋

₂

₍

_d

₋

₃

₎₎

×



(

d

−

2

)

p4

+ (

d

−

2

)(

d

−

3

)(

d

(

d

−

6

)

+

4

)

p2

+

4

(

d

−

3

)

2

(

d

−

1

)



,

and obeying the condition p

<



2

(

d

−

3

)/(

d

−

2

)

. The solution

(16) possesses the Lifshitz scaling symmetry t

→ λ

zt

,

xi

→

λ

xi

,

r

→

r

/λ

andgeneralizes thed

=

4 EYM solutionofRef.[19]

to higher dimensions. As discussed there, in this case the field equationspossessblackbranesolutionswitharegularhorizon ap-proachingthebackground

(16)

asr

→ ∞

.Weexpecttheexistence ofsimilarblackbranesolutionsford

>

4 aswell.

ReturningtothecaseofsolutionswithAdSasymptotics,itturns out convenientforthenumericalconstruction tochoose ametric ansatzoftheform

grr

=

1 N

(

r

)

,

g

=

r 2

_,

gtt

= −

N

(

r

)

σ

2

(

r

),

with N

(

r

)

=

r2 L2

−

m

(

r

)

rd−3

.

(18) InsertingthisansatzintotheEinsteinandYang–Millsequations yields four equations ofmotion6 _{for m}

₍

_r

_),

_σ

₍

_r

_),

_w

₍

_r

₎

_{and V}

₍

_r

₎

(a primedenotes _drd): m

=

2

α

2_rd−4



1 d

−

2 r2V2

σ

2

+

N w 2

₊

(

d

−

3

)

2r2 w 4



,

σ



=

2

α

2 r

σ

w 2

_,

w

+



d

−

4 r

+

N N

+

σ



σ



w

− (

d

−

3

)

w 3 r2_N

=

0

,

V

+



d

−

2 r

−

σ



σ



V

=

0 (19)

Thelastequationaboveimpliestheexistenceofthefirstintegral V

=

σ

Q

rd−2

,

(20)

withQ aconstantfixingtheelectricchargeofthesolutions. Theequationsofmotion

(19)

areinvariantunderthreescaling transformations(invariantquantitiesarenotshown):

(

I

)

σ

→ λ

σ

,

V

→ λ

V

,

(

II

)

r

→ λ

r

,

m

→ λ

d−1m

,

w

→ λ

w

,

V

→ λ

V

,

(

III

)

r

→ λ

r

,

m

→ λ

d−3m

,

L

→ λ

L

,

V

→

V

λ

,

α

→ λ

α

,

(21)

where

λ

representsthepositive(real)scalingparameter.Using

(

I

)

, wesettheboundaryvaluesofthemetricfunction

σ

(

r

)

toone,so that the metric will be asymptotically (locally)AdS. We are free to use

(

II

)

to set theasymptotic value of the magnetic potential

w

(

r

)

to an arbitrary(non-vanishing) value (equivalently, one can usethis symmetry to fix the value of the electric charge or the horizonradiusofthesolution,sayrH).Finally,thesymmetry

(

III

)

canbe used to fixthe value of theAdS radius L or thevalue of thecouplingconstant

α

;formostoftheworkinthispaperweset

α

=

1 (thuswetreatL asaninputparameter).

Denoting theposition ofthe horizonof the blackbrane solu-tionsbyrH,wehavetoimpose N

(

rH

)

=

0 (and N

(

rH

)

≥

0)while theothermetricfunctionsstaystrictlypositive.Anonextremal so-lutionhasthefollowingexpressionneartheeventhorizon:

m

(

r

)

=

r d−1 H 2L2

+

m

(

rH

)(

r

−

rH

)

+

O

(

r

−

rH

)

2

_,

σ

(

r

)

=

σ

H

+

σ



(

rH

)(

r

−

rH

)

+

O

(

r

−

rH

)

2

,

w

(

r

)

=

wH

+

w

(

rH

)(

r

−

rH

)

+

O

(

r

−

rH

)

2

,

V

(

r

)

=

V

(

rH

)(

r

−

rH

)

+

O

(

r

−

rH

)

2

,

(22) where

6 _{Oneextraequationcontainingthesecondderivativesofthemetricfunctions} m(r),σ(r)isalso found.However,onecanshowthatthisconstraintequationis implicitlysatisfiedforthesetofboundaryconditionschosen.

m

(

rH

)

=

2

α

2_Q2

(

d

−

2

)

rd_H−2

+

α

2

₍

_d

₋

₃

₎

w4H rd_H−6

,

w

(

rH

)

=

(

d

−

3

)

L2w3_H r3_H



d

−

1

−

α2L2 r4_H

(

2Q2 (d−2)r2_H(d−4)

+ (

d

−

3

)

w 4 H

)

,

V

(

rH

)

=

Q rd_H−3

,

σ



(

rh

)

= −

2

α

2

(

d

−

3

)

2L4

σ

Hw6H r7 H



d

−

1

−

α2L2 r4 H

(

2Q2 (d−2)r_H2(d−4)

+ (

d

−

3

)

w 4 H

)

,

(23)

withwH and

σ

H arbitraryconstants.

