3 December 1998

Physics Letters B 442 Ž1998. 97–101

BPS black holes in N s 2 five dimensional ADS supergravity K. Behrndt

a,1

, A.H. Chamseddine

b,c,2

, W.A. Sabra

b,d,3

a

Humboldt-UniÕersitat, ¨ Institut fur ¨ Physik InÕalidenstraße 110, 10115 Berlin, Germany Center for AdÕanced Mathematical Sciences, American UniÕersity of Beirut, Lebanon 4 c Institute for Theoretical Physics, ETH Zuerich, Switzerland Physics Department, Queen Mary and Westfield College, Mile End Road, London E1 4NS, United Kingdom b

d

Received 10 August 1998 Editor: P.V. Landshoff

Abstract BPS black hole solutions of UŽ1. gauged five-dimensional supergravity are obtained by solving the Killing spinor equations. These extremal static black holes live in an asymptotic AdS5 space time. Unlike black holes in asymptotic flat space time none of them possess a regular horizon. We also calculate the influence, of a particular class of these solutions, on the Wilson loops calculation. q 1998 Elsevier Science B.V. All rights reserved.

In the past years a considerable amount of work has been devoted to establish a duality between supergravity and super Yang Mills theories. For example the conformal field theory ŽCFT. living on the boundary of the five-dimensional anti-de Sitter space Ž AdS5 . is expected to be dual Žin certain limits. to the four-dimensional super Yang Mills theory. Since this conjecture has been made w1x and further developed in w2x, five-dimensional anti-de Sitter spaces have received a great deal of interest. The aim of this letter is to describe BPS black holes living in an asymptotic AdS5 vacuum Žfor the AdS4 case, Reissner-Nordstrom ¨ solutions have been discussed in w3x and non-abelian monopoles in w4x.. To keep these solutions as general as possible we formulate them in terms of D s 5, N s 2 supergrav1

E-mail: [email protected] E-mail: [email protected] 3 E-mail: [email protected] 4 Permanent address. 2

ity with an arbitrary prepotential, i.e. the N s 4,8 black holes appear as a special subclass for special choices of the prepotential of the N s 2 theory. Because the asymptotic vacuum should be AdS5 instead of flat Minkowski space, we gauge a UŽ1. subgroup of the SUŽ2. automorphism group, which results in a scalar potential that becomes constant at infinity. In the first part we will describe these black holes as solutions of the Killing spinor equations of gauged D s 5, N s 2 supergravity w5x and in the second part we ask for the modification of the Wilson-loop calculations as done in w6–8x. First, we briefly describe the theory of N s 2 supergravity coupled to an arbitrary number n of abelian supermultiplets. N s 2 supergravity theories in five-dimensions can be obtained, for example, by compactifying eleven-dimensional supergravity on a Calabi-Yau 3-folds w9x. The massless spectrum of the compactified theory contains Ž hŽ1,1. y 1. vector multiplets with real scalar components. Including the

0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 2 0 8 - 8

K. Behrndt et al.r Physics Letters B 442 (1998) 97–101

98

graviphoton, the theory has hŽ1,1. vector bosons. The theory also contains hŽ2,1. q 1 hypermultiplets, where hŽ1,1. and hŽ2,1. , are the Calabi-Yau Hodge numbers. In what follows and for our purposes the hypermultiplets are switched off. The anti-de Sitter supergravity can be obtained by gauging the UŽ1. subgroup of the SUŽ2. automorphism group of the N s 2 supersymmetry algebra. This gauging, which breaks SUŽ2. down to UŽ1. can be achieved by introducing a linear combination of the abelian vector fields present in the ungauged theory, i.e. Am s VI AmI , with a coupling constant g. To restore supersymmetry, gdependent and gauge-invariant terms have to be added. In a bosonic background, this amounts to the addition of a scalar potential, Žfor more details see w10,11x.. The bosonic part of the effective gauged supersymmetric N s 2 Lagrangian which describes the coupling of vector multiplets to supergravity is given by ey1 L s y 12 R y 14 GI J Fmn I F mn J y 12 g i j Em f i E mf j ey1 q 48

e mnrsl CI JK FmnI FrsJ Alk

Ž 1.

where R is the scalar curvature, Fmn s 2 E w m An x is the Maxwell field-strength tensor and e s y g is the determinant of the Funfbein e ma . 5 ¨ The physical quantities in Ž1. can all be expressed in terms of a homogeneous cubic polynomial V which defines ‘‘ very special geometry’’ w12x.

