Bertotti-Robinson-type solutions to dilaton-axion gravity

PHYSICAL REVIEW D, VOLUME 63, 124011

Bertotti-Robinson-type solutions to dilaton-axion gravity

Gerard Clement*Laboratoire de Physique Theórique LAPTH (CNRS), B.P. 110, F-74941 Annecy-le-Vieux cedex, France

Dmitri Gal’tsov†

Laboratoire de Physique Theórique LAPTH (CNRS), B.P. 110, F-74941 Annecy-le-Vieux cedex, Franceand Department of Theoretical Physics, Moscow State University, 119899, Moscow, Russia

~Received 6 February 2001; published 21 May 2001!

We present a new solution to dilaton-axion gravity which looks like a rotating Bertotti-Robinson~BR!universe. It is supported by an homogeneous Maxwell field and a linear axion and can be obtained as anear-horizon limit of extremal rotating dilaton-axion black holes. It has the isometrySL(2,R)3U(1) whereU(1) is the remnant of theSO(3) symmetry of BR broken by rotation, whileSL(2,R) corresponds to theAdS2 sector which no longer factors out of the full spacetime. Alternatively our solution can be obtained fromtheD55 vacuum counterpart to the dyonic BR universe with equal electric and magnetic field strengths. Thederivation amounts to smearing it inD56 and then reducing toD54 with dualization of one Kaluza-Kleintwo-form in D55 to produce an axion. Using a similar dualization procedure, the rotating BR solution isuplifted toD511 supergravity. We show that it breaks all supersymmetries ofN54 supergravity inD54, andthat its higher dimensional embeddings are not supersymmetric either. But, hopefully it may provide a newarena for conformal mechanics and holography. Applying a complex coordinate transformation we also derivea BR solution endowed with a NUT parameter.

DOI: 10.1103/PhysRevD.63.124011 PACS number~s!: 04.20.Jb, 04.50.1h

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I. INTRODUCTION

The discovery of AdS conformal field theory~CFT! du-alities @1# ~for a review see@2#! stimulated search for geometries containing AdS sectors. Recall that AdS geometypically arises as the near horizon~throat! limit of staticcharged Bogomol’nyi-Prasad-Sommerfield~BPS! blackholes and/orp-branes in various dimensions. The known eamples of AdS-CFT correspondence make use of geomeAdSn3K with K being some compact manifold. For rotatinblack holes or branes the near horizon limit genericallydifferent: one finds a nontrivial mixing of AdSn and K. Nev-ertheless, the asymptotic geometry relevant for hologramay remain unaffected by rotation forn>3, so the AdS-CFT correspondence applies directly. This is not so incase of AdS2 @3,4#. While for nonrotating extremal chargeblack holes the near-horizon geometry factorizes as A23S2, in the rotating case the azimuthal coordinates mix wthe time coordinate in such a way that the AdS23S2 geom-etry is not recovered asymptotically. Therefore, in highermensions the rotation parameter just adds a specific extion mode in the dual theory@5–8#, but in four dimensions itapparently leads to more serious consequences, whose nis not clear yet. Additional problems in the four-dimensioncase are related to the fact that the AdS2 holography is lesswell understood than the higher-dimensional examp@9–13# ~for some recent progress in this direction see@14#!.

Bardeen and Horowitz@3# observed that violation of thedirect product structure AdS23K due to rotation inD54 ismanifest already for vacuum Kerr black holes. In the thr

*Email address: [email protected]†Email address: [email protected]

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geometry of the extreme Kerr~and Kerr-Newman! four-dimensional black hole one finds a mixing of azimuthal atime coordinates which does not disappear in the asymptregion, but grows infinitely instead leading to the singunature of the conformal boundary. Nevertheless, the geetry still share some important features with AdSn3K space-times such as~partial! confinement of timelike geodesics andiscreteness of the Klein-Gordon particle spectrum ongeodesically complete AdS patch. But, apart from the fthat the near-horizon spacetime is no longer the direct pruct of AdS2 with something, the geometry is also plaguedcumbersomeu-dependent factors which modify the spectruof the angular Laplacian. It turns out that the spectrum ofKlein-Gordon field contains a continuous sector which ehibits superradiance inherited from Kerr. All this substatially complicates the analysis and no definitive concluswas gained in@3# about the possible relevance of such gometries in holography.

Here we present another geometry containing the A2sector mixed nontrivially with the rest of the spacetimwhich has the advantage of not being afflicted byu factors. Itcan be obtained as the near-horizon limit of extremal rotatdilaton-axion black holes@solutions to the Einstein-Maxwelldilaton-axion ~EMDA! theory#. This is therefore a nonvacuum solution which is supported by a homogeneous Mwell field @similarly to the Bertotti-Robinson ~BR!spacetime# and a linear axion. The rotation breaks theSO(3)symmetry of BR so that the solution is only axially symmeric. Meanwhile, as in the Bardeen-Horowitz case, tSL(2,R) symmetry of the AdS component still holds, so thfull isometry group isSL(2,R)3U(1).

We show that this geometry has a nontrivial connectwith BR via a nonlocal dimensional reduction mechanis@15# involving dualization of Kaluza-Klein two-forms in or

©2001 The American Physical Society11-1

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GERARD CLEMENT AND DMITRI GAL’TSOV PHYSICAL REVIEW D 63 124011

der to generate higher-rank antisymmetric forms. Startwith the dyonicD54 BR solution with equal strengths fothe electric and magnetic components one finds its puvacuumD55 counterpart~the KK dilaton is not excited inthe symmetric dyon case!. This solution can be smeared inthe sixth dimension providing the ‘‘BR6’’ vacuum geometrThen one performs dimensional reduction back to fivemensions dualizing the Kaluza-Klein two-form and reintepreting the resulting three-form as a Neveu-Schwarz–NevSchwarz~NS-NS! field. Finally going to four dimensions viathe usual KK reduction one recovers the EMDA theocounterpart of the BR6 which coincides with our neahorizon limit of rotating EMDA black holes.

