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13 July 2000

Physics Letters B 485 Ž2000. 301–307 www.elsevier.nlrlocaternpe

Magnetic and dyonic black holes in D s 4 gauged supergravity A.H. Chamseddine, W.A. Sabra Center for AdÕanced Mathematical Sciences (CAMS) and Physics Department, American UniÕersity of Beirut, Lebanon Received 19 April 2000; accepted 26 May 2000 Editor: L. Alvarez-Gaume´

Abstract Magnetic and Dyonic solutions are constructed for the theory of abelian gauged N s 2 gauged four dimensional supergravity coupled to vector multiplets. The solutions found preserve 1r4 of the supersymmetry. q 2000 Published by Elsevier Science B.V.

1. Introduction There has been lots of interest in the study of solutions of gauged supergravity theories in recent years w1–9x. Anti-de Sitter ŽAdS. black holes solutions and their supersymmetric properties, in the context of gauged N s 2, D s 4 supergravity theory, were first considered in w10,11x. One of the motivations for the renewed interest is the fact that the ground state of gauged supergravity theories is AdS space-time and therefore, solutions of these theories may have implications for the AdSrconformal field theory correspondence proposed in w12x. The classical supergravity solution can provide some important information on the dual gauge theory in the large N Žthe rank of the gauge group.. An example of this is the Hawking–Page phase transition w13,14x. Also, the proposed AdSrCFT equivalence opens the possibility for a microscopic understanding of the Bekenstein–Hawking entropy of asymptotically anti-de Sitter black holes w15,16x. Moreover, AdS spaces are E-mail addresses: [email protected] ŽA.H. Chamseddine., [email protected] ŽW.A. Sabra..

known to admit ‘‘topological’’ black holes with some unusual geometric and physical properties Žsee for example w17x.. Of particular interest in this context are black objects in AdS space which preserve some fraction of supersymmetry. On the CFT side, these supergravity vacua could correspond to an expansion around non-zero vacuum expectation values of certain operators. Although lots is known about solutions in ungauged supergravity theories w18x, the situation is different for the gauged cases. Our main purpose in this paper is to obtain supersymmetric solutions for a very large class of supergravity theories coupled to vector multiplets. The theories we will consider are four dimensional abelian gauged N s 2 supergravity theories coupled to vector multiplets. Electrically charged solutions of these theories, asymptotic to AdS 4 space-time, were constructed in w7x. In this paper we will be mainly concerned with the construction of magnetic as well as dyonic solutions. Our work in this paper is organized as follows. Section two contains a brief review of D s 4, N s 2 gauged supergravity where we collect formulae of

0370-2693r00r$ - see front matter q 2000 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 6 5 2 - 3

A.H. Chamseddine, W.A. Sabrar Physics Letters B 485 (2000) 301–307

302

special geometry w20x necessary for our discussion and for completeness we present the electrically charged solutions of w7x. In section three, supersymmetric magnetic and dyonic solutions are constructed together with their Killing spinors. Finally, our results are summarized and discussed.

where UA s DAV s Ž EA q 12 EA K . V s

f AI . hA I

ž /

In general, one writes MI s NI J LJ ,h A I s NI J f AJ and the metric can then be expressed as g A B s y2ŽIm N .I J f AI f BJ. Moreover, special geometry implies the following useful relation

2. Gauged supergravity and special geometry IJ

g A B f AI f BJ s y 12 Ž Im N . y LI LJ . The construction of BPS solutions of the theory of ungauged and gauged N s 2 supergravity coupled to vector supermultiplets, relies very much on the structure of special geometry underlying these theories. The complex scalars z A of the vector supermultiplets of the N s 2 supergravity theory are coordinates which parametrize a special Kahler manifold. ¨ The Abelian gauging is achieved by introducing a linear combination of the abelian vector fields AmI of the theory, Am s k I AmI with a coupling constant g, where k I are constants. The couplings of the fermifields to this vector field break supersymmetry which in order to maintain one has to introduce gauge-invariant g-dependent terms. In a bosonic background, these additional terms produce a scalar potential w19x V s g 2 g A Bk I k J f AI f BJ y 3 k I k J LI LJ .