'

GI J s y

E

ž

dcm s Dm q

ž

Ef i

/

.

y 4dm nG r . Fnr

I

/

dl i s

ž

E X I G mn FmnI y

3 8 i

i 2

g i j G mEm f j q 32 igVI E i X I e

/

Ž 4. where e is the supersymmetry parameter and Dm is the covariant derivative. The spherically symmetric BPS electric solutions can be obtained by solving for the vanishing of the gravitino and gaugino supersymmetry variation for a particular choice for the supersymmetry parameter. We impose the projection operator condition on the spinor e

Ž 5.

where a 2 q b 2 s 1 and this breaks N s 2 supersymmetry to N s 1. We briefly 6 describe the procedure of obtaining solutions preserving N s 1 supersymmetry. First we start with an ansatz for the metric and gauge field ds 2 s ye 2Vdt 2 q e 2W Ž dr 2

Ž 2.

where the functions U,V,W and f are functions of r, and Ž u , f , c . are the polar coordinates of the 3-sphere. As solution of the gauge field equations we find

Ž 3.

e 2U X I s 13 HI

For Calabi-Yau compactification V s 16 CI JK X I X J X K s X I X I s 1.

nr

A It s ey2 U X I

E Ei '

X I Ž Gm

qf 2 r 2 Ž du 2 q sin2u d f 2 q cos 2u d c 2 . . ,

Ž lnV . < Vs 1 ,

g i j s GI J E i X IE j X J < Vs 1 ,

8

q 12 g Gm X I VI y 32 igVI AmI e ,

E

2 EXI EXJ

i

e s Ž ia G 0 q bG 1 . e ,

q g 2 VI VJ Ž 6 X I X J y 92 g i jE i X IE j X J .

1

Since we are interested in finding BPS solutions in the gauged theory, we display the supersymmetry transformation of the Fermi fields in a bosonic background

V is the intersection form, X I and X I correspond to the size of the 2 and 4-cycles and CI JK are the intersection numbers of the Calabi-Yau threefold.

where HI is a harmonic functions which depends on the electric charge q I . The supersymmetry variation

5

The signature Žyqqqq. is used. Antisymmetrized indices are defined by: w ab x s 12 Ž aby ba..

6

More detailed analysis will be given in w11x.

K. Behrndt et al.r Physics Letters B 442 (1998) 97–101

of the gaugino and the time component of the gravitino imply the following relations e 2V s ey4U f 2 , asy

1 f

1

e 2W s e 2U

f

2

constant spinor. Thus, inserting all terms in our ansatz one obtains

1

f 2 s 1 q g 2 r 2 e 6U

Ž 1 q g 2 r 2 V 2 . dt 2 dr 2

2r3

qV

b s y gre 3U , f

,

y4 r3

V ds 2 s yV

,

99

1qg2r2 V

2

qr 2 Ž du 2 q sin2u d f 2 q cos 2u d c 2 .

where we used some relations of special geometry analog to the derivation in w13x. The time and spatial components of the gravitino transformation imply differential constraints on the Killing spinor. These are

Ft Im s yEm Ž V

y1

YI. ,

V s 16 CI JK Y I Y J Y K ,

1 2

CI JK Y J Y K s HI s 3VI q

Ž E t y ig . e s 0 ,

ž

Er y

ž ž ž

Eu q

i

ž

2f i 2

1

/

q 3U X G 0 y

r

1 1 2

ž

r

qUX

//

/

G 012 e s 0 ,

/ /

n

i 1 Ec q cos uG 014 q sin uG 24 e s 0. 2 2

2

1 3Y

I

where V s e 3U , one obtains the following solution 7

ese

igt y

e

1

i 2

G 01 2 u

e

2

G 23 f

i

ey 2 G 014 c w Ž r .

1

wŽ r. s

1 V y2

'gr Ž 'f q 1 y 'f y 1 G1 .

2

1

=e

2

(

r

H dr

ž

1

1 q

r

3

2 2

VX V

where f s 1 q g r V

2

/ Ž1yi G . e 0

ž

VsH s 1q

Going to the rescaled coordinates y

q r2

n

/

2

,

n s 0,1,2,3 .