Using the same mechanism of generation of antisymmric forms, we uplift the new solution into eleven-dimensionsupergravity where it is supported by a nontrivial four-forThis is based on the correspondence between eidimensional vacuum gravity with two commuting Killinvector fields and a consistent 21316 three-block truncationof D511 supergravity@16#. To apply this technique one hafirst to smear BR6 in two additional dimensions and thendualization of the KK two-form in six dimensions to get thfour-form which is then used to reconstruct the four-formD511 supergravity. In doing this there are two optionsthe choice of the Killing vectors which lead to two supegravity solutions with different four-forms but the same mric.

Since rotation breaks the BPS condition for extremdilaton-axion black holes, it can be expected that our rotaBR solution is not supersymmetric in the sense ofD54, N54 supergravity. To check this, we consider the purelygebraic equation for variation of dilatino and show that tvariation is nonzero. We also check by a direct computatthat theD511 embedding of our solution is not supersymmetric in the sense ofD511 supergravity. Nevertheless wargue that the rotating BR geometry provides an interesnew arena for conformal mechanics and holography.

II. NEAR-HORIZON LIMIT OF ROTATING EMDA BLACKHOLES

Consider the Einstein-Maxwell-dilaton-axion~EMDA!theory which is a truncated version of the bosonic sectoD54, N54 supergravity. The action describes the gravicoupled system of two scalar fields: dilatonf and ~pseudo-scalar! axion k, and an Abelian vector fieldAm :

S51

16pE d4xAugu H 2R12]mf]mf11

2e4f]mk]mk

2e22fFmnFmn2kFmnFmnJ , ~2.1!

where1 Fmn5 12 EmnltFlt , F5dA. The black hole solu-

tions to this theory were extensively studied in the rec

1Here Emnlt[ugu21/2«mnlt, with «1234511, wherex45t is thetime coordinate.

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past@17–22#. These depend on six real parameters, the coplex massM5M1 iN @where N is the Newman-Unti-Tamburino ~NUT! parameter#, electromagnetic chargeQ5Q1 iP and axion-dilaton chargeD5D1 iA constrainedby D52Q 2/2M, and the rotation parametera. The generalblack hole metric is of the form

ds25D2a2 sin2u

S~dt2vdw!22SS dr2

D1du2

1D sin2u

D2a2 sin2udw2D , ~2.2!

with

D5~r 2r 2!~r 22M !1a22~N2N2!2,

S5r ~r 2r 2!1~a cosu1N!22N22 , ~2.3!

v52

a2 sin2u2D@ND cosu1a sin2u„M ~r 2r 2!

1N~N2N2!…#,

and

r 25M uQu2

uMu2, N25

N

2Mr 2 . ~2.4!

The vector field may be parametrized by two scalar elec(v) and magnetic~u! potentials defined by

Fi05] iv/A2,

e22fFi j 1kF i j 5~S sinu!21e i jk]ku/A2. ~2.5!

The potentialsv andu and the axion-dilaton field are give~after adapting the formulas of@19# to the conventions of@20#! by

v5A2ef`

SRe@Q~r 2r 22 id!#,

u5A2ef`

SRe@Qz`~r 2r 22 id!#,

z[k1 ie22f5z`r1D* z*

r1D*, ~2.6!

where

d5a cosu1N2N2 , r5r 2Mr 2

2M2 id, ~2.7!

and the~physically irrelevant! asymptotic value of the axiondilaton fieldz`[k`1 ie22f` will be chosen later for convenience.

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BERTOTTI-ROBINSON-TYPE SOLUTIONS TO . . . PHYSICAL REVIEW D63 124011

The metric~2.2! has two horizons located at the zerosr5r H

6 of the functionD:

r H65M1r 2/26A~ uMu2uDu!22a2. ~2.8!

The extremal solutions correspond to the case

uDu5uMu2a, ~2.9!

where these two horizons coincide, withS.0 ~without lossof generality we assumea.0). In this case, using

r 2r 25r 2r H1aM

uMu, ~2.10!

we can rewrite the metric as

ds25SD sin2u

Gdt22SS dr2

D1du2D2

G

S~dw2Vdt!2,

~2.11!

with

G5D@~h14g!sin2u24N2 cos2u#

14~g2aN cosu!2 sin2u,

S5h12g, V52~Nh cosu1ag sin2u!/G, ~2.12!

D5~r 2r H!2, h5D2a2 sin2u,

g5M ~r 2r H!1a~ uMu1N cosu!.

On the horizonr 5r H , the metric functions occurring inEq. ~2.12! simplify to

GH54a2uMu2 sin2u, VH51/2uMu,

SH52a~ uMu1N cosu!2a2 sin2u. ~2.13!

It follows that in the static casea50, the horizon reduces toa point, so that the question of near-horizon limit becommeaningless. But for rotating extremal black holesa5” 0 onefinds a nontrivial limiting solution. In this case, before takinthe near-horizon limit, let us first transform to a frame cotating with the horizon:

w[w2VHt. ~2.14!

In this frame, the differential angular velocity is, near thorizon,

V5V2VH522aM~r 2r H!sin2u

G1O~D!. ~2.15!