The AdSboundary isreached asr

→ ∞

.We are interested in configurations with w

(

r

)

→

w0

=

0,such that the magnetic field ontheboundaryisnonvanishing,F_{i j}I J

= −

1_gˆw2₀

δ

_[iI

δ

_j]J.A straightfor-wardbutcumbersomecomputation leadstothefollowinggeneral asymptoticexpression ofthe solutions asr

→ ∞

(note the pres-enceoflog termsforanoddvalueofthespacetimedimension): m

(

r

)

=

M0

+

α

2_Ld−5_wd−1 0 d

−

2

×

log

(

r L

)



6

δ

d,5

−

40

δ

d,7

+

567 4

δ

d,9

+ . . .



+

α

2

(

d

−

3

)

(

d

−

5

)

w 4 0rd−5

(

d

−

6

)

−

2

α

2L2

(

d

−

3

)

2

(

d

−

6

)

(

d

−

5

)

2

₍

_d

₋

₇

₎

w 6 0rd−7

(

d

−

8

)

+ . . . ,

σ

(

r

)

=

1

−

4 3

α

2_w6 0log2

(

r L

)

L4 r6

δ

d,5

−

α

2

(

d

−

3

)

2L4w60 3

(

d

−

5

)

2_r6

(

d

−

6

)

+ . . . ,

w

(

r

)

=

w0

+

J rd−3

+

wd₀−2Ld−3 rd−3

×

log

(

r L

)



−δ

d,5

+

3

δ

d,7

−

27 4

δ

d,9

+ . . .



−

d

−

3 d

−

5 w3₀L2 2r2

(

d

−

6

)

+

3

(

d

−

3

)

2 8

(

d

−

5

)(

d

−

7

)

w5₀L4 r4

(

d

−

8

)

+ . . . ,

V

(

r

)

=

V0

−

Q rd−3

+ . . . ,

(24)

The seriestruncatesfor anyfixed dimension,with newterms entering atevery new even value ofd, as denoted by the step-function (

(

x

)

=

1 provided x

≥

0, andvanishes otherwise). The constants w0,M0,V0 and J intheaboveexpressions arefree pa-rameterswhicharefixedbynumerics.

AsintheAbeliancase,weexpecttheparameterM0 toencode themassdensityofthesolutions,whichisstillgivenby

(11)

. How-ever, a rigorousproof ofthis statement israther difficult,due to the complicatedasymptoticbehavior of themetric functions. For

d

=

5,aregularizedboundaryenergy–momentumtensorandmass arefoundbyincludingin

(1)

thefollowingmattercounterterm IYM_ct

= −

log

(

r L

)



∂M d4x



−

hL 4 F I J abF I J ab

_.

₍₂₅₎

Fig. 2. Theprofilesofatypicald=6 Einstein–Yang–Millsisotropicblackbrane so-lutionareshownasafunctionsoftheradialcoordinater.

We have found that the boundary counterterm (10) regularizes alsothemassofthed

=

6 solutions.Inbothcases,thisresultsin theexpression

(11)

forthemassdensityoftheblackbranes.(Note that (11)holdsalsoford

=

4,inwhich casenomatter countert-ermisnecessary.)However,theabovesimplecountertermfailsto regularizealldivergenciesintheexpressionofM ford

>

6.Thusa moregeneralmattercountertermthan

(10)

isrequiredinthed

>

6 case. Wefind thatfor anyd

≥

4, themassof thesolutions com-putedbyintegratingthefirstlawequation

(12)

,coincideswiththe relation

(11)

withgoodaccuracy.

Otherquantities whichenterthethermodynamicsofthe solu-tionsaregivenby

AH

=

rd_H−3

,

TH

=

1 4

π

N 

₍

_r H

)

σ

(

rH

),

=

V0

,

Qe

=

α

2 4

π

Q

.

