View metadata, citation and similar papers at core.ac.uk brought to you by CORE
provided by CERN Document Server
Anti-de Sitter space and black holes
M´aximo Ba˜nados1,2,3, Andr´es Gomberoff2 and Cristi´an Mart´ınez2 1Departamento de F´ısica Te´orica, Universidad de Zaragoza, Ciudad Universitaria 50009, Zaragoza, Spain, 2Centro de Estudios Cient´ıficos de Santiago, Casilla 16443, Santiago 9, Chile, 3Departamento de F´ısica, Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chile
topological black hole has an interesting causal struc- Anti-de Sitter space with identified points give rise to ture displayed by a D − 1 Kruskal (or Penrose-Carter) black-hole structures. This was first pointed out in three diagram, as opposed to the usual 2-dimensional picture dimensions, and generalized to higher dimensions by Amin- [14]. Indeed, the metric in Kruskal coordinates has an ex- neborg et al. In this paper, we analyse several aspects of plicit SO(D−2,1)×SO(2) invariance, as opposed to the the five dimensional anti-de Sitter black hole including, its Schwarzschild black hole having an SO(1, 1) SO(D 1) relation to thermal anti-de Sitter space, its embedding in × − invariance. Accordingly, the topology of this black hole a Chern-Simons supergravity theory, its global charges and D−1 holonomies, and the existence of Killing spinors. is M ×S1 with S1 being the compactified coordinate, and M n denotes n−dimensional conformal Minkowski space. Just as for ordinary black holes, in the Euclidean D−1 I. INTRODUCTION formalism the Minkowski factor becomes < .This 2 topology has to be compared with M × SD−2 arising in usual situations like the D-dimensional Schwarzschild Despite of the “unphysical” properties of coupling a black hole. negative cosmological constant to general relativity, anti- We shall start (Sec. II) by reviewing the main prop- de Sitter space has a number of good properties. It has erties of the four dimensional topological black hole. Of been shown to be stable [1], and to possess positive en- particular interest is the Euclidean black hole obtained ergy representations [2] (see [3] for a review of further by identifications on Euclidean anti-de Sitter space. We properties of anti-de Sitter space). Recently, anti-de Sit- shall exhibit in section II E the explicit relation between ter space has appeared in a surprising new context. Mal- the Euclidean manifold considered in [5], as the Euclidean dacena [4] has conjectured that the large N limit of cer- sector of the topological black hole. We then introduce tain super conformal theories are equivalent to a string the five dimensional black hole metric (Sec. III), and theory on a background containing the direct product of study its embedding in a Chern-Simons theory in Sec. anti-de Sitter space (in five dimensions) times a compact IV. The black hole has a non-trivial topological struc- manifold. The conjecture raised in [4] has been inter- ture and its associated holonomies are calculated in Sec. preted as an anti-de Sitter holography in [5] by noticing V. Finally, we shall consider other possible solutions with that the boundary of anti-de Sitter space is conformal exotic topologies following from an ansatz suggested by Minkowski space. In this sense, the relation between con- the topological black hole in D dimensions. formal field theory in four dimensions and type IIB string theory on adS5 × M5 is analog to the relation between three dimensional Chern-Simons theory and 1+1 confor- II. THE FOUR DIMENSIONAL TOPOLOGICAL mal field theory [6]. BLACK HOLE The relationship between 5D anti-de Sitter space and conformal field theory is particularly interesting in the context of quantum black holes. The reason is that, in We shall start by considering the four dimensional case. 2+1 dimensions, the adS/CFT correspondence has pro- For a exhaustive analysis and classification of the prop- vided interesting proposals to understand the 2+1 black erties of anti-de Sitter space with identifications see [16]. hole entropy [7,8]. One may wonder whether there exists Here we shall review and further develop the three dimen- black holes analogous to the 2+1 black hole in five di- sional causal structure introduced in [14] and, in partic- mensions. This is indeed the case. It is now known that ular, the form of the metric in Kruskal coordinates. identifying point in anti-de Sitter space, in any number of dimensions, gives rise to black hole structures. These A. Identifications and causal structure black holes, often called “topological black holes”, were first discussed in three dimensions in [10,11], in four di- mensions in [12], and in higher dimensions in [13,14]. Fur- In four dimensions anti-de Sitter space is defined as the ther properties have been studied in [15]. universal covering of the surface A topological black hole can be defined as a space- 2 2 2 2 2 2 time whose local properties are trivial (constant cur- −x0 + x1 + x2 + x3 − x4 = −l . (1) vature, for example) but its causal global structure is This surface has ten Killing vectors or isometries and that of a black hole. The higher dimensional (D>3) define the anti-de Sitter group.