ž

/

Ž 1.

The meaning of the various quantities in Ž1. is as follows. Roughly one defines a special Kahler mani¨ fold in terms of a flat Ž2 n q 2.-dimensional symplectic bundle over the Kahler–Hodge manifold, with the ¨ covariantly holomorphic sections Vs

LI , MI

ž /

Ž 3.

The supersymmetry transformations for the gauginos and the gravitino in a bosonic background i i d cm s Dm q Ta bg a bgm y ig k I AmI q g k I LIgm e , 4 2

ž

/

dl A s i g mEm z A q i GrsA g rg s y gg A Bk I f BI e ,

ž

/

Ž 4.

where Dm is the covariant derivative, Dm s Ž Em y 14 vma bga b q 2i Qm ., vma b is the spin connection and Qm is the Kahler connection, which locally can be ¨ represented by Qsy

i 2

Ž EA Kdz A y EA Kd z A . .

Ž 5.

The quantities Tmn and GrsA are given by Tmn s 2 i Ž Im NI J . LI FmnJ Grn A s yg A B f BI Ž Im NI J . Frn J .

Ž 6.

Electrically charged BPS solutions for the N s 2, D s 4 supergravity with vector multiplets were constructed in w7x and are given by

I s 0, PPP ,n; ds 2 s y Ž e 2U q g 2 r 2 ey2 U . dt 2

DAV s Ž EA y 12 EA K . V s 0,

1 q

Že

obeying the symplectic constraint i ² V < V : s i Ž LI MI y LI MI . s 1.

Ž 2.

2U

q g 2 r 2 ey2U .

dr 2

q ey2 U r 2 Ž du 2 q sin2u d f 2 . , ey2 U s Y I HI ,

The scalar metric appearing in the potential can then be expressed as follows

i Ž FI Ž Y . y FI Ž Y . s HI ,

g A B s yi ²UA < UB : ,

A It s e 2U Y I ,

Y IsY I

Ž 7.


where HI s k I q qrI , and q I are the electric charges. These electric BPS solutions break half of supersymmetry where the Killing spinor e satisfies

e s Ž ag 0 q bg 1 . e ,

Ž 8.

where as

1

(1 q g

b s yi

2 2 y4U

First we concentrate on the purely magnetic BPS solutions. Therefore, we take the vector potential to have only one non-vanishing component, i.e., FufI s yq I sin u .

(1 q g

2 2 y4U

.

Ž 9.

r e

The solution for the Killing spinor is given by igt

1

2

1 y

g 0g 1g 2 u y

1

Using the above ansatz for the gauge fields and the spin connections in Ž13., the supersymmetry transformations for the gravitino Ž4. give

Ž 10 .

i q ge Ak I LIg 0 e , 2

where e 0 is an arbitrary constant spinor and T s H r 21r X Ž1 y r XEr X UŽ r X .. dr X .

dcu s Eu q 12 eyBg 1 g 2 q

3. Supersymmetric solutions

dcr s Er q

In this section, we find magnetic and dyonic BPS solutions Žwith constant scalars. of the theory of abelian gauged N s 2 supergravity coupled to vector supermultiplets. The solutions found preserve a quarter of supersymmetry. We consider the following general form for the metric ds s ye

2

2B

2

dt q e dr q r

2

Ž du

2

g 2 g 3 f UqT

e 2 e ' e e 2 = Ž 'f y 1 y ig 1'f q 1 . Ž 1 y g 0 . e 0 ,

2A

i

dc t s E t y 12 AX e Ay Bg 0 g 1 q

2 gr

2

Ž 14 .

r e

grey2 U

eŽ r. s

3.1. Magnetic solutions

AfI s q Icos u ,

,

303

2

2

q sin u d f

2

i 2

i 2

e A T23 g 2 g 3 g 0

T23 rg 3 q

T23 g 2 g 3 g 1 e B q

i 2

i 2

grk I LIg 2 e ,

g Ž k I LI . g 1 e B e ,

dcf s Ef q 12 cos ug 2 g 3 q 12 eyB sin ug 1 g 3

ž

qr sin u yT23 g 2 q

i 2

g k I LIg 3 y ig k I AfI e ,

/

..