(26)

Solutions interpolating between the near horizon expansion

(22) and the far field asymptotics (24) are constructed numer-ically, using a standard Runge–Kutta ordinary differential equa-tions solver. In this approach we evaluate the initial conditions atr

=

rH

+

10−5, forglobaltolerance 10−14,adjusting for shoot-ing parameters and integrating towards r

→ ∞

(thus we have restrictedourstudytothe regionoutsidetheeventhorizon).The equationswereintegratedforallvaluesofd betweenfourandten; thussimilarsolutionsareexpectedtoexistforanyvalueofd.

For a given d, we have considered a range of values for

(

rH

,

wH

,

Q

)

,theparameters

σ

H and M0

,

V0

,

J resultingfromthe numericaloutput.SinceEqs.(19)areinvariantunderthe transfor-mation w

→ −

w,onlyvaluesof wH

>

0 areconsidered.Also,we havestudiedmainlythecasewheretheAdSlengthscaleissetto one, L

=

1.The profile ofa typical d

=

6 non-Abeliansolutionis shownin

Fig. 2

.(Therewe havedisplayed alsothemassfunction densitym

(

r

)

regregularizedviathecounterterm

(10)

.)

We havefound that the nA solutions sharemostof the basic propertiesoftheEinstein–Maxwellconfigurationsdiscussedabove. In particular, thepresence of an electric charge doesnot change qualitativelythe generalpicture.Also, a numberofbasicfeatures oftheseblackholesaresimilartothoseoftheknownd

=

4 (purely magnetic) configurationsin

[8]

.Thiscan be understoodby notic-ing that, forour choice of the ansatz, the magnetic and electric potentials interact only via the spacetime geometry. As a result, theseblackbranescan bethought ofasnonlinearsuperpositions

ofpurelyelectricReissner–Nordström–AdSsolutions(i.e. thelimit

w0

=

0 in (6), (7)) andpurely magneticnA configurations7 with

V

(

r

)

=

0.Thiscanbeseenin

Fig. 3

,whereweplottheevent hori-zonareaandthemassofd

=

5 solutions,forseveral(fixed)values oftheelectriccharge;notethatinthatplotthequantitiesare nor-malizedw.r.t. tothemagneticfieldontheboundary,asdefinedby

(13),whichremaininvariantunderthetransformation(ii)in

(21)

. Onecaneasilyseethatthecorrespondingq

=

0 curvesaregeneric. Also,asintheAbeliancase,wehavenoticedtheexistenceofd

>

5 solutions withanegative totalmass,M0

<

0,see

Fig. 2

(solutions withM0

=

0 doalsoexist).

However,thelimitingbehavior oftheEYMsolutionisvery dif-ferentfromtheAbeliancase,thelimit TH

→

0 beingsingularthis time.Thiscanbeunderstoodbynoticingthatthenon-linearityof the YM equation forthe magnetic potential implies the absence ofaAdS2

×

Rd−2 nearhorizongeometryasasolutionofthefield equations.

5. Non-Abelian black branes in odd dimensions with a Chern–Simons term

Inodd spacetimedimensions,the usualgauge fieldaction can be augmentedwithaChern–Simons (CS)term. Suchaterm typi-cally enters the actionofgauged supergravities, the caseof

N =

8

,

d

=

5 modelwithagaugegroupSO

(

6

)

,beingperhapsthemost interesting.8

The expression of the CS Lagrangean for the case d

=

5 dis-cussedinwhatfollows,is9

LCS

=

κ

I1···I6



FI1I2

∧

_FI3I4

∧

_AI5I6

− ˆ

g FI1I2

∧

_AI3I4

∧

_AI5J

∧

_AJ I6

+

2 5g

ˆ

2_AI1I2

∧

_AI3J

∧

_AJ I4

∧

_AI5K

∧

_AK I6



,

(27)

with

κ

anarbitraryparameter,theCScouplingconstant.10

Onecaneasilyshow thattheAbelianconfiguration

(7)

still re-mains a solution in the presence of a CS term11; however, the situationisdifferentfornon-Abelianfields.Thesesolutionscanbe studiedwithinthesameansatz

(15)

,

(18)

;theequationsformetric functionsm

(

r

)

,

σ

(

r

)

arestillvalid,sincetheCStermdoesnot con-tribute tothe energy–momentumtensor, whilethe equationsfor thegaugepotentialscontainnewtermsencodingadirect interac-tionbetweenmagneticandelectricpotentials:

w

+



d

−

4 r

+

N N

+

σ



σ



w

− (

d

−

3

)

w 3 r2_N

−

κ

(

d 2

₋

₁

₎

(

d

−

2

)

wd−3V N

σ

rd−4

=

0

,

(28)

7 _{Oneinteresting feature isthe absence ofsolutionswith nodes ofthe} mag-neticpotential.Thiscanbeanalyticallyprovenbyintegratingtheequationforw,

(Nσrd−4_w₎_{= (d−3)w}3_σrd−6_,betweenr

Handsomer;obtainingw>0 forevery

r>rH.Inasimilarway,onecanprovethatthemetricfunctionσ(r)monotonically

increasestowardsitsasymptoticvalue.