1 We shall identify points on this space along the boost upwards. The cone (3) has two pointwise connected branches, called Hf and Hp, defined by 2 r+ 2 r+ 2 2 ξ = (x3∂4 + x4∂3),ξ= (−x3+x4), (2) l l2 0 2 2 Hf : x =+ x1+x2, (5) where r+ is an arbitrary real constant. The norm of ξ 0 q 2 2 can be positive, negative or zero. This is the key prop- Hp : x = − x1 + x2 . (6) erty leading to black holes [10,11]. Identifications along a q rotational Killing vector produce the conical singularities Similarly, the hyperboloid (4) has two disconnected discussed in [17] which do not exhibit horizons. branches that we have called Sf and Sp, In order to see graphically the role of the identifica- tions, we plot parametrically the surface (1) in terms of 0 2 2 2 Sf : x =+ l +x1+x2, (7) the values of ξ2. 0 q 2 2 2 Sp : x = − l + x1 + x2 . (8) q The Killing vector ξ is spacelike in the region contained in-between Sf and Sp,itisnullatSf and Sp, and it is timelike in the causal future of Sf and in the causal past of Sp. We now identify points along the orbit of ξ.The1- dimensional manifold orthogonal to Fig. 1 becomes com- 2 pact and isomorphic to S1. The region where ξ < 0 becomes chronologically pathological because the identi- fications produce timelike curves. It is thus natural to Sf remove it from the physical spacetime. The hyperboloid is thus a singularity because timelike geodesics end there. In that sense Sf and Si represent the future and past sin- H gularities, respectively. f x0 Once the singularities are identified one can now see that the upper cone Hf defined in (3) represents the fu- ture horizon. Indeed, Hf coincides with the boundary of the causal past of lightlike infinity. In other words, all particles in the causal future of Hf can only hit the sin- gularity. Because infinity is connected in this geometry, as opposed to the usual Schwarzschild black hole having Hp two disconnected asymptotic regions, the lower cone does not represent a horizon in the usual sense. Note, how- ever, that the lower cone does contain relevant invariant Sp information. For example, all particles in the causal past of Hp come from the past singularity. In summary, we have seen that by identifying points in anti-de Sitter space one can produce a black hole with an FIG. 1. Anti-de Sitter space. Each point in the figure rep- unusual topology. The causal structure is displayed by a 2 2 three dimensional Penrose diagram (see fig 2), as opposed resent the pair x3,x4 with x3 − x4 (the norm of the Killing vector) fixed. to the usual two dimensional picture. Since, each point in Fig. 1 represents a circle, the topology of this black hole 3 n is M ×S1, where we denote by M conformal Minkowski 2 2 First we consider the surface ξ = r+ giving rise to the space in n dimensions. null surface
2 2 2 x0 = x1 + x2. (3) Then we consider the surface ξ2 = 0 which is represented by the hyperboloid
2 2 2 2 x0 = x1 + x2 + l . (4)
In Fig. 1, the ‘time’ coordinate x0 is drawn along the z-axis. Hence, future directed light cones are oriented
2 3 ϕ +2πn, and the resulting topology clearly is M × S1. With the help of (9), it is clear that the Kruskal diagram
Future Singularity associated to this geometry is the one shown in Fig. 1. 3 Thus, the metric (11) represents the M × S1 black hole written in Kruskal coordinates. The metric (11) has some differences with the usual
Schwarzschild–Kruskal metric that are worth mention-
8 8 + r + r ing. First of all, infinity is connected. There is only one asymptotic region with the topology of a cylinder ×S1 and thus, in particular, only one patch of Kruskal coordinates is needed to cover the full black hole space- time. Second, the metric has an explicit SO(2, 1)×SO(2) symmetry. The presence of the SO(2, 1) factor is a con- sequence of the three dimensional character of the causal structure. One could project down the Penrose diagram (Fig. 2) to a flat diagram, as done in [12,16], but this is not natural and makes the causal structure more compli- Past Singularity cated. The issue of trapped surfaces in the four dimensional topological black hole was studied in [16] where it was proved that there exists a non-trivial