Ž 15 .

Ž 11 . and for purely magnetic solutions we have

The vielbeins of this metric can be taken as e t0 s e A , e 0t s eyA ,

e 1r s e B , e1r s eyB ,

eu2 s r , e 2u s

ef3 s r sin u , 1 r

,

e 3f s

T23 s 2 i Ž Im NI J . LI F23J . 1

r sin u

,

Ž 12 . and the spin connections for the above metric are

v t01 s AX e Ay B ,

We will find supersymmetric configuration admitting Killing spinors which satisfy the following conditions

g1e sie ,

vu12 s yeyB ,

g2g3 e s ie .

vf13 s yeyB sin u ,

Because of the double projection on the spinor e , it is independent of the angular variables u and f . With the conditions Ž16., one obtains from the van-

vf23 s ycos u .

Ž 13 .

Ž 16 .


304

ishing of the gravitino supersymmetry transformations in Ž15. the following equations X yB

1 2

y Ae

i q 2

y 12 eyB y 1 2

i 2

1 2

eyB s yiT23 r q gr Ž k I LI . s gr q I

T23 q g Ž k I L . s 0,

T23 r q 12 g Ž k I LI . r s 0,

Ž 17 .

E t e s 0, Eu e s 0, 2

T23 e B y 12 g Ž k I LI . g 1 e B e s 0,

Ef e s 0.

Ž 18 .

ž

/

A s yB,

q

i 2 g

then from the first two equations in Ž17. one obtains

eyB s yiT23 r q gr Ž k I LI . .

Ž 19 .

An obvious solution to the above equations can be obtained if we set Ž k I LI . to a constant say k I LI s 1. Moreover, from the third relation in Ž17., one gets the following ‘‘quantization relation’’ .

2g

Therefore, as a solution for the holomorphic sections we take real LI and imaginary MI , with LI s 2 g q I .

Ž 20 .

Using the symplectic constraint Ž2., I

I

I

i Ž L MI y L MI . s iL Ž MI y MI . s 2 iL MI s 1.

Ž 21 . Then this relation together with Ž20. implies that the magnetic central charge Zm s MI q I s y 4ig , and therefore we obtain MI q r

2

I

i s

2 gr 2

,

/

The above transformation vanishes provided

Ž F23 I y iLI T23 . s 0, g 2

Im N I Jk J q g k J LJ LI s 0.

Ž 23 .

Using our solution it can be easily seen that the above equations are satisfied. In summary, we have obtained a BPS magnetic solution preserving a quarter of supersymmetry for the theories of abelian gauged N s 2, D s 4 supergravity theories with vector multiplets. This solution is given by ds 2 s y

I

T23 s 2 i Ž Im NI J . LI F23J s y2

Ž Fmn I y iLI Tmn . g mg n

q Im N I Jk J q g k J LJ LI e . 2

E Ž eyB . s iT23 q g Ž k I LI . ,

kI q s

Ž 22 .

f AI dl a A s ig mEm z A Ž EA q 12 EA K . LI

ž

1

.

dl A s i g mEm z A q i GrsA g rg s y gg A Bk I f BI e ,

If one takes the ansatz

I

2 gr

where Grn A s yg A B f BI Ž Im NI J . Frn J, g A B is the inverse Kahler metric and f BI s Ž E B q 12 E B K . LI. To ¨ demonstrate the vanishing of the gaugino supersymmetry variations for the choice of e , it is more convenient to multiply with f AI . This gives using our solution and the relations following from special geometry Ž3.,

and

i

1

Finally one has to check for the vanishing of the gaugino supersymmetry transformation for our solution. The gaugino transformation is given by

cos u y g Ž k I AfI . s 0,

Er y

and from Ž19. we obtain the following solution

ž

2

1 2 gr

q gr

/

dt 2 q

ž

1 2 gr

y2

q gr

/

dr 2

q r 2 Ž du 2 q sin2u d f 2 . , LI s 2 g q I ,

AfI s q Icos u .

