PHYSICAL REVIEW D VOLUME 55, NUMBER 6 15 MARCH 1997
Enhancement of supersymmetry near a 5D black hole horizon
Ali Chamseddine* Theoretishe Physik, ETH-Ho¨nggenberg, CH-8093 Zu´rich, Switzerland
Sergio Ferrara† Theory Division, CERN, 1211 Geneva 23, Switzerland
Gary W. Gibbons‡ DAMTP, Cambridge University, Silver Street, Cambridge CB3 9EW, United Kingdom
Renata Kallosh§ Physics Department, Stanford University, Stanford, California 94305-4060 ͑Received 29 October 1996͒
dϪpϪ2 Geometric Killing spinors which exist on AdS pϩ2ϫS sometimes may be identified with supersym- metric Killing spinors. This explains the enhancement of unbroken supersymmetry near the p-brane horizon in d dimensions. The corresponding p-brane interpolates between two maximally supersymmetric vacua, at infinity and at the horizon. A new case is studied here: pϭ0, dϭ5. The details of the supersymmetric version of the very special geometry are presented. We find the area-entropy formula of the supersymmetric 5D black holes via the volume of S3 which depends on charges and the intersection matrix. ͓S0556-2821͑97͒00406-2͔
PACS number͑s͒: 04.70.Dy, 04.50.ϩh, 11.30.Pb
I. INTRODUCTION Remarkably, the supersymmetry near the five- dimensional ͑5D͒ black hole horizon has not yet been stud- The enhancement of supersymmetry near the p-brane ho- ied. Moreover, the unbroken supersymmetry of the 5D black rizon is an interesting phenomenon. The corresponding holes was established in ͓5͔ only for black holes of pure p-branes interpolate between two maximally supersymmetric Nϭ2, dϭ5 supergravity without vector multiplets. In par- d dϪpϪ2 vacua: at infinity and AdS pϩ2ϫS near the hori- ticular, the Strominger-Vafa 5D black holes ͓6͔ and the ro- zon ͓1–3M͔. The generic reason for the enhancement of super- tating generalization of them ͓7͔ have not yet been embedded symmetry is the following. Any anti–de Sitter space as well into a particular 5D supersymmetric theory and the unbroken as any sphere admit Killing spinors which we will call geo- supersymmetry of either solution has not been checked di- metric Killing spinors. The relation of this geometric Killing rectly. An analogous situation holds with the more general spinor, which has a dimension of a Dirac spinor, to the Kill- 5D static and rotating black holes found in ͓8͔. There are ing spinor of unbroken supersymmetry requires further various indications, however, that these solutions have un- investigation. In theories with Nϭ2 supersymmetry the broken supersymmetry. unbroken supersymmetry of the p-brane has dimension The purpose of this paper is to find out whether the five- of one-half of the Dirac spinor. If near the horizon the dimensional black holes near the horizon show the enhance- dimension of the supersymmetric Killing spinor becomes ment of the supersymmetry. Specifically, we will try to iden- that of the Dirac spinor, we have an enhancement of super- tify the supersymmetric Killing spinors admitted by black symmetry. In some cases it has already been established that holes near the horizon with the geometric Killing spinors of 3 the Killing spinor defined by the zero mode of the gravitino AdS 2ϫS . We will also derive the area formula for generic transformation at the near horizon geometry of the p-brane solutions in Nϭ2 theory interacting with an arbitrary num- solutions coincides with the geometric spinors. These cases ber of vector multiplets as the volume of the S3 and express 2 include dϭ4, pϭ0 with AdS 2ϫS ͓1,2͔, dϭ10, pϭ3 with it as the function of charges and the intersection matrix. 