8 _{NotethatasimpleEYMCStheorydoesnotseemtocorrespondtoaconsistent} truncationofanysupergravitymodel.However,weexpectthatthebasicproperties ofoursolutionswouldholdalsointhatcase(seeRef.[23]forastudyofnAinof theN=4+,d=5 gaugedsupergravitymodel,whichcontainsalsoaCSterm).

9 _{TheexplicitexpressionoftheCSLagrangeanford=7}

,9 canbefounde.g. in Ref.[20].

10 _{Thevalueofκisfixedinsupersymmetrictheories,butinthisworkwetreatκ} asafreeinputparameter.

11 _{Notethesituationchangesforanisotropic dyonicAbelianblackbranes,inwhich} casetheinclusionofaU(1)CStermleadstovarietyofnewinterestingproperties, seee.g.[4].

Fig. 3. The reduced area aHand massμare shown as functions of the reduced temperature tH for d=5 isotropic black branes in Einstein–Yang–Mills theory.

Fig. 4. Left: Theasymptoticvalueofthemagneticgaugepotentialw0isshownasafunctionofitsvalueattheeventhorizonw(rH)ford=5 solutionsinEinstein–Yang–

Mills–Chern–Simonstheory.Right: ThescaledhorizonareaAH isshown–forseveralvaluesofκ–asafunctionofthescaledtemperatureTH ford=5 Einstein–Yang–

Mills–Chern–Simonssolutionswithavanishingmagneticfieldontheboundary.

V

=

σ

rd−2

(

Q

+

κ

(

d2

−

1

)

(

d

−

2

)

w

d−2

_),

₍₂₉₎

withQ anintegrationconstant.

Byusinga similar approachto that described above, wehave studiedfamilies ofd

=

5 solutionsoftheEYM–CSmodelina sys-tematicway.12

TheEYM–CSsolutionspossessanearhorizonexpansionsimilar to

(22)

,whiletheirleadingordertermsinthefarfieldexpression isstillgivenby

(24)

,

κ

enteringthroughthelowertermsonly.

Wehavefoundthatallbasicpropertiesofthesolutionswithout aCS termare retainedin thiscase. However, some newfeatures occursaswell,themostinterestingbeingtheexistenceof config-urationswithw

(∞)

=

0.

Foraspecialsetofeventhorizondata,onefindssolutionswith vanishing magneticfield onthe AdS boundary (although w

(

r

)

is nonzero in the bulk). From (24), this implies that in this case, asr

→ ∞

the massfunctionm

(

r

)

approachesafinite value.This featureisillustratedin

Fig. 4

(left),whereweplotw0,the asymp-toticvalueofthemagneticgaugepotentialw,asafunctionofthe valueofthemagneticpotentialonthehorizonforfixed valuesof

12 _{Weexpectthe propertiesofthefive-dimensional solutionstobegeneric.} In-deed,thisissupported bythepreliminaryresultswehavefoundforEYMCS solu-tionsind=7 spacetimedimensions.

κ

,

Q

,

rH andL (thespecialvalue ofw

(

rH

)

whichcorrespondto

w

(

∞)

=

0

)

aremarkedwithdots).

Naively, this resembles the solutions describing holographic

p-wavesuperconductors andsuperfluids whichhave been exten-sivelystudiedinrecentyears,startingwiththeseminalwork

[10]

. However, theoverallpictureisratherdifferentfortheEYMCS so-lutions obtained here. First, in contrast to the EYM solutions of Refs.[10,15,16],theseconfigurationsdonotemergeasa perturba-tionoftheRN–AdSAbeliansolution.13_{Second,thegeneralpattern}

oftheEYMCSblackbraneswithavanishingmagneticfieldonthe boundary isdifferent fromtheone corresponding to nA configu-rations withouta CS term. Forexample,asseen in

Fig. 4

(right), the EYMCSblack braneswithgiven

(

α

,

κ

)

form two branchesof solutions. These branches extend up to a maximal value of the Hawkingtemperatureandhorizonarea,wheretheyjoin(notethat the quantitiesplottedare scale invariant under(ii) in(21)by an appropriatecombinationwiththeelectriccharge).