3.2. Dyonic solutions In this section we generalize the previous discussion to include electric charges and thus obtaining


dyonic solutions. From the outset we set A s yB in the ansatz Ž11. as well as the condition k I LI s 1. The supersymmetric transformation for the gravitino in this case gives X y2 B

1 2

dc t s E t q B e

g 0g 1 q

i 2

Using the conditions Ž25., then from the vanishing of the transformations Ž24. we obtain the equations 1 y 2b

yB

T23 g 2 g 3 g 0 e

i

a 2b 1

i

q T01 g 1 eyB y ig k I A It q geyBg 0 e , 2 2

2b

dcu s Eu q 12 eyBg 1 g 2 q

i 2

dcr s Er q

i 2

i 2

2

2b

T23 r q

T01 r

BX eyB q

y

i i y T01 rg 0 g 1 g 2 q grg 2 e , 2 2

i

eyB y

eyB y

a

T23 rg 3

305

i 2

1 b

i 2

a

T01

b

rq

1 2

gr s 0,

s 0,

T23 y

i 2

a

T01

BX ey2 B y g k I A It q

b

i

q 12 g s 0,

T01

2

1 b

eyB s 0,

Ž 26 .

together with

E t e s 0, Eu e s 0,

T23 g 2 g 3 g 1 e B

Er y 12 T23 g 1 e B q

i i q T01 g 0 e B q gg 1 e B e , 2 2

i 2

T01 g 0 e B q

i 2

gg 1 e B e s 0,

Ef e s 0.

Ž 27 .

As an ansatz for the gauge and scalar fields we take 1 2

1 yB 2

dcf s Ef q cos ug 2 g 3 q e

sin ug 1 g 3 A It s

qr sin u

ž

i 2

T23 g 2 g 3 y

i q gg 3 y ig k I AfI e , 2

i 2

/

T01 g 0 g 1 g 3

F01I s

Ž 24 .

where T23 s 2 iIm NI J L

r

Z e LI r2

AfI s q Icos u ,

, ,

F23I s y

qI

I

where Z e s LI Q I , is the electric central charge, Q I being the electric charge. Therefore we obtain

F23J , T01 s 2 iIm NI J LI F01J s yi

The dyonic solutions can now be easily obtained by modifying one of the supersymmetry breaking conditions Ž16. and imposing the following conditions on the Killing spinors

Zm

Ž 25 .

clearly the coefficients a and b must satisfy the condition a 2 q b 2 s 1.

r

2

Ze r2

i s

2 gr 2

,

.

From Ž26. we obtain the following solution for the metric and the functions a and b, ey2 B s J12 q J22 . J1 s iT01 r s k I A It s

Ž iag 0 q bg 1 . e s i e ,

LI s 2 gq I ,

,

r2

T23 s 2 iIm NI J LI F23J s y2

T01 s 2 iIm NI J LI F01J .

g2g3 e s ie ,

Z e LI

Ze r

.

J2 s gr y iT23 r s gr q 2 i a s J1 e B ,

b s J2 e B .

Zm r

s gr q

1 2 gr

Ž 28 .


306

From the vanishing of the gaugino transformations in the presence of electric charges, we get the conditions Ž23. together with the new condition

Ž F01 I y iLI T01 . s 0. All these conditions can be seen to be satisfied for our ansatz. Finally, we summarize dyonic solutions of N s 2, D s 4 gauged supergravity with vector multiplets by ds 2 s y

q

žž žž

gr q

gr q

2

1 2 gr 1 2 gr

/ /

q

Z e2 2 r

2

q

Z e2 r2

/ /

One can also consider flat transverse space with vanishing gauge fields. Clearly one finds the solution which is locally AdS4 . However, one may wish to compactify the Ž u , f . sector to a cylinder or a torus, considering thus a quotient space of AdS4 . This identification results in AdS4 quotient space preserving half of the supersymmetries. 3.4. Killing spinors We now derive an expression for the Killing spinors of our solutions. From Ž27., one obtain the following equations for the Killing spinors

dt 2 y1

E t e s 0,

dr 2

Eu e s 0,

q r 2 Ž du 2 q sin2u d f 2 . , LI s 2 gq I ,

Af s q Icos u ,

ž

Z e s LI Q I .