apparent hori- zon. We would like to point out here that if one de- FIG. 2. Penrose Diagram fines trapped surfaces as suggested by the three dimen- sional Kruskal diagram, then the apparent horizon is not present. We shall say that a closed spatial surface is trapped B. Kruskal coordinates if its evolution along any light like curve diminishes its area. Conversely, we say that the surface is not trapped 3 So far we have not displayed any metric. The M × S1 if there exists at least one light like curve along which black hole can be best described in terms of Kruskal co- the surface increases its area. Since every point in Fig. ordinates. Let us introduce local coordinates on anti-de 1 represents a circle, trapped surfaces are circles in this Sitter space (in the region ξ2 > 0) adapted to the Killing black hole and not spheres or tori. vector used to make the identifications. We introduce The existence of trapped surfaces (as defined above) in the 4 dimensionless local coordinates (yα,ϕ)by the black hole geometry can be easily checked using the Kruskal coordinates. At each point in Fig. 1 labeled by 2lyα coordinates (y0,y1,y2) there is a circle of radius r(yα), xα = 2 ,α=0,1,2 2 0 1 − y and inside the region Hf of Fig. 1, y < 0andy > lr r+ϕ 0. From (10) we see that a variation in the coordinates x3 = sinh , (9) yα(α =0,1,2) produces the following variation in r r+ l lr r+ϕ 2r x4 = cosh , α + α δr(y )= yαδy . (12) r+ l (1 − y2)2 with Since we move along a light like curve (δy)2 =0.Itis α 1+y2 then easy to see that the product yαδy is negative (just 2 2 2 2 α r = r+ ,y=−y0+y1+y2. (10) take a basis where y =(1,0,0)). Thus in the region 1−y2 which is at the causal future of the horizon (inside Hf ) Before the identifications are made, the coordinate ranges the circles are trapped. are −∞ <ϕ<∞and −∞ 2 2 2 l (r + r+) 2 2 2 2 2 ds = 2 (−dy0 + dy1 + dy2)+r dϕ , (11) r+ C. Schwarzschild metric 2 2 and the Killing vector reads ξ = ∂ϕ with ξ = r .The We have seen that anti-de Sitter spacetime leads natu- quotient space is thus simply obtained by identifying ϕ 3 ∼ rally to the existence of a M ×S1 black hole, and we have 3 found an explicit form for the metric in Kruskal form. A horizon, has the higher dimensional structure described natural question now is whether one can find a global set above. To pass to the region U<0 one would probably of coordinates for which the metric takes Schwarzschild need to make an analytic extension (Kruskal extension) form. We shall now prove that locally one can find such of (16). coordinates but they do not cover the full manifold, not There is another change of coordinates of (11), which even the full outer region. Indeed, the black hole space- has the advantage of covering the entire exterior of the time is not static. Minkowskian black hole geometry: This has a close analogy with the Schwarzschild black hole situation. In that case, the spherically symmetric co- y0 = f sinh(r+t/l), ordinates cover the outer manifold and the metric looks y1 = f cos θ cosh(r+t/l), static there. The maximal extension covers the full man- y = f sin θ cosh(r t/l), (17) ifold but it is not static. In our case, the maximal exten- 2 + sion is not static and covers the full manifold (and only where 0 <θ<2π,−∞ y2 = f cos θ, (13) 1/2 with f(r)=[(r−r+)/(r+r+)] and the ranges 0 < D. The vacuum solution θ<π,−∞ 4 E. Euclidean black hole and thermal anti-de Sitter previous discussion shows that one could keep ϕ real and space instead make the transformation ξ → iξ. This leads to an incomplete spacetime whose maximal extension give The black hole constructed above has an Euclidean sec- rise to the three dimensional Kruskal diagram shown be- tor which can be obtained from Euclidean anti-de Sitter fore. The idea of using the azimuthal coordinate as time space with identified points. Indeed, consider the surface was first discussed in [13]. 