5 7 AdS 5ϫS , dϭ11, pϭ2 with AdS 3ϫS , and dϭ11, pϭ5 For the d-dimensional manifold which is a product space 3 ϫ dϪpϪ2 with AdS 7ϫS ͓3͔. The near horizon geometry of these AdS pϩ2 S the geometric Killing spinors are given by dϪpϪ2 p-branes is known to be maximally supersymmetric. In the product of Killing spinors on AdS pϩ2 and S .On dϪpϪ2 dϭ4 the integrability condition for the Bertotti-Robinson ge- AdS pϩ2 and on S the Killing spinor equations are 2 ometry near the black hole horizon AdS 2ϫS was proved in ˆ ,using the fact that this geometry is conformally flat and ٌa͑x͒ϵٌ͑aϩc1˜␥a͒͑x͒ϭ0, aϭ0,1,...,pϩ1 4,2͔͓ that the graviphoton field strength is covariantly constant. ͑1͒
ˆ .͑y͒ϵٌ͑␣ϩc2␥␣͒͑y͒ϭ0, ␣ϭpϩ2,...,dϪ1␣ٌ *Electronic address: [email protected] ͑2͒ †Electronic address: ferraras@cernvm.cern.ch ‡ Electronic address: [email protected] Here ␥␣ is the ␥ matrix of the (dϪpϪ2)-dimensional Eu- § Electronic address: [email protected] clidean space and the commutator ͓˜␥a ,˜␥b͔ equals
0556-2821/97/55͑6͒/3647͑7͒/$10.0055 3647 © 1997 The American Physical Society 3648 CHAMSEDDINE, FERRARA, GIBBONS, AND KALLOSH 55
Ϫ͓␥a ,␥b͔ where ␥a is the ␥ matrix of the II. SUPERSYMMETRIC VERY SPECIAL GEOMETRY 5؍pϩ2)-dimensional Minkowski space. IN d) The integrability conditions for the geometric Killing The action for dϭ5, Nϭ2 supergravity coupled to spinors defined above are Nϭ2 vector multiplets has been constructed by Gu¨naydin, Sierra, and Townsend ͓11͔. The bosonic part of the action ˆ ˆ cd 2 c d d c a ,ٌb͔͑x͒ϭ0 Rab ϭ4͑c1͒ ͑␦a ␦b Ϫ␦a ␦b ͒, has been adapted to the very special geometry ͓12͔ and theٌ͓ ⇒ ͑3͒ compactification of 11D supergravity down to five dimen- sions on Calabi-Yau threefolds ͓13͔ with Hodge numbers (h(1,1) ,h(2,1)) and topological intersection form CIJK . For ,͒␥ ␦␦ ␦͑y͒ϭ0 R ␥␦ϭϪ4͑c ͒2͑␦ ␥␦ ␦Ϫ͔ ˆٌ, ˆٌ͓ ␣  ⇒ ␣ 2 ␣  ␣  our purpose, it will be extremely useful to adapt the full ͑4͒ action and the supersymmetry transformation laws of ͓11͔ to that of very special geometry. In what follows we will give and it is satisfied by the geometries of the anti–de Sitter the detailed derivation of some formulae reported in ͓9͔. m I i i space and sphere. The full Killing spinor is The fields of the theory are e ,A , ,r ,r , where Iϭ0,1,...,h(1,1)Ϫ1, iϭ1,..., h(1,1)Ϫ1, and rϭ1,2. The index r on the spinors is raised and lowered with the sym- ⑀͑x,y͒ϭ͑x͒͑y͒ ͑5͒ plectic metric ⑀rs and will be omitted. The Nϭ2, dϭ5 supersymmetric Lagrangian describing and it forms a Dirac spinor in a given dimension d.Onthe the coupling of vector multiplets to supergravity is deter- other hand, the Killing spinor of unbroken supersymmetry of mined by one function which is given by the intersection the p-brane solution near the horizon is defined by the cor- form on a CY threefold: responding supersymmetry transformation of the gravitino. If supersymmetric Killing spinor near the horizon can be iden- 1 ϭ C XIXJXK. ͑7͒ tified with the geometric one of the maximally supersymmet- V 6 IJK ric product space, we have an enhancement of supersymme- try: The action is