Let us now put

r 2r H[lx, cosu[y, t[r 0

2

lt ~r 0

252auMu!,

~2.16!

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and take the limitl→0. In this limit the extreme metric inthe rotating frame reduces to

ds25r 02F ~a1ny1by2!S x2dt22

dx2

x2 D 2a1ny1by2

12y2dy2

212y2

a1ny1by2~dw1mxdt!2G , ~2.17!

where b5a/2uMu, a512b, m5M /uMu, n5N/uMu (m21n251), and we have relabelled the coordnates (t ,w)→(t,w).

Just as the extreme Kerr or Kerr-Newman geometriesEinstein-Maxwell theory @3#, the near-horizon geometr~2.21! admits four Killing vectors

L15x]x2t] t ,

L25xt]x21

2~x221t2!] t1mx21]w ,

L35] t , ~2.18!

L45]w ,

generating the groupSL(2,R)3U(1). Indeed, the metric~2.17! becomes, in the NUT-less casen50,

ds25r 02F ~a1by2!S x2dt22

dx2

x2 D 2a1by2

12y2dy2

212y2

a1by2~dw1xdt!2G . ~2.19!

This is similar in form to the extreme Kerr-Newman neahorizon metric@3,23#:

dsEM2 5r 0EM

2 F ~aEM1bEMy2!S x2dt22dx2

x2 D2

aEM1bEMy2

12y2dy22

12y2

aEM1bEMy2

3~dw1mEMxdt!2G , ~2.20!

with r 0EM2 5M21a2, bEM5a2/(M21a2), aEM51

2bEM , mEM52aM/(M21a2), whereM25a21Q 2. Thetwo extreme geometries~2.19! and ~2.20! become identicalin the neutral caseQ50 (M5a) and reduce to the extremKerr geometry witha5b51/2.

Now let us take the limita→0 in the NUT-less near-horizon geometry~2.19!, while keepingr 0

252aM fixed. Inthis manner we arrive at the metric

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ds25x2dt22dx2

x22

dy2

12y22~12y2!~dw1xdt!2

~2.21!

~we have scaledr 02 to unity!. This metric is remarkably simi-

lar in form to the Bertotti-Robinson metric@the near-horizongeometry of the extreme Reissner-Nordstro¨m black hole, i.e.,the static limita→0 of Eq. ~2.20!#,

ds25x2dt22dx2

x22

dy2

12y22~12y2!dw2. ~2.22!

However, unlike the BR metric, the BREMDA metric~2.21!is nonstatic, and invariant only under the groupSL(2,R)3U(1). The t,x coordinates do not cover the full AdS hyperboloid. The geodesically complete manifold is coveredanother coordinate patch in which case the metric reads~wepreserve the same symbols for radial and azimuthal coonates!

ds25~11x2!dt22dx2

11x22

dy2

12y2

2~12y2!~dw1xdt!2. ~2.23!

Another useful coordinate system~also incomplete! is givenby

ds25~x221!dt22dx2

x2212du22

dy2

12y2

2~12y2!~dw1xdt!2. ~2.24!

Now proceed along the same lines with the matter fieFrom the last Eq.~2.6!, the dilaton and axion fields reduce othe horizon to

e22fH5e22f`2a~ uMu1Ny!2a2~12y2!

~M1D !21~N1A1ay!2,

kH5k`12e22f`D~N1ay!2AM

~M1D !21~N1A1ay!2. ~2.25!

For N50 and in the limita→0, these become~after neglect-ing terms of ordera2)

e22fH5e22fàM

P2,

kH5k`2e22f`M

P2~A2ay!. ~2.26!

Now we choose for convenience

z`522PQ* /r 02 ~2.27!

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(r 0252aM), leading to the BREMDA dilaton and axion

fields

fH50, kH5cosu, ~2.28!

irrespective of the original values ofQ and P ~provided PÞ0).

The determination of the near-horizon behavior of tgauge field is more involved. With the choice~2.27!, thescalar potentials for the NUT-less extreme black hole are

v5A2ef`

S@Q~r 2r H1a!1Pa cosu#,

u52A2e2f`

S

2M ~M2a!

P~r 2r H1a!. ~2.29!

First, we must transform these to the rotating and rescacoordinate frame (w, t ):

S dw

d tD 5S 1 2VH

0 l/r 02 D S dw

dt D . ~2.30!

From Eq.~2.5! we obtain the transformation laws

] i v5r 0

2/l

S2D sin2u/G22V2 H S S2D sin2u

G22VV D ] iv

1VHe2fD sinu

Ggi j e

jk~]ku2k]kv !J ,

~2.31!

] i u5r 0

2/l

S2D sin2u/G22V2 H S S2D sin2u

G22VV D ] iu

2VHe2fD sinu

Ggi j e

jk@~e24f1k2!]kv2k]ku#J( i , j 5r ,u). Then, using

] rS.2M , ]uS522a2 sinu cosu, ~2.32!

we evaluate the derivatives in Eq.~2.31! near the horizon,keeping only the leading terms ina ~only the partial deriva-tives relative tou contribute in this order!:

] r vH.2r 0

lcosu, ]uvH.

r 0

l~r 2r H!sinu,

] r uH.r 0

lsin2u, ]uuH.

r 0

l~r 2r H!2 sinu cosu.

~2.33!

From these we obtain the near-horizon potentials

vH52r 0x cosu,

uH5r 0x sin2u, ~2.34!

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again irrespective of the original values of the electric amagnetic charges. We have checked that these, togetherthe near-horizon metric~2.21! and axion-dilaton fields~2.28!, solve the field equations as given in@20#. From thepotentials~2.34!, after settingr 0 to unity, we recover thenear-horizon gauge fields

F1452y

A2, F2352

1

A2, F2452

x

A2,

F1352xy

A2, F145

y

A2, F2352

1

A2~2.35!