Er q

1 2r

q igg 1 e B e s 0,

/

Ef e s 0. 3.3. Solutions with hyperbolic and flat transÕerse spaces Clearly the spherical solutions we found represent naked singularities. However, one can obtain extremal purely magnetic solutions with event horizons if the two sphere is replaced with the quotients of the hyperbolic two-space H 2 . In this case, it can be shown that one obtains the following solitonic dyonic solution

ž

ds 2 s y gr y

1 2 gr

2

/

ž

dt 2 q gr y

1 2 gr

Using the method of w10x, one obtain the following solution for the Killing spinor

eŽ r. s

ž(

eyB q gr q

(

yg 0

1 2 gr

eyB y gr y

1 2 gr

/

=P Ž yig 1 . P Ž yig 23 . e 0 ,

y2

/

dr 2

q r 2 Ž du 2 q sinh2u d f 2 . . AfI s q Icosh u ,

Ž 29 .

LI s 2 gq I .

As the Killing spinors do not depend on the coordinates u , f of the transverse hyperbolic space, one could also compactify the H 2 to a Riemann surface Sn of genus n, and the resulting solution would still preserve one quarter of supersymmetry. Whereas the spherical magnetic solution contains a naked singularity, the hyperbolic magnetic black hole solution is a genuine black hole solution which has an event horizon at r s rqs 1rŽ g'2 .. In the near horizon region, the metric reduces to the product manifold AdS2 = H 2 and thus supersymmetry is enhanced w21x.

where e 0 is a constant spinor and where we have used the notation P Ž G . s 12 Ž1 q G ., where G is an operator satisfying G 2 s 1. For the purely magnetic solution, i.e., Q I s 0, the Killing spinor reduces to the simple form

eŽ r. s

(

gr q

1 2 gr

P Ž yig 1 . P Ž yig 23 . e 0 .

It must be mentioned as was observed in w10x that for the purely magnetic solution, eyB and e Ž r . are invariant under r

™ 2 g1 r , 2

and under such a transformation, the spatial component of the metric is conformally rescaled. Clearly


the Killing spinors for the hyperbolic case are similar and can be obtained by replacing gr q 21gr by gr y 21gr everywhere w2x.

4. Discussion In summary, we have obtained magnetic and dyonic BPS solutions of the theory of abelian gauged N s 2 supergravity coupled to a number of vector supermultiplets. These solutions preserve a quarter of supersymmetry. Due to the quantization of the magnetic central charge for these solutions, the metric of the magnetic solutions takes a universal form for any choice of the prepotential and for any number of vector multiplets Žcharges.. The dyonic solutions depend also on the electric central charge. We note that our solution is expressed in terms of the holomorphic sections and the central charges of the theory and therefore independent of the existence of a holomorphic prepotential. A subclass of solutions of N s 2 supergravity Žfor a particular choice of prepotential. are actually also solutions of supergravity theories with more supersymmetry,Ži.e. N s 4 or N s 8 supersymmetries.. The spherically symmetric solutions has the common feature of representing naked singularities. However, it is known from the work of w11x that the spherical Kerr–Newman–AdS solution is both supersymmetric and extreme. This means that one can obtain supersymmetric extremal black holes in an AdS background if the black holes are rotating. Thus the situation in anti-de Sitter background seems to be the opposite of that of the asymptotically flat. Whereas the spherical BPS magnetic black holes contain a naked singularity, the hyperbolic black holes have an event horizon. Supersymmetric higher genus solution black holes with magnetic charges preserve the same amount of supersymmetry which is enhanced near the horizon as has been demonstrated in w21x for the pure supergravity theory without vector multiplets. Here we found that the situation remains the same in the presence of vector multiplets. In summary, in order to get genuine black holes for the theories we have considered in this paper, one should allow for rotations as well as

307

nonspherical symmetry. This will be reported on in a future publication.