1 1 1 ͑x͒ϭ0 eϪ1 ϭϪ RϪ ¯ ⌫D Ϫ G F IFJ ˆٌ ˆ 2 2 4 IJ ⌿Mϭ0 ٌM⑀͑x,y͒ϭ0 . ͑6͒ L␦ ͪ ͑y͒ϭ0 ˆٌ ͩ⇒ ⇒ ␣ i k⌫l g ͒j ץϪ ¯i͑g ⌫D ϩ⌫ 2 ij kj li To study this issue we will work out some details of dϭ5, Nϭ2 supergravity coupled to Nϭ2 vector multiplets 1 i i ץjϪ ¯⌫⌫ץi ץ Ϫ g in the framework of very special geometry, see Sec. II. This 2 ij 2 i formulation of the dϭ5 theory is particularly adapted to dϭ11 supergravity compactified on Calabi-Yau manifold 1 3 2/3 i ¯i I characterized by the intersection number C . In Sec. III ϩ tI,i ⌫ ⌫ Fϩ 1/3 ABC 4ͩ4ͪ 16•6 we will identify the supersymmetric Killing spinors admitted by black holes near the horizon with the geometric Killing J K L ,k 3i 3 ϫ͑gijtIϪ9CJKLt,it,jt,ktI ͒Ϫ 1/3 tI spinors of AdS 2ϫS . We will also perform a detailed deri- 16•6 vation of the area formula of the five-dimensional black Ϫ1 holes ͓9͔ in terms of the volume of S3 which we evaluate in e ϫ͑¯ ⌫ FI ϩ2¯FI ͒ϩ ⑀ terms of the charges and intersection number matrix. In Sec. 48
IV we will consider a special case of the Nϭ2 theory inter- I J K acting with one vector multiplet, which comes from the trun- ϫCIJKFFAϩ••• , ͑8͒ cation of Nϭ4, dϭ5 supergravity. This will allow us to I I embed some of the known black holes into supersymmetric where we use dots for four-fermionic terms. Here t ϭt () theories and study the enhancement of supersymmetry near are the special coordinates subject to the conditions the horizon. We also study the Nϭ4 theory interacting with tIt ϭ1, C tItJtKϭ1, ͑9͒ arbitrary number n of Nϭ4 vector multiplets as a particular I IJK example of Nϭ2 theory interacting with nϩ1, Nϭ2 vector and tI are the ‘‘dual coordinates’’ multiplets and intersection matrix C0ijϭij , where ij is the Lorenzian metric on (1,n). The duality invariant area- J K J I IJ tIϭCIJKt t ϵCIJt , t ϭC tJ. ͑10͒ entropy formula for the Nϭ4 theory is truncated to Nϭ2 theory as it was done before in four-dimensional theory in We have introduced a notation ͓10͔. In this way one describes the theory with very special K IJ I geometry O(1,1)ϫO(1,n)/O(n). In the discussion section CIJϵCIJKt , C CJKϭ␦ K . ͑11͒ we explain the relation between the enhancement of super- symmetries and finiteness of the area of the horizon and The metric is derived from the prepotential ͑7͒ through the I ( Xץ/ץIϵץ) speculate about other cases as yet not known. relation 55 ENHANCEMENT OF SUPERSYMMETRY NEAR A 5D . . . 3649
1 III. NEAR HORIZON GEOMETRY J͑ln ͉͒ ϭ1 ͑12͒ AND SUPERSYMMETRYץIץ GIJϭϪ 2 V V Double-extreme black holes which have 1/2 of unbroken I I ͑ and X is related to t by supersymmetry and have constant moduli ͓14͔͒ in five di- I 1/3 I mensions have the geometry of the extreme Tangherlini so- X ϭ6 t ͉ ϭ1 . ͑13͒ V lution ͓15͔. It is a 5D analogue of the extreme Reissner- Nordstrom metric: Finally, the metric gij is given by 2 2 2 Ϫ2 J K I J K r0 r0 gijϭGIJX ,iX ,jϭϪ3CIJKt t ,it ,j . ͑14͒ ds2ϭϪ 1Ϫ dt2ϩ 1Ϫ dr2ϩr2d⍀2 ͫ ͩ r ͪ ͬ ͫ ͩ r ͪ ͬ 3 I We can express GIJ in terms of t through the equations ͑20͒ 61/3 3 and G ϭϪ C Ϫ t t ͑15͒ IJ 2 