~with x15x andx25y5cosu), deriving from the gauge potentialsA352y/A2, A452xy/A2.

In view of the discussion in@24# one could wonderwhether this Maxwell field is consistent with theSL(2,R)3U(1) isometry ~2.18!. In Einstein-Maxwell theory it iswell known that the Einstein equations imply the followinrelation for the Lie derivative of the Maxwell field@25#:

LjFmn5CFmn ~2.36!

~wherej is the Killing vector! with some constantC. TheMaxwell field therefore shares the spacetime symmetryC50, which is the case for the BR solution. Our theoincludes a scalar field, so this theorem is not applicablerectly, but a direct check shows that the Lie derivative ofMaxwell tensor along all four Killing vectors~2.18! is alsozero. Therefore the Maxwell field~2.35! shares theSL(2,R)3U(1) symmetry of BREMDA.

Finally, passing to more general coordinates containinfree parameterb, we can write the BREMDA solution afollows2:

ds25~x21b!dt22dx2

x21b2du22sin2u~dw1xdt!2,

A5Amdxm52cosu

A2~dw1xdt!, k5cosu. ~2.37!

For b50 this coincides with the solution derived abovethe limiting procedure~Poincare´ coordinates on AdS sector!,for positive nonzerob ~usually set tob51) one has a coordinate patch covering the full AdS hyperboloid. For compason consider the near-horizon limit of the Kerr solution@3#.Setting in Eq.~2.19! r 051,a5b51/2 and passing to similacoordinates, we obtain

2Let us here mention that in@26# the low-energy limit of a certainconformal field theory was shown to correspond to a formal nehorizon limit of Kerr-NUT solutions of EMDA, with metric andmatter fields different from Eq.~2.37!. However one can show thathese fields@given in Eq.~3.4! of @26## do not solve the field equations of dilaton-axion gravity.

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ds251

2~11cos2u!F ~x21b!dt22

dx2

~x21b!2du2G

22 sin2u

11cos2u~df1xdt!2. ~2.38!

In both cases the mixing of the azimuthal and time coornates does not vanish asx→`. Both metrics coincide in theequatorial plane, but differ foru5” p/2. The BREMDA ge-ometry is simpler due to the absence of cumbersome angfactors, and apparently is more suitable for the search oholographic dual. We return to this question in a separpublication @27#, while here we discuss other geometricaspects of the new solution related to its embeddingshigher dimensions.

III. DÄ4 EMDA FROM DÄ6 EINSTEIN GRAVITY

Let us now show howD54 EMDA theory can be derivedfrom the purely vacuum Einstein theory in six dimensionThis may be hinted from the following considerations. Dmensional reduction of stationaryD54 EMDA to three di-mensions leads to a gravity coupled sigma-model withtarget space isometry groupSp(4,R) @28,29#, while dimen-sional reduction of the 6D vacuum gravity to three dimesions gives a sigma model with theSL(4,R) target spacesymmetry@30#. It was shown in@15# that a consistent truncation of theSL(4,R) sigma model to theSp(4,R) one ex-ists, i.e., any stationary solution ofD54 EMDA gravity hasa D56 vacuum gravity counterpart and vice versa. Hereshow that this holds not only for stationary, but for all soltions of the two theories. This duality is essentially nonlocit involves dualization of the Kaluza-Klein two-form in thintermediate five dimensions.

Let us start with the action

S52E d6xAug6uR6 , ~3.1!

denoting the 6-dimensional coordinates asxm,h,x, and makethe assumption of two commuting spacelike Killing vecto]h ,]x . In any number of dimensions, the Kaluza-Klein dmensional reduction

dsn112 5e22cfdsn

22e2(n22)cf~dh1Cmdxm!2 ~3.2!

gives

Augn11uRn115AugnuFRn2~n21!~n22!c2~]f !2

11

4e2(n21)cfF2~C!12c¹2fG . ~3.3!

For n55,c51/A6, this leads, after dualizing the 2-formF(C) to a 3-formH5dK,

r-

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Fmn~C!521

6Aug5ue2afemnlrsHlrs , ~3.4!

to the reduced action for 6-dimensional sourceless gravi

S55E d5xAug5uF2R512~]f !211

12e2afH2G , ~3.5!

with a54A2/3.In Eq. ~3.5! we recognize the action of 5-dimension

gravity coupled to a dilaton and a 3-form, as written down@15#. In this paper it was observed that, under the assumpof two commuting Killing vectors]4 and ]5, this theoryreduces to a 3-dimensionals model with theSL(4,R) sym-metry group. Here we see that this symmetry follows direcfrom theSL(4,R) symmetry of sourcelessD56 gravity with3 Killing vectors, which is a special case of sourcelen-dimensional gravity with (n23) Killing vectors, as dis-cussed by Maison@30#.

In a second step, the 5-dimensional theory~3.5! with aspacelike Killing vector]x is further reduced by the KaluzaKlein ansatz

ds525e22s/A3ds4

22e4s/A3~dx1Dmdxm!2,

K5Kmndxm`dxn1Emdx`dxm, ~3.6!

to the action

S45E d4xAug4uF2R412~]f!21~]c!211

2e4f~]k!2

21

4e2(c2f)F2~D !2

1

4e22(c1f)F2~E!2

k

4@F~D !F~E!

1F~E!F~D !#G , ~3.7!

where

f5A2

3f2A1

3s, c5A2SA1

3f1A2

3s D ,

~3.8!

andk is the dual of the 3-form

H[dK2D`F~E!52e4f* dk. ~3.9!