2 2 2 2 2 2 The relation between thermal anti-de Sitter and the x0 + x1 + x2 + x3 − x4 = −l , (21) black hole is already present in the three dimensional where we identify points along the boost in the plane situation. Indeed, consider the spinless three dimensional x3/x4. Following the same steps as before one obtains black hole the Euclidean Kruskal metric, 2 2 2 2 2 dr 2 2 2 2 ds (r+)=(−r +r )dt + + r dϕ , (28) l (r + r+) + 2 2 2 2 2 2 2 2 −r+ + r ds = 2 (dy0 + dy1 + dy2)+r dϕ , (22) r+ with r+ ≤ r<∞,0≤t<βand 0 ≤ ϕ<2π.The with period of t is fixed by demanding the absence of coni- 2 1+y cal singularities in the r/t plane and gives β =2π/r+. r = r+ . (23) 1−y2 We shall now prove that there exists a global change of coordinates that maps (28) into the three dimensional The ranges are 1 y ,y ,y 1with0 y2 <1. − ≤ 0 1 2 ≤ ≤ thermal adS space. After the identifications are done, the coordinate ϕ has First define r = r cosh ρ. This maps (28) into the range 0 ≤ ϕ<2π. The full Euclidean manifold is + thus mapped into a three dimensional solid ball (0 ≤ 2 2 2 2 2 2 2 2 2 ds1 = r+ sinh ρdt + dρ + r+ cosh ρdϕ . (29) y < 1) times S1. The boundary is then the 2-sphere 2 y =1timesS1. Since 0 ≤ t<2π/r+, we can define θ = r+t which has 0 The metric (22) can be put in a more familiar form period 0 ≤ θ<2π. We also define t = r+ϕ which has 0 1 2 0 by making the change of coordinates {ϕ, y ,y ,y }→ the period 2πr+. Finally we define r =sinhρobtaining {ϕ, r,˜ θ, ξ}: 02 0 dr y = f sin θ sin ξ (24) ds2(β0)=(1+r02)dt02 + +r02dθ2, (30) 2 1+r02 y1 = f sin θ cos ξ (25) 2 0 0 y = f cos θ, (26) with 0 ≤ r < ∞,0≤t <2πr+ and 0 ≤ θ<2π. √ The key element in this transformation is the permu- 2 2 2 with f =˜r/(l + l +˜r ). One obtains tation between the angular and time coordinates. It dr˜2 should be evident that for the Minkowskian black hole ds2 = N 2 (r dϕ)2 + +˜r2(dθ2 +sin2θdξ2) , (27) + N 2 this transformation is not possible. The permutation be- tween t and ϕ can be interpreted as a modular trans- with N 2 =1+˜r2/l2. This metric clearly represents formation that maps the boundary condition A0 = τAϕ Euclidean anti-de Sitter space with a “time” coordinate into A0 =(−1/τ)Aϕ [9]. It is then not a surprise that t = r ϕ. Actually, since ϕ is compact, 0 ≤ ϕ<2π, + the black hole had a temperature β =2π/r+, while (30) (27) should be called thermal anti-de Sitter space. The 0 has a temperature β =2πr+. Indeed, the corresponding metric (27) is the four dimensional version of the mani- modular parameters are related by a modular transfor- fold considered in [5]. The fact that we have end up with mation [9]. The line elements (28) and (30) have recently identified adS is not surprising since this was our starting been investigated in the context of string theory in [18]. point. The metric (27) represents the Euclidean sector of our 3 M × S1 black hole. We can now look at the black hole III. THE FIVE DIMENSIONAL TOPOLOGICAL from a different point of view. Suppose we start with Eu- BLACK HOLE clidean anti-de Sitter space with metric (27). Now we ask what is the hyperbolic, or Minkowskian sector, associ- We have seen in the last section that by identifying ated to this line element. The most natural continuation points in anti-de Sitter space one can produce a black to Minkowski space is obtained by setting ϕ → iϕ and 1 hole with many of the properties of the 2+1 black hole. yields Minkowskian anti-de Sitter space . However, our The topology of the causal structure is however different. In this section we shall describe how this procedure can be repeated in five dimensions. Actually, the black hole exists for any dimension [14] but the five dimensional case 1 Note that after continuing ϕ to imaginary values one is has some peculiarities that makes it worth of a separate forced to unwrap it in order to eliminate the closed timelike study, specially in the Euclidean sector. curves. 