ͩ IJ 2 I Jͪ J i 2ͱϪgGIJFtrϭqI , ϭconst. ͑21͒ and the inverse metric acting on vectors is The horizon is at rϭr0 where the parameter r0 defining the 2 horizon as well as the constant values of moduli depend on GIJϭϪ ͑CIJϪ3tItJ͒, GIKG ϭ␦I . ͑16͒ 61/3 KJ J charges of the vector fields and on the topological intersec- tion form CIJK . Our study of the near horizon geometry will We will need, in what follows, the relation between the de- allow us to determine this dependence. I Note that the area of the horizon of the black hole is given rivative of the special coordinate (t ),i and that of the dual by the volume of the three-dimensional sphere coordinate tI,i . Using the fact that CIJK is a symmetric nu- merical tensor we have 2 3 Aϭ2 r0 . 1 t ϭ2C ͑tJ͒ tKϭ2C ͑tJ͒ , ͑tJ͒ ϭ CIJt . The area formula of 5D black holes was found in ͓9͔ from I,i IJK ,i IJ ,i ,i 2 I,i the observation that the unbroken supersymmetry near the ͑17͒ I black hole horizon requires that the central charge Zϳt qI Differentiating Eqs. ͑9͒ we get has to be extremized in the moduli space, i.e., near the hori- iZϭ0. This leads to the area formula in the formץ zon C tItJ͑tK͒ ϭ0, t ͑tI͒ ϭt tIϭ0. ͑18͒ Aϳ(q q CIJ )3/4, where CIJ is the inverse of iZϭ0ץ͉ IJK ,i I ,i I,i I J K CIJϭCIJKt . In the derivation of this area formula it was The supersymmetry transformation laws are assumed that the unbroken supersymmetry of the black hole 1 solution is enhanced near the horizon. This will be proved m m ␦e ϭ ¯⑀⌫ , now. It will be also explained why the enhancement of su- 2 persymmetry near the horizon requires the extremization of the central charge for describing the area formula. By exactly i ˆ ˆ I solving the Killing equations for the near horizon geometry ␦ϭD͑͒⑀ϩ 1/3 tI͑⌫ Ϫ4␦ ⌫ ͒F ⑀ 8•6 we will be able to justify the area formula suggested in ͓9͔ and find the explicit area formula including the numerical 1 1 ϩ g ⌫ ⑀¯i⌫jϪ⌫ ⑀¯i⌫j factor in front of it. 12 ijͩ4 Near the horizon at r r0 and one can exhibit the 3 →ˆ AdS 2ϫS geometry using rϭ(rϪr0) 0 ¯i j ¯i j → Ϫ⌫ ⑀ ⌫ ϩ2⑀ ⌫ , ͪ 2 Ϫ2 2rˆ 2rˆ ds2ϭϪ dt2ϩ drˆ2ϩr2d⍀2 . ͑22͒ ͩ r ͪ ͩ r ͪ 0 3 1 0 0 ␦AI ϭ61/3 tI ¯⑀⌫ iϩi¯⑀tI , 2 ,i ͩ ͪ Since we deal with the product space we may use aϭ0,1 for the coordinates of the AdS space and ␣ϭ2,3,4 for the co- 1 3 2/3 i 2 k l I j ordinates of the three-sphere ͑in tangent space͒. The vector ⑀ ץ ⌫iϭ␦ ⌫ki lϩ tI,i⌫ ⑀FϪ gij␦ 4ͩ4ͪ 2 field ansatz near the horizon becomes
3i 3 J ϩ tI tJ tK Ϫ3⑀¯jkϩ⌫⑀¯j⌫k 2͑r0͒ GIJFabϭ⑀abqI . ͑23͒ 16 ,i , j ,kͩ Let us use this ansatz in the fermionic part of supersymmetry 1 ϩ ⌫ ⑀¯j⌫k , transformations with all vanishing fermions and constant 2 ͪ moduli. We start with gaugino and keep only relevant terms
i 1 3 2/3 ␦iϭ ¯⑀i. ͑19͒ ␦ ϭ t ⌫⑀FI ϭ0. ͑24͒ 2 i 4 ͩ 4ͪ I,i 3650 CHAMSEDDINE, FERRARA, GIBBONS, AND KALLOSH 55