Remarkably, the equations of motion for the fieldsD, E, andc deriving from the action~3.7!

¹m@e2(c2f)Fmn~D !1kFmn~E!#50,

¹m@e22(c1f)Fmn~E!1kFmn~D !#50, ~3.10!

¹2c11

4e22f@e2cF2~D !2e22cF2~E!#50

are consistent with the ansatz

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c50, Dm5Em[A2Am , ~3.11!

which reduces the action~3.7! to the action~2.1! of EMDA~this is similar to the reduction of five-dimensional KaluzKlein theory to self-dual Einstein-Maxwell theory!.

This two-step reduction of 6-dimensional vacuum gravcan be summarized in a direct reduction from 6 to 4 dimsions. From Eq.~3.4!,

Fm5~C!5]mC552e24fHm5]mk, ~3.12!

wherem51, . . . ,4, sothat the 5-dimensional 1-formC re-duces according to

C5Cmdxm1kdx. ~3.13!

The two successive Kaluza-Klein ansa¨tze ~3.2! ~for n55, c51/A6) and~3.6! can be combined into

ds625e2cds4

22ec22f~dx1Dmdxm!22ec12f~dh1Cmdxm

1kdx!2. ~3.14!

Finally, we compute

Fmn~C!521

2Aug5ue2afemnlr5Hlr5

51

2Aug4uemnlr@e22(c1f)Flr~E!2e24fDtH

tlr#

5Fmn~B!1Fmn~kD !, ~3.15!

with the definition@solving the second equation~3.10!#

Fmn~B![e22(c1f)Fmn~E!2kFmn~D !. ~3.16!

Accordingly we can rewrite the double Kaluza-Klein ansa~3.14! as

ds625e2cds4

22ec22fu22ec12f~z1ku!2, ~3.17!

with

u[dx1Dmdxm, z[dh1Bmdxm. ~3.18!

Taking into account Eq.~3.11!, it follows that the ansatzfor reducing 6-dimensional vacuum gravity to EMDA mabe written

ds625ds4

22e22fu22e2f~z1ku!2,

u[dx1A2Amdxm, z[dh1Bmdxm, ~3.19!

Fmn~B![A2@e22fFmn~A!2kFmn~A!#.

IV. DÄ6 VACUUM COUNTERPART OF BREMDA

Using the machinery of the preceding section, we cshow that BREMDA is dual to the six-dimensional vacuusolution whose standard KK reduction gives the usualD54 dyonic BR solution with equal electric and magne

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strengths. From Eqs.~2.28! and ~2.35! we obtain~note thatthe coordinate transformationt,x,y,w→t,x,u,w reversesorientation, so that accordingly we must change the signthe axion!

F14~B!512y2, F23~B!522y, F24~B!522xy,~4.1!

leading~in a suitable gauge! to the 1-form

B52y2dw1x~12y2!dt. ~4.2!

It follows that the 6-dimensional line element correspondto Eq. ~2.37! is ~with b52c2)

ds625~x22c2!dt22

dx2

x22c22

dy2

12y22~12y2!~dw1xdt!2

2~dx2xydt2ydw!22~dh1xdt2ydx!2. ~4.3!

This may be rearranged to the more compact form

ds625~x22c2!dt22

dx2

x22c22

dy2

12y22~12y2!dx2

2~dw1xdt2ydx!22~dh1xdt2ydx!2, ~4.4!

which is explicitly symmetric between the two Killing vectors ]w and ]h , and enjoys the higher symmetry grouSL(2,R)3SO(3)3U(1)3U(1).

Equation~4.4! represents the Bertotti-Robinson solutioof 6-dimensional vacuum gravity. A simpler formachieved by making ap/4 rotation in the plane of the twoKilling vectors (]w ,]h) and relabeling the third spacelikKilling direction according to

dw521

A2~dh1dx !, dh5

1

A2~dh2dx !, dx5dw.

~4.5!

This leads to

ds625~x22c2!dt22

dx2

x22c22

dy2

12y22~12y2!dw2

2~dx2A2xdt1A2ydw !22dh2, ~4.6!

which is the trivial 6-dimensional embedding of th5-dimensional Bertotti-Robinson metric

ds525~x22c2!dt22

dx2

x22c22

dy2

12y22~12y2!dw2

2~dx2A2xdt1A2ydw!2. ~4.7!

Remarkably, this is exactly the solution whose foudimensional Kaluza-Klein reduction is the Einstein-MaxwBertotti-Robinson solution~2.22!; for more details see theAppendix.

12401

of

g

-l

The metric~4.3! may also be dimensionally reduced reltively to the Killing vectors]h and] t ~instead of]x). Choos-ing c251, rearranging Eq.~4.3! as

ds625~x221!~dw2ydx!22

dx2

x2212

dy2

12y22~12y2!dx2

2~dt1xdw2xydx!22~dh1xdt2ydx!2, ~4.8!

and relabeling the Killing directions according tow→t→x→2w, one obtains the equivalent 6-dimensional metric

ds625~x221!~dt1ydw!22

dx2

x2212

dy2

12y22~12y2!dw2

2~dx1xdt1xydw!22~dh1xdx1ydw!2. ~4.9!