5 A. The spinless 5D black hole We start considering Euclidean anti-de Sitter space. In five dimensions, this space has three commuting Killing We shall not repeat here the geometrical construction vectors. In this section we shall write the induced met- of the 5D black hole (it can be found in [14]). We only ric in a set of coordinates for which those isometries are quoteheretheformoftheKruskalmetricwhichisa manifiest. natural generalization of (11), Euclidean anti-de Sitter space in five dimensions is de- fined by 2 2 2 l (r + r+) α β 2 2 ds = dy dy η + r dϕ , (31) 2 2 2 2 2 2 2 2 αβ x + x + x + x + x − x = −l . (36) r+ 0 1 2 3 4 5 where α =0,1,2,3, −∞ 2 2 2 l 2 2 2 dΩ3=−sin θdt + 2 (dθ +sin θdχ ). (35) r+ C. A black hole with two times The ranges are 0 <θ<π/2, 0 ≤ χ<2π,0≤ϕ<2π The key step in producing the black hole, in both the and r+ 6 2 2 2 2 2 2 ϕ 2 2 2 2 2 ds =cos θ[l N dt + r (dϕ + N dt) ]+ (r − r+)(r − r ) N 2 = − , +sin2θ[l2N2dχ2 + r2(dϕ + N ϕdχ)2] l2r2 2 2 2 2 ϕ r− (r − r+) r − r+ −2 2 2 2 N = − 2 , (48) + N dr + l 2 2 dθ , (42) r+ r r+ + b and with 2 2 2 2 2 2 2 2 2 r = r+ cosh ρ − r− sinh ρ. (49) 2 (r − r+)(r + b ) N = 2 2 , (43) l r Now we identify points along the coordinate ϕ: ϕ b r+b N = − 2 , (44) r+ r ϕ ∼ ϕ +2πn, n ∈ Z. (50) and the radial coordinate r is related to ρ by The metric (47) with the identifications (50) defines the 2 2 2 2 2 five dimensional Minkowskian rotational black hole. An r = r+ cosh ρ + b sinh ρ. (45) important issue, that we shall not attack here, is to find the maximal extension associated to this geometry. One We now identify the points ϕ ϕ +2πn obtaining the ∼ can imagine the form of the Penrose diagram in the rotat- Euclidean rotating black hole in five dimensions. ing case as a cylinder containing an infinite sequence of The metric (42) has a remarkable structure. The lapse alternating asymptotic and singularity regions. However, N 2 and shift N ϕ functions are exactly the same as those the explicit construction of the maximal extension has es- of the 2+1 black hole. The section θ = 0 represents a cape us. This problem is of importance because, as it has rotating 2+1 black hole in the three dimensional space been pointed out in [16], the topological black hole is not r, t, ϕ, while the section θ = π/2 represents a rotating static and therefore the metric (47) cannot be complete. 2+1 black hole in the space r, χ, ϕ. There is also an ex- Indeed, it is easy to prove that the replacement t˜→ it˜ in plicit symmetry under the interchange of t χ.This ↔ (41) produces an hyperbolic metric with does not cover metric can only exists in the Euclidean sector, or in a the full Minkowskian anti-de Sitter surface. Note finally spacetime with two times. Indeed, if one continues back that the section θ = 0 corresponds exactly to a rotating to a Minkowskian time by t it, one also needs to → 2+1 black hole. The metric (47) has a horizon located at change χ → iχ in order to maintain the metric real. r = r+. D. Minkowskian rotating black hole E. The extreme and zero mass black hole The Euclidean black hole that we have produced in the To find a extreme black hole from the general rotating last section cannot be continued back to Minkowskian case (47), we define the non-periodic coordinate σ by the space. In order to produce a black hole by identifications 2 2 2 2 on Minkowskian anti-de Sitter space we need to make relation lθ = σ r+ − r−, and take the limit r+ → r− = identifications along a different Killing vector. Let us go a2. The resultingq metric is back to the metric (41), make the replacement t˜ → it˜, and redefine the coordinates as r2 − a2 l2r2dr2 ds2 = ∓2 dtdϕ + l (r2 − a2)2 r2 − r2 r ϕ t˜= + − t + − 2 2 2 2 2 r − a 2 2 l r+ l + r + a σ dϕ r l2 ϕ˜ = + ϕ l +(r2 −a2)(dσ2 + σ2dχ2). (51) χ˜ = χ (46) ϕ The ∓ sign arises from the quotient r−/r+ in N (see with r+ and r− two arbitrary real constants with dimen- (48). This metric represents the extreme black hole. It sions of length. This converts (36) into has only one charge and one horizon and it also obtain- able from anti-de Sitter space by identifications. How- ds2 =cos2θ[−N2l2dt2 + r2(dϕ + N ϕdt)2]+N−2dr2 ever, as in 2+1 dimensions, the Killing vector needed is 2 2 not the same as in the non-extreme case. We shall ex- 2 r − r+ 2 2 2 + l (dθ +sin θdχ ) plicitly prove this below and when we compute the group r2 − r2 + − element g associated to the extreme black hole. The zero 2 2 r − r− 2 2 2 mass black hole can be obtained from (51) by letting + 2 2 r+ sin θdϕ , (47) r+ − r− a =0andϕ→ϕ±t. with 7 IV. EUCLIDEAN GLOBAL CHARGES IN FIVE this theory. Indeed, there is a curious cancellation of the DIMENSIONS contributions to the charges coming from the Einstein- Hilbert and Gauss-Bonnet