We study the possibility that the zero mode of this equation What remains to be done to make the extremization of the is given by the full size spinor ⑀ without linear constraints on central charge consistent is to check what happens with the it, i.e., that the unbroken supersymmetry is indeed enhanced gravitino: does the gravitino transformation rule admit the near the horizon. This is possible, provided that Killing spinor of the full supersymmetry in our background? And what are the conditions of that? Using the ansatz for the vector fields near the horizon ͑23͒ and taking into account IJ tI,iG qJϭ0. ͑25͒ that we have a product space we get the following form of the gravitino transformations ͑26͒: The enhancement of unbroken supersymmetry near the hori- ,zon which can be deduced from the gaugino part of super- ␦ ϭٌ͑ˆ ͒ϵٌ͑ ϩc˜␥ ͒ϭ0, aϭ0,1 symmetry, can be represented as the condition of the mini- a a a a mization of the central charge, as found in ͓9͔. Let us derive c ˆ this here in a more detailed way. The gravitino transforma- ␦␣ϭٌ͑␣͒ϵ ٌ␣Ϫ ␥␣ ϭ0, ␣ϭ2,3,4, ͑33͒ tion ͩ 2 ͪ where i I ˜␥ ϭi⑀ ␥b, ⌫aϭ␥aI, ⌫ ϭi␥ ␥ ␥ , ͑34͒ ␦ϭD͑͒⑀ϩ 1/3 tI͑⌫ Ϫ4␦ ⌫ ͒F ⑀ϭ0 a ab ␣ 0 1 ␣ 8•6 ͑26͒ and shows that the graviphoton field strength is given by the 1 tIq I I linear combination of vector fields and moduli tIF , and cϭϪ 2/3 3 , ͑35͒ I 6 r0 therefore the central charge is proportional to t qI . From Eq. 25 we get ͑ ͒ and the ␥ matrices satisfy
͕˜␥ ,˜␥ ͖ϭ2 , ϭ͑Ϫ,ϩ͒, ˜␥ ϭϪ␥ , t CIJϪ3tItJ q ϭ0. ͑27͒ a b ab ab [ab] [ab] I,i͑ ͒ J ͑36͒ Using Eqs. ͑17͒ and ͑18͒ we can conclude that ͕␥␣ ,␥͖ϭ2␦ab , ␣ϭ2,3,4. ͑37͒
Zϭ0. ͑28͒ This is a special example of the general case presented in ץ t ͑CIJϪ3tItJ͒q ϭ2͑tJ͒ q ϭ0 I,i J ,i J ⇒ i the Introduction. We have identified the supersymmetric Thus we have derived the condition of minimization of the Killing spinor with the geometric Killing spinor. The inte- I grability conditions for the geometric Killing spinors defined central charge Zϭt qI in the moduli space from the require- ment that the gaugino supersymmetry transformation for above are constant moduli has the full size spinor ⑀ as a zero mode, 4 ͑tIq ͒2 i.e., from the condition of enhancement of supersymmetry ˆ ˆ cd I c d d c ,͒ a ,ٌb͔͑x͒ϭ0 Rab ϭ 4/3 6 ͑␦a ␦b Ϫ␦a ␦bٌ͓ near the black hole horizon. The central charge has to be ⇒ 6 r0 independent of i. Let us consider some useful identities of ͑38͒ the real special geometry which are valid only near horizon. ␦␥ Note that ٌ͓ˆ ,ٌˆ ͔͑y͒ϭ0 R ␣  ⇒ ␣ 1 tIq ͒2 ͑ I ␥ ␦ ␦ ␥ .͒ ␦ ␣␦ZϭgijtI tJ q q ϵ⌸IJq q ϭ0. ͑29͒ ϭϪ 4/3 6 ͑␦␣ ␦ Ϫ ץZ ץgij i j ,i ,j I J I J 6 r0 Using Eq. ͑18͒ we find that ͑39͒
This can be contracted to give us the Ricci tensors on IJ 3 ⌸ tIϭ0 ͑30͒ AdS 2 and on S :
I 2 I 2 and we may look for the combination which is orthogonal to 4 ͑t qI͒ 2 ͑t qI͒ IJ IJ I J R ϭ , R ϭϪ ␦ , tI in the form ⌸ ϭl(C Ϫt t ). To get the coefficient l we ab 64/3 r6 ab ␣ 64/3 r6 ␣ 2/3 I J ij 0 0 use gijϭ6 t ,it , jGIJ and contract it with g : ͑40͒
with the result that the radii of AdS and S3 are related: ij 2/3 IJ 2 gijg ϭh͑1,1͒Ϫ1ϭ6 ⌸ GIJ , ͑31͒ 4 1 R ϭϪ R͑3͒. ͑41͒ which leads to lϭϪ 3 . Thus we conclude that near the ho- ͑2͒ 3 rizon where the central charge is moduli independent we have an identity This relation restricts the properties of geometric Killing spinors. They would exist without any relation between these ͓͑CIJϪtItJ͒q q ͔ ϭ0. ͑32͒ iZϭ0 two product geometries. However, supersymmetric Killingץ I J 55 ENHANCEMENT OF SUPERSYMMETRY NEAR A 5D . . . 3651