Ž 8.

Obviously, the first case Ž n s 0. defines the AdS5 vacuum with no black hole. The cases of n s 1,2 correspond to black holes with a singular horizon and they appear naturally as BPS solutions of N s 4,8 supergravity. In both cases the scalars are either zero or blow up near the horizon. The last case Ž n s 3. is an example of a BPS black hole of N s 2 supergravity, which seems to have a regular horizon at r , 0. However this coordinate system is misleading. Defining

r2sr2qq 0

r2 Ž 7.

Note that the constant parts in the harmonic functions are given by VI , which fixes the UŽ1. that has been gauged. The only deviation from the ungauged case w15x comes via the function f 2 s 1 q g 2 r 2 V 2 . This term however changes completely the singularity structure of the black hole solution. To investigate this in more detail we may consider simple cases were V can be written as

es0 ,

i 1 Ef q sin uG 013 y cos uG 23 e s 0 , 2 2

X IsV

qI

Ž 6.

and e 0 is an arbitrary

Ž 9.

one finds ds 2 s ye 2Vdt 2 q ey2 VDy1 d r 2 q D r 2 d V 3 , 2n

e s H˜ 2V

3

3yn 2 2

qg r D ,

D s H˜

3

,

H˜ s 1 y

q

r2

7

The Killing spinors for a general AdS p = S q geometry are also discussed in w14x.

Ž 10 .

K. Behrndt et al.r Physics Letters B 442 (1998) 97–101

100

In the ungauged case Ž g s 0. the horizon is at H˜ s 0 Žor r 2 s q ., which is regular in the case n s 3 or D s 1. But taking into account the gauging the horizon disappeared Ž e " 2V is finite at r 2 s q for n s 3. and the singularity at r s 0 becomes naked. For the other cases Ž n s 1,2. the horizon becomes singular. For n s 1 the singular horizon is infinitely far away, i.e. a light signal Ž ds 2 s 0. would need infinite time to reach any finite distance Žnull singularity.. But for n s 2 the distance to the singular horizon is finite. This is different to the ungauged case Ž g s 0., where all singular cases have null horizons. Note also, the naked singularity at r s 0 for n s 3 Ži.e. D s 1. is only a finite distance away! Certainly, this makes this solution rather suspisious and to overcome this situation one should consider the non-extremal case. Let us nevertheless ask, what is the influence of this black hole on the Wilson loops as calculated in w6–8x. For this we calculate the Nambo-Goto action for open strings that are attached to the asymptotic boundary Ss

1 2pa X

Hdt d s(
ab

<,

g a b s Ea XEb X N GM N

Ž 11 . where GM N is the 5d metric. For the worldsheet coordinates we choose the gauge

tst ,

ssu

Ž 12 .

where u is the polar angle in V 3 Žsee Fig. 1 and Eq. Ž7... Obviously, the string will stretch inside the AdS space and thus its position is given by a function f Ž u , r . s 0, where r is the radial coordinate. This defining equation can also be expressed as r s r Ž s s u .. For the induced metric we find therefore gtt s Et X MEt X N GM N s G 00 s ye 2V

Fig. 1. This is our geometry: a open string attached with the endpoints at asymptotic boundary, which is ‘‘perturbed’’ by a BPS black hole in the middle. The dotted line should indicate the horizon.

use the fact that the Lagrangian does not depend explicitly on u and therefore r 2 e 2VD cs . Ž 15 . 2 Ž r X . Dy1 q r 2 e 2VD

(

The constant c can be determined by going at the extremum r 0 where r X s 0 i.e. c 2 s Ž r 2 e 2V D . rs r 0 . Ž 16 . Ž . In addition it follows from 15 that dr ds s 1 2 2V rD e V r e Dy1 c2 dy s 3 2n n d yyl d 4 3 2 2 g r 0 y Ž y y l. y y1 1yl

(

)

2

gss s Es rEs r Grr q Guu s Ž r X . ey2 VDy1 q r 2D

ž

/

Ž 17 .

Ž 13 . .2

and thus, Ss

1 2pa X

Hd s dt(Ž r . X

2

Dy1 q r 2 e 2VD .

Ž 14 .