terms. The issue of global charges requires to fix the asymp- We then consider a supergravity theory constructed as totic conditions and the class of metrics that will be con- a Chern-Simons theory for the supergroup SU(2, 2|N) sidered in the action principle. We shall make a slight [19] which is the natural extension of the three dimen- generalization of the metric (47) by considering again sional supergravity theory constructed in [20]. Remark- Euclidean anti-de Sitter space with the metric (41) and ably, the black hole also solves the equations of motion making the change of coordinates following from this action and as such the mass and an- gular momentum are different from zero. t˜= At + bϕ, In five dimensions, a Chern-Simons theory is defined by a Lie algebra [J ,J ]=fC J possessing and invariant ϕ˜ = Bt + r+ϕ, (52) A B AB C fully symmetric three rank tensor gABC. The Chern- with Simons equations of motion then read, 1 B C A = r β + bΩ, g F ∧F = 0 (56) + 2 ABC B = −bβ + r+Ω. (53) where F = dA + A∧A is the Yang-Mills curvature for the A This class of metrics contain the black hole in the case Lie algebra valued connection A = A JA.Thesolutions 2 2 that we have considered have F =0andthusA=g−1dg B = 0 which implies Ω = βb/r+, A = β(r+ +b )/r+ and (52) reduces to (46). The parameter β allows us to fix 0 < where g is a map from the manifold to the gauge group. t<1. For the black hole β is fixed by the demand of no However, due the identifications the black hole map g is 2 2 not single valued. In other words, if A represents the flat conical singularities at the horizon as β = r+/(b + r+). We consider here, however, only the asymptotic metric connection associated to the black hole, then the path following from (52) with β and Ω fixed but arbitrary. ordered integral along the non-trivial loop γ generated A convenient choice for the tetrad is by the identifications P exp γ A is different from one. In three dimensions, where the equations are simply 1 B H e = l sinh ρ cos θ(Adt + bdϕ) gABF = 0, it is now well established that the Hilbert e2 = l sinh ρ sin θdχ space of a Chern-Simons theory is described by a con- formal field theory [6]. For manifolds with a boundary, e3 = l sinh ρdθ the underlying conformal field theory is a Chiral WZW 4 e = ldρ model [21] whose spectrum is generated by Kac-Moody 5 e = l cosh ρ(Bdt + r+dϕ). (54) currents. From the point of view of classical Chern- Simons theory, these results can be derived by studying The corresponding non-zero components of the spin con- global charges [22,23]. nection are It is remarkable that part of the results valid on three dimensions are carried over in the generalizations to 13 ω = − sin θ(Adt + bdϕ) higher odd-dimensional spacetimes. Indeed, it has been ω14 =coshρcos θ(Adt + bdϕ) proved in [24] that in the canonical realization of the 45 gauge symmetries the constraints satisfy the algebra of ω = − sinh ρ(Bdt + r+dϕ) the WZW theory found in [25]. The WZW theory is ω23 =cosθdχ 4 4 a natural generalization to four dimensions of the usual 24 ω =coshρsin θdχ WZW theory. A key property of the WZW4 theory is ω34 =coshρdθ. (55) the need of a K¨ahler form (an Abelian closed 2-form). This two form, which greatly facilitates the issue of global With these formulas at hand we can now compute the charges, appears naturally in five dimensional supergrav- value of the different charges associated to this metric. ity [19,26]. Global charges can easily be constructed from the Chern-Simons equations of motion (56). Let EA be the A. Global charges in Chern-Simons gravity Chern-Simons equations of motion and let δAA be a per- turbation of the gauge field not necessarily satisfying the We shall now consider the five dimensional topologi- linearized equations. Using the Bianchi identity ∇F A = cal black holes embedded in a Chern-Simons formulation 0 and the invariant property of gABC (∇gABC =0)itis for gravity and supergravity in five dimensions [14]. We direct to see that δEA is given by, shall consider first a SO(4, 2) Chern-Simons theory which B C δE = ∇(g F ∧δA ). (57) represents the simplest formulation of gravity in five di- A ABC mensions as a Chern-Simons