spinors require relation between geometries. The curvature 1 ␦e mϭ ¯⑀i⌫m , can be calculated also directly from the metric ͑22͒: 2 i 1 ˆ ͑/3͒ Ϫ͑2/3͒ 4 2 ␦iϭD͑͒⑀iϩ e FijϪ e G⍀ij Rabϭ 2 ab , R␣ϭϪ 2 ␦␣ . ͑42͒ 6ͩ 2ͱ2 ͪ r0 r0 j ϫ͑⌫ Ϫ4␦ ⌫ ͒⑀ ϩ•••, Comparing Eqs. ͑40͒ with Eqs. ͑42͒, we can express r via 1 1 0 ij Ϫ/3 ¯[i j] ij¯k the values of the moduli near the horizon and electric charges ␦AϭϪ e ⑀ ⌫ ϩ ⍀ ⑀ ⌫k ͱ3 ͩ 4 ͪ as 1 ϪieϪ͑/3͒ ¯⑀[i j]ϩ ⍀ij¯⑀k , 1 4 k r4ϭ ͓͑tIq ͒ ͔2. ͑43͒ ͩ ͪ iZϭ0ץ I 64/3 0
1 3 This gives us the area of the horizon 2/3 i ͱ i ␦Bϭ e ¯⑀ ⌫iϪ ¯⑀ i , ͱ6 ͩ 2 ͪ 2 Aϭ22r3ϭ ͕͑tItJq q ͒ ͖3/4. ͑44͒ iZϭ0ץ I J 3 0 i 1 ⑀iϩץ ⌫ Near the horizon at iZϭ0 we can rewrite it as ␦iϭϪ 2ͱ3 2ͱ3 ץ
2 1 ϭ IJ 3/4 /3 Ϫ2/3 j ,•••iZϭ0͖ ͑45͒ ϫ e Fijϩ e G⍀ij ⌫ ⑀ ϩץA ͕͑C qIqJ͒ 3 ͩ ͱ2 ͪ
using identity ͑32͒. ͱ3 ␦ϭ ¯⑀i . ͑47͒ Thus we have combined the supersymmetry analysis with 2 i the analysis of the geometry. In this way we have confirmed
the structure of the area formula of five-dimensional black Here ⍀ij is a symplectic matrix. The truncation of Nϭ4 holes in Nϭ2 theories obtained in ͓9͔ and found the exact supergravity to Nϭ2 supergravity interacting with one vec- numerical coefficient in front of it for supersymmetric five tor multiplet is carried out by keeping i ,i with (iϭ1,2) dimensional black holes with finite area of the horizon in the only and A 12ϭϪA 34ϵ 1 A , B , g and .Byin- Nϭ2 theory. 2 specting the supersymmetry transformations of the truncated fields it is not difficult to see that this is a consistent trunca- tion. The bosonic action becomes1 2؍SUPERGRAVITY TO N 4؍IV. TRUNCATION OF N
In some cases it is useful to consider those Nϭ2 theories 1 1 1 Ϫ1 ͑2/3͒ Ϫ͑4/3͒ which can be obtained by truncation from Nϭ4 theories. We e ϭϪ RϪ e FF Ϫ e GG L 2 4 4 will first study the truncation of pure Nϭ4 supergravity and later generalize the result to the case of Nϭ4 supergravity 1 eϪ1 ͒2ϩ ⑀F F B . ͑48͒ ץ͑ interacting with arbitrary number of vector multiplets as it Ϫ 6 was done before in 4D theories in ͓10͔. 4ͱ2 We focus on a a special example of the double extreme The supersymmetry transformations of the fermionic fields five-dimensional black hole known as the Strominger-Vafa with vanishing fermion are black hole ͓6͔. The Lagrangian which allows a supersymmet- ric embedding of this black hole is obtained most easily by i ␦ ϭD ͑͒⑀Ϫ ͑⌫ Ϫ4␦ ⌫͒ the truncation of Nϭ4 supergravity in dϭ5 constructed by 12 Awada and Townsend ͓17͔. The bosonic part of the action is 1 ͑/3͒ Ϫ͑2/3͒ ϫ e FϪ e G ⑀, ͩ ͱ2 ͪ 1 1 1 eϪ1 ϭϪ RϪ e͑2/3͒F ijF Ϫ eϪ͑4/3͒ G G 2 4 ij 4 i 1 L /3 ⑀Ϫ ⌫ ͑e Fץ ⌫ ϭϪ␦ 2ͱ3 4ͱ3 1 eϪ1 2 ij Ϫ2/3 ,⑀͒ ϩ ⑀ F FijB , ͑46͒ ϩͱ2e G͒ץ͑ Ϫ 6 4ͱ2
1 where i, jϭ1,...,4andG is the field strength of B and This action is equivalent upon rescalings to the action of Nϭ2 ij ij F is the field strength of A . The supersymmetry trans- supergravity interacting with one vector multiplet as presented in formation laws are the Appendix of ͓16͔. 