Following Eq.~3.19!, this may be reduced to the solution oEMDA:

ds425~x221!~dt1ydw!22

dx2

x2212

dy2

12y22~12y2!dw2,

A5x

A2~dt1ydw!, f50, k5x, ~4.10!

with B52x2dt2(x221)ydw @k may for instance be obtained by solving the Maxwell equationsdF(B)50]. ThisBertotti-Robinson-NUT solution may be obtained from tBREMDA solution ~2.37! by the correspondence@which isan isometry of the 5-dimensional Bertotti-Robinson met~4.7!#

t↔ iw, x↔2y, ds2→2ds2, A→2 iA.~4.11!

The dimensional reduction of Eq.~4.4! to Eq. ~4.10! breaksthe full symmetry group of the 6-dimensional BertotRobinson solution toSO(3)3U(1).

Remarkably, this BR-NUT solution may also be obtainas a near-horizon limit of near-extremal static black hsolutions of EMDA with NUT charge. Such near-extremblack holes are defined by the condition that

~ uMu2uDu!2[uMu21uDu22uQu2[l2c2 ~4.12!

is small. Puttingr 2M2r 2/2[lx, we obtain for the metricfunctions in Eq.~2.2!

D5l2~x22c2!, S52luMuS M

uMux1cD1D.

~4.13!

So the limitl→0 will yield a Bertotti-Robinson-like metriconly for M50. In this case, rescaling times byt→(r 0

2/l)t asin Eq. ~2.16!, with now

r 0252lN, ~4.14!

we obtain the limiting 4-dimensional metric

1-7

e

b

k

tionl-

g

b-s


ds25r 02Fx22c2

c~dt1ydw!2

2c

x22c2@dx21~x22c2!dV2#G , ~4.15!

which is identical with Eq.~4.10! after scalingr 02 to unity

and choosing without loss of generalityc51 ~one can al-ways rescalex→cx and ds2→cds2), as this constructiongoes along only forc5” 0. Likewise, choosing

z`52iQQ* /cr02 , ~4.16!

we obtain from Eq.~2.6! ~after reversing the signs of thpseudoscalarsk and u as explained above! the limiting~gauge transformed and rescaled! scalar potentials

f50, k5x

c, v5

r 0

Acx, u5

r 0

Ac

x22c2

c, ~4.17!

in agreement with Eq.~4.10!. Again, we note that this limit isindependent of the original values ofQ and P, providedQÞ0.

V. ELEVEN-DIMENSIONAL SUPERGRAVITY

The idea to generate higher rank antisymmetric formsdualizing the KK two-forms was generalized toD511 su-pergravity as follows@16#. Starting with the Lagrangian

S(11)5E d11xA2gH R(11)21

234!F [4]

2 J2

1

6E F [4]`F [4]À[3] , ~5.1!

we use the following three-block ansatz for theD511 met-ric:

ds(11)2 5g2

1/2dabdzadzb1g31/3d i j dyidyj

1~g2g3!21/4g(6)mndxmdxn, ~5.2!

where all variables depend only on the six coordinatesxm

anda,b51,2; i , j 51,2,3; m,n50, . . . ,5. Thethree-formpotential A[3] is reduced to its six-dimensional pull-bacB[3] (x), the one-formA[1]5Amdxm and the scalark(x),with

Amz1z25Am~x!, Ay1y2y35k~x!. ~5.3!

For the four-form one has

F [4]5G[4]1F [2]2 ` Vol~2!1dk` Vol~3!, ~5.4!

whereG[4]5dB[3] ,F [2]5dA[1] . After reduction to six di-mensions we obtain the theory governed by the action

12401

y

S(6)5E d6xAug(6)u H R(6)2e2f

2~¹k!22

1

2~¹f!2

23

16~¹c!22

1

2e2fF 1

2!e(3/4)cF [2]

2

11

4!e2(3/4)cG[4]

2 G J 1Chern-Simons terms,

~5.5!

where two new scalar fields are introduced via

ln g252

3f2c, ln g3522f. ~5.6!

It is easy to see thatk,f form a cosetSL(2,R)/SO(1,1),while F [2] and the six-dimensional dualG[2]5* G[4] ~whichis also a two-form! can be combined into theSL(2,R) dou-blet

F[2]5ec/2F [2]1 ie2c/2G[2] , ~5.7!

transforming underSL(2,R) as follows:

z→ az1b

cz1d, z5k1 ie2f, ad2bc51,

F[2]→~cz1d!F[2] , c→c1const. ~5.8!

The multiplet of matter fields in theD56 action is thesame as that which may be obtained from compactificaof D58 vacuum gravity. Moreover, the action which folows from theD58 Einstein action with the metric ansatz

ds(8)2 5gmn~dzm1Am

mdxm!~dzn1Anndxn!

1e2 ~1/4! cgmndxmdxn,

ec5detuugmnuu, ~5.9!

(m,n51,2) leads exactly to the theory~5.5! after the identi-fication of variables

gmn5ec/2S ef kef

kef e2f1k2efD ,

dA[1]m 5F [2]

m ,

F [2]1 1kF [2]

2 5e2f23c/4G[2] . ~5.10!

More precisely, the field equation forB[3] becomes a Bian-chi identity for A[1]

1 and vice versa. Note, that interchanginthe D58 Killing vectors, i.e., relabelingAm

1 ↔Am2 we will

get differentD511 field configurations.Now we can construct a solution toD511 supergravity

which is dual to the eight-dimensional vacuum metric otained from BR6~4.6! smeared in two flat extra dimension

1-8

rre

e

t ih

ne

treh

l

e

ald

e-ht-

ec-

tric,4

ure


ds825~x22c2!dt22

dx2

x22c22

dy2

12y22~12y2!dw2

2~dx2A2xdt1A2ydw!22dh22dz122dz2

2 .

~5.11!

This solution possesses several commuting Killing vectofrom which one can choose any pair to be used in KKduction back to six dimensions.