Interestingly(andinstrongcontrasttothepureEYMcase dis-cussedabove),thelimit TH

→

0 correspondstoextremalsolutions 13 _{Thatis, whentreating w(r) asa smallperturbation around the electrically} chargedRN–AdSblackbrane,onefindsthatthesolutionoftheYM–CSlinearized equation(28)possessesanessentiallogarithmicsingularityatthehorizon.However, wehaveverifiedthattheEYMCShairysolutionswithw(∞)=0 are thermodynam-icallyfavouredovertheRN-AdSAbelianconfigurations,i.e.theyminimizethefree energyforthesameQ,TH.

witharegular horizon. Suchconfigurations possessan AdS2

×

R3 nearhorizongeometry,with14

ds2

=

v1

(

dr2 r2

−

r 2_dt2

₎

₊

_v 2d



32

,

and w

(

r

)

=

w0

,

V

(

r

)

=

qr

,

(30) (wheretheredefinitionr

−

rH

→

r isimplicitly assumed)and v1

=

2 3



8 L2

−

α

2_w2 0 16

κ

2

₍

_Q

₊

₈

_κ

_w3 0

)

2



−1

,

v2

= −

4

κ

(

Q

+

8

κ

w3₀

)

w0

,

and q

=

v1

(

Q

+

8

κ

w 3 0

)

v3₂/2

.

(31)

Given

κ

,

α

andL,thisconfigurationpossessesonesingle free pa-rameter,theconstants Q ,w0satisfyingthealgebraicequation

512

κ

2

₍

_Q

₊

₈

_κ

_w3

0

)

2

+

α

2L2w30

(

Q

−

4

κ

w30

)

=

0

.

(32) Wenote thatthe overallpicturepossesses anontrivial depen-denceonthevalueoftheCScouplingconstant,withtheexistence of a minimal value of

κ

allowing for a vanishing magnetic field ontheboundary.We hopeto returnelsewherewithasystematic studyoftheEYMCSconfigurations,inamoregeneralcontext.

6. Conclusions

Inthiswork wehave constructedisotropicblack branesinan

AdSd backgroundpossessingbothelectricandmagnetic SO

(

d

+

1

)

non-Abelianfields. The solutions were obtainedby using a com-binationofanalyticalandnumericalmethods.Severalbasic prop-ertiesof thesesolutions ind

>

4 canhardly be anticipated from thestudyoftheirfour-dimensional counterparts.Forexample,the magnetic field of the EYM solutions does not vanish asymptoti-cally. As a result their mass – defined in the usual way – al-ways diverges. However, solutions with a finite mass exist – in odd spacetime dimensions – when supplementing the action by aChern–Simonsterm.

Therearevariouspossiblenaturalextensionsofthiswork. Per-haps the most interesting one would be to study the transport propertiesofthesesolutions.Investigationofthethermodynamics oftheblackbranesis anotherimportantproblem.Here we men-tion only that the heat capacity is always positive for the EYM blackholes inacanonical ensemble.Asa result,these configura-tionsarealwaysthermodynamicallylocallystable,afeatureshared withthevacuumsolutions.Finally,note thatthe YMansatzused in thiswork is not the mostgeneral one leading to an isotropic blackbrane; forinstance the components ofthe connection (15)

14 _{Thisconfigurationcanbegeneralizedforany(odd)d≥5;however,therelations} aremuchmorecomplicatedinthegeneralcase.

taketheirvaluesinthealgebraofSO

(

d

−

1

)

×

U

(

1

)

andnotinthe fullalgebraofSO

(

d

+

1

)

.ThefullySO

(

d

+

1

)

YMansatzcanbe writ-tenintermsoftwomagneticpotentialsandtwoelectricpotentials, andisexpectedtoleadtoamorecomplicatedpicture.15

Acknowledgements

We acknowledge M.Ortaggio forbringing Ref. [24] to our at-tention. The work of Y.B. was supported in part by an ARC con-tract AUWB-2010/15-UMONS-1.E.R. gratefullyacknowledges sup-port from the FCT-IF programme and CIDMA strategic project UID/MAT/04106/2013.

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15 _{Thed=5 EYMCScounterpartoftheseconfigurationswithasphericalhorizon} topologyhavebeenstudiedin[14,21];seealso[22,20]forthe=0 limitofthese solutions.