3652 CHAMSEDDINE, FERRARA, GIBBONS, AND KALLOSH 55 and we have omitted the symplectic index on the spinors ͑to 32/3 i tIq ϭ ͓q ͑q ͒2͔1/3 ͑56͒ avoid confusion with the index on in the previous sec- I 27/6 0 1 tion͒. Our action can be compared with the action used in ͓6͔ if and we rewrite it as follows: 1 1 8 Q Q 2 6 I 3 2 ͑ H F͒ 1 1 1 r0ϭ 2 ͕͑t qI͒hor͖ 11/2 ͑Ϫq0͒ q1ϭ 2 , eϪ1 ϭ ϪRϪ e͑2/3͒FЈ FЈϪ eϪ͑4/3͒GЈ GЈ 6 ⇒ 2 L 2 ͩ 4 4 ͑57͒ Ϫ1 1 e and ͒2ϩ ⑀FЈ FЈ BЈ , ͑50͒ ץ͑ Ϫ 3 8 ͪ 2 1/2 2 q q 2 3 0 1 where Aϭ2 ͑r0͒ ϭ . ͑58͒ 2 ͩ ͱ2 ͫͱ2ͬ ͪ FЈϭͱ2F, GЈϭͱ2G, BЈϭͱ2B. ͑51͒ Thus we have reproduced the Strominger-Vafa area for- mula as an example of our general area formula We observe a discrepancy in the kinetic term for the scalar field which is, however, harmless since the solution has a 2 Q ͑Q ͒2 constant moduli field. Thus the extreme black hole of 2 3 IJ 3/4 H F , iZϭ0͖ 8ͱץAϭ2 ͑r0͒ ϭ ͕͑C qIqJ͒ Strominger-Vafa can be embedded into the Nϭ2 supergrav- 3 ⇒ 2 ity interacting with one constant vector multiplet. ͑59͒ To make contact with the formalism of previous section and therefore this black hole solution fits into our consider- dealing with the general case of the very special geometry, ation of the very special geometry. we make the following identifications: iϭ1, Iϭ0,1 and For the general case of n, Nϭ4 vector multiplets with ϫ iϭ1, Iϭ0,1, A 1ϵA , A 0ϵB , duality group O(1,1) O(5,n)/͓O(5)x͔O(n) the formula for the largest eigenvalue of the central charge, extremized in the moduli space was presented in ͓10͔. This translates into the G ϭe2/3, G ϭeϪ͑4/3͒, C ϭ2ͱ2, g ϭ1, 11 00 011 11 Nϭ4 duality invariant area formula
1 8ͱ2Q 2 1 F QH͑QF͒ ϭ , q1ϭϪ , q0ϭ2ͱ2QH . ͑52͒ Aϭ8 , ͑60͒ ͱ3 ͱ 2
2 Comparing the supersymmetry transformations we must where QH is the singlet charge and (QF) is the O(5,n) identify Lorenzian norm of the other 5ϩn charges QF . Upon truncation to Nϭ2 theory this gives a very special t 2 4 2/3ץ t 1 4 2/3ץ 1 /3 0 Ϫ2/3 geometry with CIJK intersection of the form 1 ϭϪ e , 1 ϭϪ e . ( ͱ3ͩ3ͪ (I,Jϭ0,1, ...,nϩ1) and (iϭ1,..., nϩ1ץ ͱ3 ͩ3ͪץ ͑53͒ C0ijϭij C , ͑61͒ To satisfy the relations of special geometry and in particular IJK⇒ͩ0 otherwiseͪ I to have (t ),itIϭ0 we get where ij is the Lorenzian metric of O(1,n) and the very 2/3 2/3 4 2 3 special geometry is O(1,1)ϫO(1,n)/O(n). Upon truncation t ϭϪ e/3, t1ϭ eϪ/3, 1 ͩ3ͪ 3ͩ 4ͪ the area has the same form as in Eq. ͑60͒,
2 1 4 2/3 2 3 2/3 QH͑QF͒ QH͑QiijQj͒ Ϫ2/3 0 2/3 Aϭ8 ϭ8 , ͑62͒ t0ϭ e , t ϭ e . ͑54͒ ͱ 2 ͱ 2 ͱ2 ͩ 3ͪ ͱ3ͩ4ͪ where again Q is the singlet charge and (Q )2 is the The enhancement of supersymmetry near the horizon is pro- H F IJ O(1,n) Lorenzian norm of the other 1ϩn charges Qi . )G qJϭ0 and we get the fixed value ofץ/ tIץ) vided by the moduli near the horizon: V. DISCUSSION 1 q 1 1 1 2 1/3 Thus we have given a complete description of 5D static e ϭϪ , t q1ϭ 1/6 1/3 ͓q0͑q1͒ ͔ , ͱ2 q0 2 3 black holes near the horizon which can be embedded into Nϭ2 and Nϭ4 supergravity with arbitrary number of vector 1 multiplets. They all show enhancement of supersymmetry t0q ϭ t1q . ͑55͒ 0 2 1 near the black hole horizon. In the Nϭ2 as well as in the Nϭ4 case, the unbroken supersymmetry ͑1/2 in Nϭ2 and I Now we can express the combination t qI near the horizon 1/4 in Nϭ4) is doubled. As the result in all cases the dimen- required for the entropy as sion of the Killing spinor of unbroken supersymmetry is the 55 ENHANCEMENT OF SUPERSYMMETRY NEAR A 5D . . . 