Choosingz15x,z25h, we have

A15A2~ydw2xdt!, A250. ~5.12!

Transforming to theD511 variables we obtain

g25g351, f5c5k50,

F [4]5A2 Vol~2!`~dy`dw2dx`dt!. ~5.13!

For the different order of vector fieldsz15h,z25x one ob-tains the sameg2 ,g3 and zero scalarsf,c,k but a differentfour-form:

F [4]5* A2~dy`dw2dx`dt!, ~5.14!

where a star denotes theD56 Hodge dual. In both cases thD511 metric is a trivial smearing of theD56 metric:

ds112 5ds6

22dx622•••2dx10

2 . ~5.15!

VI. BREAKING OF SUPERSYMMETRY

Now let us discuss the issue of supersymmetry. As iwell known, the Bertotti-Robinson solution preserves all tsupersymmetries ofD54,N52 supergravity@31,32#. TheBardeen-Horowitz solution~2.38! is a vacuum one, so it cabe probed forN51 supersymmetry. The result is negativno geometric Killing spinors exist. Our solution~2.23!should be tested in the context ofD54, N54 supergravity,the relevant equations coming from the supersymmevariation of the dilatino and gravitino. The variation of thdilatino leads to a purely algebraic equation, which in tcase of a vanishing dilaton reads

~gm]mk1 iA2smnF mn2

!e50, ~6.1!

whereF2 is the anti-self-dual part of the Maxwell tensor.Substituting herek5cosu and the Maxwell tensor~2.35!

one obtains the equation

M ~u!e50, M5gu sinu2~cosu1 i !~suw2 is t x!.~6.2!

The determinantudetM u5sin4u, so that there is no nontriviasolution to Eq.~6.1!, i.e., the BREMDA bosonic solutionbreaks all the supersymmetries ofD54, N54 supergravity.Similarly, one can show that the Bertotti-Robinson endowwith NUT ~4.10! is not supersymmetric either.

Now consider theD511 embedding. OurD511 solutionis related to the four-dimensional BREMDA in a nonlocway, since it is obtained using dualizations in the interme

12401

s,-

se

:

ic

e

d

i-

ate dimensions. Soa priori it is not clear whether it is non-supersymmetric in the supergravity sense.

It was shown@16# that theD511 Killing spinor equationfor the 32-component Majorana spinore (11) ensuring thevanishing of the supersymmetry variation of the gravitino

DMe (11)11

288~GM

NPQR28dMN G PQR!FNPQRe (11)50

~6.3!

for the 21316 block truncation considered above corrsponds to the purely geometric equation for the eigdimensional dual:

S ]m21

4vm

absabD e850. ~6.4!

We use the flat gamma matrices, anda,b,m are the tetradand coordinate indices, respectively. Here the spin conntion vm

ab has to be calculated for the spacetime~5.2!. There-fore to explore the supersymmetry in theD511 supergravitysense we have to check whether the correspondingD58solution admits covariantly constant spinors.

The nonzero spin connection one-forms for the me~5.11! read ~we use tetrad indices and numbering 0,1,2,3for t,j,u,w,x, with x5cosx andy5cosu!:

v015cosudw1dx/A2, v045dj/A2,

v1452sinhjdt/A2, v235coshjdt2dx/A2,

v2452sinudw/A2, v345du/A2. ~6.5!

A direct substitution in Eq.~6.4! gives a system of matrixequations which should satisfy the integrability conditions

Rabmnsabe850, ~6.6!

where the mixed coordinate-tetrad components of theD58Riemann tensor are introduced. Writing them as curvattwo-formsVab5Rmn

abdxm`dxn, one finds the following non-zero quantities:

V015sinudu`dw21

2sinhjdt`dj,

V0251

2sinudj`dw,

V0351

2du`dj,

V0451

A2cosu sinhjdt`dw1

1

2sinhjdt`dx,

V1251

2sinu sinhjdt`dw,

1-9

te

r-

vittti-

rth

d

uziorveS

uae

ckthinnd

nsf

nges

imi-ureVi-

-to

rtlso

ofssin

tordi-

ofireby,

eirro-


V1351

2sinhjdu`dt,

V1451

A2coshjdt`dj1

1

A2cosudj`dw1

1

2dj`dx,

V2352sinhjdt`dj11

2sinudu`dw,

V2451

A2coshjdu`dt2

1

A2cosudu`dw2

1

2du`dx,

V34521

A2sinu coshjdt`dw2

1

2sinudw`dx. ~6.7!

All ten two-forms are independent, so one obtains ten ingrability conditions from which the following five:

s04e85s03e85s14e85s24e85s34e850 ~6.8!

are obviously inconsistent~sigma-matrices do not have kenels!. Therefore the BREMDA geometry is not supersymmetric in the sense ofD511 supergravity either.

VII. CONCLUSION

We have presented a new solution to dilaton-axion grain four dimensions which is a rotating version of the BertoRobinson metric. It breaks theSO(3) symmetry of the latterbut preserves theSL(2,R) symmetry of the anti–de Sittesector. The metric arises as the near-horizon limit ofcharged rotating axion-dilaton black hole~in the theory withone vector field! and is supported by nontrivial vector anaxion fields. It looks simpler than the near-horizon Kerr~orKerr-Newman! metric due to the absence of additional anglar factors, while preserving the same mixing of the amuthal and time coordinates induced by rotation. It is imptant to note that the AdS sector does not factor out easymptotically. Moreover, in contrast to the case of Ad23S2, the conformal boundary is now a singular 112 space.