Following arguments given by Maldacena w7x, we

l s qrr 02

Žsee where y s Ž rrr 0 , and d s 3 also Fig. 1.. Furthermore we consider here only the simplest case where e 2V s g 2r 2D Ži.e. we neglect the first term., which is a good approximation for the region 0 < q - r 2 . Integrating this equation yields the function r s r Ž s s u . that determines the posi2 Ž3y n .

K. Behrndt et al.r Physics Letters B 442 (1998) 97–101

tion of the string in the AdS space. We can also calculate the distance between both endpoints on the boundary 8 uL

H0 2 (g

Ls2

ds 2

ss

101

r 02 the energy remains finite although the string comes close to the singular horizon and it scales with the charge or the BPS mass of the black hole E ; q. It is interesting to note that the energy is independent of g.

d

Ž 1 y l.

s

2

Acknowledgements

g dy

`

=

H1

n

d

3

2

y Ž y y l.

(

.

2n

y

3

d d Ž y y l. y Ž 1 y l. Ž 18 .

There are some interesting things to notice. First, for n s 3 Ž d s 0. all l dependence drops out and L ; 1rg becomes independent of r 0 and the charge q, it scales only with cosmological constant. Thus it coincides with the case without black hole. Secondly, for d / 0 the integral is finite if l - 1, i.e. the string is away from the horizon. However, if the horizon comes close to string Ž l ™ 1. the integral becomes divergent for n s 1. However taking the pre-factor into account one finds that in this limit L behaves like L ; 1g Ž1 y l.1y d r2 . Therefore, the string endpoints approach each other L ™ 0 for q ™ r 02 Žsee Fig. 1. if the horizon becomes large enough. This is different from the Žneutral. Schwarzschild black hole, where L ™ ` if the horizon comes close to the string w8x. Finally, one may insert the solution Ž17. in the action and calculate the energy Es

T 2p

r 02

`

H1 dy y n r6 Ž y y l .

=

(y

2 n r3

d

dr4

Ž y y l. y Ž 1 y l.

1 y d

'y Ž 19 .

where the last term is the subtraction of the infinite self-energy Žsee w7,8x.. Obviously, for l ™ 1 or q ™ 8

Note, we are dealing here with a different asymptotic geometry of R = S3 Žwhere Ris the time..

We thank P. Townsend for bringing to our attention the Ref. w16x, which deals with black hole solutions in Žanti. de Sitter background without vector multiplets.

References w1x J. Maldacena, The large N limit of superconformal field theory and supergravity, hep-thr9711200. w2x S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Gauge theory correlators from non-critical string theory, hep-thr9802109; E. Witten, Anti-de Sitter space and holography, hepthr9802150 w3x L.J. Romans, Nucl. Phys. B 383 Ž1992. 395, hep-thr9203018. w4x A.H. Chamseddine, M.S. Volkov, Phys. Rev. D 57 Ž1998. 6242, hep-thr9711181; Phys. Rev. Lett. 79 Ž1997. 3343, hep-thr9707176. w5x A.H. Chamseddine, H. Nicolai, Phys. Lett. B 96 Ž1980. 89, and unpublished notes. w6x S.-J. Rey, Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity, hepthr9803001. w7x J. Maldacena, Phys. Rev. Lett. 80 Ž1998. 4859, hepthr9803002. w8x S.-J. Rey, S. Theissen, J.-T. Yee, hep-thr9803135; A. Brandhuber, N. Itzhaki, J. Sonnenschein, S. Yankielowicz, hep-thr9803138, hep-thr9803263. w9x A.C. Cadavid, A. Ceresole, R. D’Auria, S. Ferrara, Phys. Lett. B 357 Ž1995. 76. w10x M. Gunaydin, G. Sierra, P.K. Townsend, Nucl. Phys. B 242 ¨ Ž1984. 244; B 253 Ž1985. 573. w11x K. Behrndt, A.H. Chamseddine, W.A. Sabra, to appear. w12x B. de Wit, A. Van Proyen, Phys. Lett. 293 Ž1992. 94. w13x A.H. Chamseddine, W.A. Sabra, Phys. Lett. B 426 Ž1998. 36, hep-thr9811161. w14x H. Lu, C.N. Pope, J. Rahmfeld, A Construction of Killing spinors on S n , hep-thr9805151. w15x W.A. Sabra, Mod. Phys. Lett. A 13 Ž1998. 239, hepthr9708103 w16x L.A.J. London, Nucl. Phys. B 434 Ž1995. 709.