3653 dimension of the Dirac spinor admitted by the anti–de Sitter Killing spinors with the geometric ones for the 5D and 6D space and the sphere. strings. The significance of the enhancement of supersymmetry From the perspective of this study it would be interest- near the horizon is related to the fact that in all cases when ing to have a more careful look into configurations with dϪ2 we have found finite horizon area of supersymmetric solu- AdS 2ϫS geometries near the horizon. These would have dϪ2 tions there was also the enhancement of supersymmetry finite nonvanishing area related to the volume of the S present. This concerns all four- and five-dimensional static sphere. Such configurations of the Reissner-Nordstrom- Tangherlini type are known to solve equations of motion of black holes where we deal with the near-horizon geometries 2 2 3 Einstein-Maxwell theory in any dimension. However in AdS 2ϫS and AdS 2ϫS , respectively. The area in both cases is given by the volume of the S2 and S3 sphere. dϾ5 they do not seem to have a supersymmetric embedding, In more general situations when the near horizon geom- at least such embeddings have not been found so far. dϪpϪ2 In conclusion, we have studied the mechanism of en- etry is given by AdS pϩ2ϫS , the area of the horizon is given by the volume of the SdϪpϪ2 sphere times the volume hancement of unbroken supersymmetry near the 5D black of the torus of dimension p, which for the supersymmetric hole horizon and we have found the entropy-area formula for solutions shrinks to zero near the horizon. As the result, in solutions of Nϭ2 supergravity interacting with arbitrary dϪpϪ2 number of vector multiplets. the class of solutions with AdS pϩ2ϫS near-horizon geometries one cannot expect finite area of the horizon per ACKNOWLEDGMENTS unit volume except for black holes. Examples of such con- figurations with enhancement of supersymmetry and still We are grateful to Gary Horowitz, Andrew Strominger, vanishing area of the horizon include: dϭ10, pϭ3; Amanda Peet, Toma´s Ortı´n, Arvind Rajaraman, Paul dϭ11, pϭ2, pϭ5. This geometric observation matches Townsend, and Wing Kai Wong for stimulating discussions. the recent analysis ͓19͔ of the cases of the vanishing entropy The work of S.F. was supported in part by U.S. DOE Grant in which a duality invariant expression in terms of integral No. DE-FGO3-91ER40662 and by European Commission charges does not exist. In all these situations the extremum TMR program No. ERBFMRX-CT96-0045. G.W.G. would of the central charge does not occur at finite rational values like to thank N. Straumann for hospitality and the Swiss of the moduli. National Science Foundation for financial support at ITP, Few more configurations with near-horizon geometry Zurich where part of this work was carried out. R.K. was dϪpϪ2 2 AdS pϩ2ϫS are known ͓18͔. They include AdS 3ϫS supported by the NSF Grant No. PHY-8612280 and by the 3 describing a 5D magnetic string and AdS 3ϫS describing a Institute of Theoretical Physics ͑ETH͒ in Zurich and CERN 6D self-dual string. It has not been established yet whether where part of this work was done. they have enhancement of supersymmetry near the horizon. A calculation of the type which we have performed here for 5D black holes is required to identify the supersymmetric 2We are grateful to G. Horowitz for this observation.
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