The new metric was shown to be related to the usdyonic BR solution with equal electric and magnetic chargafter uplifting it to six dimensions and then coming baalong a different reduction scheme. In this procedureaxion emerges via dualization of one of the Kaluza-Kletwo-forms. Using a similar reasoning, we were able to fitwo solutions ofD511 supergravity with nontrivial four-form fields whose dimensional reduction~including dualiza-tions at intermediate steps! gives our solution.

This solution is not supersymmetric, whether in the seof the originalD54, N54 supergravity, or in the sense ohigher dimensional embeddings. But it still looks promisifrom the point of view of holography. Indeed, it preservsome features of AdS23S2 found also for the Kerr throat@3#,

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-

-

y

e

---n

ls

e

e

and it is not plagued by superradiance as the latter. Prelnary considerations show that, in spite of the singular natof the boundary, the asymptotic symmetry contains therasoro algebra@27#. Also it is likely to provide a new versionof conformal mechanics of the type studied recently@33,34#.We will discuss these issues in a separate publication.

ACKNOWLEDGMENTS

We would like to thank K. Bronnikov, J. Fabris, L. Palacios, and S. Solodukhin for discussions. D.G. is gratefulLAPTH Annecy for hospitality and to CNRS for suppowhich made possible this collaboration. His work was asupported in part by the RFBR grant 00-02-16306.

APPENDIX

In this appendix we discuss the near-horizon limitstatic black hole solutions of 5-dimensional sourceleKaluza-Klein theory, i.e., 5-dimensional vacuum Einstegravity

S52E d5xAug5uR5 , ~A1!

together with the assumption of a spacelike Killing vec]/]x5. The 5-dimensional metric may be reduced to 4mensions by the Kaluza-Klein dimensional reduction

ds525e22s/A3ds4

22e4s/A3~dx512Amdxm!2, ~A2!

with the reduced action

S5E d4xAug4u$2R412]ms]ms2e2s/A3FmnFmn%.

~A3!

The general static, NUT-less black hole solutionKaluza-Klein theory was derived by Gibbons and Wiltsh@35#, and generalized to rotating black hole solutionsRasheed@36#. We will consider only static black holeswhich depend on 3 parametersM ~mass!, S ~scalar charge!,Q andP ~‘‘electric’’ and ‘‘magnetic’’ charge! constrained by

Q2

S1M /A31

P2

S2M /A35

2S

3. ~A4!

The corresponding 5-dimensional metrics, as well as th4-dimensional reductions, have two regular horizons pvidedM21S22P22Q2>0. The condition of extremality isthereforeM21S25P21Q2. However for more generalitywe shall consider near-extremal black holes, with

M21S22P22Q2[l2c2 ~A5!

small.The black hole solutions are

1-10

r-

aps

iteo-

the

inntly

fndlar

lds

eld

d-in


ds525

f 2

Bdt22AS dr2

f 21du21sin2udw2D

2B

A S dx512Q

B~r 2M1M 2!dt12P cosudw D 2

,

~A6!

where the metric functionsf ,A,B are given by

f 25~r 2M !22l2c2,

A5 f 212M 2~r 2M !1M 2

M~M 1M 21l2c2!, ~A7!

B5 f 212M 1~r 2M !1M 1

M~M 1M 21l2c2!,

and

M 6[M6S

A3, Q25

M 1~M 12 2l2c2!

2M,

P25M 2~M 2

2 2l2c2!

2M. ~A8!

Puttingr 2M[lx, we shall take the near-extremal, neahorizon limit l→0 such that the two horizonsr 5r 6[M6lc approach each other while the radial coordinateproaches the event horizonr 1 . Four-dimensional sectionw5const of the 5-dimensional metric~A6! being similar inform to the rotating metric~2.11!, with the electric potentialA4 playing the part of the angular velocity, to obtain a finlimit we again must first transform to a frame ‘‘nearly cortating’’ with the horizon, through a gauge transformation

dx55dx512QM2

B~0!dt, ~A9!

with B(0)[B(r 2M50), leading to the ‘‘electric’’ field inthe new gauge

A452QM2

B~0!B~r 2M !S M 1M 2

M1r 2M1O~l2c2! D ,

~A10!

Y

n,

12401

-

and rescale both time and the fifth coordinate, throughtransformations

r 2M[lx, cosu[y, t[AA0B0

lt , x5[A2Px,

~A11!

where (A0 ,B0)5 lim(l→0)(A,B). Taking the limitl→0, us-ing the identities

2B0P2

A02

5Q2

P2

A0

B0

M 22

M 12

51, ~A12!

and relabeling the time coordinatet→t, we finally obtain the5-dimensional near-horizon metric

A022ds5

25~x22c2!dt22dx2

x22c22

dy2

12y22~12y2!dw2

2~dx2A2xdt1A2ydw!2. ~A13!

Again, as in the case of EMDA, all the static Kaluza-Kleblack holes have the same near-horizon limit, independeof the values of the electric and magnetic chargesQ5” 0 andP5” 0.

In Eq. ~A13! we recognize the Kaluza-Klein version othe dyonic Bertotti-Robinson solution with equal electric amagnetic charges. The vanishing of the Kaluza-Klein scafield s is due to the fact that the electric and magnetic fie

F1452F23521

A2~A14!

being equal in magnitude, the source term in the scalar fiequation

¹2s521

2A3e2A3sFmnFmn ~A15!

vanishes. Accordingly the isometry group is the direct prouct of that of the Bertotti-Robinson spacetime with the Klecircle, i.e.,SL(2,R)3SO(3)3U(1).

,’’

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Bertotti-Robinson-type solutions to dilaton-axion gravity