References w1x K. Behrndt, A.H. Chamseddine, W.A. Sabra, Phys. Lett. B 442 Ž1998. 97. w2x M.M. Caldarelli, D. Klemm, Nucl. Phys. B 545 Ž1999. 434. w3x D. Klemm, Nucl. Phys. B 545 Ž1999. 461. w4x K. Behrndt, M. Cvetic, W.A. Sabra, Nucl. Phys. B 553 Ž1999. 317. w5x M.J. Duff, J.T. Liu, Nucl. Phys. B 554 Ž1999. 237. w6x J.T. Liu, R. Minasian, Phys. Lett. B 457 Ž1999. 39. w7x W.A. Sabra, Phys. Lett. B 458 Ž1999. 36. w8x A.H. Chamseddine, W.A. Sabra, Magnetic strings in five dimensional gauged supergravity theories, hep-thr9911195, to appear in Phys. Lett. B. w9x D. Klemm, W.A. Sabra, Supersymmetry of Black Strings in Ds 5 Gauged Supergravities, hep-thr0001131. w10x L.J. Romans, Nucl. Phys. B 383 Ž1992. 395. w11x A. Kostelecky, M. Perry, Phys. Lett. B 371 Ž1996. 191. w12x J.M. Maldacena, Adv. Theor. Math. Phys. 2 Ž1998. 231; Int. J. Theor. Phys. 38 Ž1999. 1113; E. Witten, Adv. Theor. Math. Phys. 2 Ž1998. 253; S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Phys. Lett. B 428 Ž1998. 105; O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri, Y. Oz, hep-thr9905111. w13x S.W. Hawking, D.N. Page, Commun. Math. Phys. 87 Ž1983. 577. w14x E. Witten, Adv. Theor. Math. Phys. 2 Ž1998. 505. w15x A. Strominger, JHEP 9802 Ž1998. 009. w16x J.D. Brown, M. Henneaux, Commun. Math. Phys. 104 Ž1986. 207. w17x R.B. Mann, Class. Quantum Grav. 14 Ž1997. L109; D. Klemm, L. Vanzo, Phys. Rev. D 58 Ž1998. 104025. w18x K.P. Tod, Phys. Lett. B 121 Ž1983. 241; Class. Quantum. Grav. 12 Ž1995.; S. Ferrara, R. Kallosh, A. Strominger, Phys. Rev. D 52 Ž1995. 5412; S. Ferrara, R. Kallosh, Phys. Rev. D 54 Ž1996. 1514; S. Ferrara, R. Kallosh, Phys. Rev. D 54 Ž1996. 1525, 1801; K. Behrndt, W.A. Sabra, Phys. Lett. B 401 Ž1997. 258; W.A. Sabra, Mod. Phys. Lett. A 12 Ž1997. 2585; Nucl. Phys. B 510 Ž1998. 247; K. Behrndt, D. Lust, ¨ W.A. Sabra, Nucl. Phys. B 510 Ž1998. 264; K. Behrndt, G.L. Cardoso, B. de Wit, D. Lust, ¨ T. Mohaupt, W.A. Sabra, Phys. Lett. B 429 Ž1998. 289; W.A. Sabra, Mod. Phys. Lett. A 13 Ž1998. 239; A. Chamseddine, W.A. Sabra, Phys. Lett. B 426 Ž1998. 36. w19x L. Andrianopoli, M. Bertolini, A. Ceresole, R. D’Auri, S. Ferrara, P. Fre’, T. Magri, J. Geom. Phys. 23 Ž1997. 111. w20x B. Craps, F. Roose, W. Troost, A. Van Proeyen, Nucl. Phys. B 503 Ž1997. 565. w21x S. Cacciatori, D. Klemm, D. Zanon, w` Algebras, Conformal Mechanics, and Black Holes, hep-thr9910065.