theory. As we shall see all Now, let λA a Killing vector of the background configura- A A charges associated to the topological black hole vanish in tion (∇λ = 0), then the combination λ δEA is a total 8 derivative, but since for the black hole R˜AB =0,δQ =0andthus the charge vanishes, up to an additive fixed constant. A A B C λ δEA = d(gABC λ F ∧δA ), (58) This result shows a curious cancellation in the value of A the charges associated to the Einstein-Hilbert and Gauss- and thus conserved d(λ δEA) = 0. Hence, for every A Bonnet terms. One can prove that the charges associated Killing vector λ there is one conserved current. We to the theory containing only the Hilbert term (plus Λ) now consider a manifold with the topology Σ ×< and are divergent quantities. We have just proved that the we assume that Σ has a boundary denote by ∂Σ. The full Lagrangian has zero charges, this implies a cancel- integral λAδE is thus independent of Σ and provides Σ a lation between the contributions to the global charges achargeat∂Σequalto R coming from the first and last two terms in (61). A B C δQ(λ)= gABCλ F ∧δA . (59) Z∂Σ This formula gives the value for the variation of Q.The C. Charges in the SO(4, 2) × U(1) theory. problem now is to extract the value of Q from (59). For this the specific form of the boundary conditions is nec- Now we consider the particular gauge group SO(4, 2)× essary. U(1). This group arises in the Chern-Simons super- gravity action in five dimensions. The orthogonal part SO(4, 2) is just the anti-de Sitter group in five dimensions B. The SO(4, 2) theory. Zero charges. while the central piece, U(1), is necessary to achieve su- persymmetry [19,26,28]. It is remarkable that the factor Consider now the case on which the Lie algebra is U(1) provides a great simplification in the analysis of the SO(4, 2) generated by JAB. Note the change in notation, canonical structure as well as the issue of global charges. each pair (A, B) correspond to A in (56). This algebra The coupling between the geometrical variables and the ˜AB ˜ indeed has a fully-symmetric invariant three rank tensor, U(1) field, denoted by b,issimplyR ∧ RAB ∧ b.The namely, the Levi-Cevita form ABCDEF . The equations equations of motion are modified in this case to of motion read AB CD ABCDEF R˜ ∧ R˜ = R˜EF ∧ K, (63) ˜AB ˜CD AB ABCDEF R ∧ R =0. (60) R˜ ∧ R˜AB =0, (64) The link with general relativity is achieved when one de- where K = db. As before, we identify ea = lωa6 which fines the ωa6 component of the gauge field as the viel- implies T a = lR˜a6 = Dea. Note that the topological bein: ea = lωa6. (The arbitrary parameter l, with di- black hole also solves the above equations of motion be- mensions of length, is introduced here to make ea adi- cause they have constant curvature (R˜ab = 0) and zero mensionful field and will be related to the cosmological torsion (T a =0). K is left arbitrary. constant.) Once this identification is done, the compo- Applying the formula (59) to this set of equations of nent Ra6 of the curvature becomes equal to the torsion: motion, using the boundary conditions R˜AB =0,which T a = lR˜a6 = Dea with D the covariant derivative in the are satisfied by the black hole, and fixing the value of b spin connection ωab. (and hence K = db), we obtain the value of δQ, The equations (60) can be shown to come from a Lagrangian containing a negative cosmological constant AB 2 δQ = K∧δω λAB . (65) (−1/l ), the Einstein-Hilbert term, and a Gauss-Bonnet ∂Σ term (which in five dimensions is not a total derivative), Z where λAB is a Killing vector of the background config- √ AB L = −g[kR2 +(1/G)R +Λ], (61) uration ∇η =0. Note that K is left arbitrary at the boundary (it ap- where R2 denotes the Gauss-Bonnet combination. The pears in the equations of motion multiplied by R˜AB). couplings k, ΛandGare not arbitrarily but linked by Hence it is consistent with the dynamics to fix its value the SO(4, 2) symmetry [19,27]. The topological black at the boundary. In terms of ea and ωab this formula hole solves the above equations simply because it has reads, zero torsion and constant spacetime curvature. These two conditions imply R˜AB = 0 and thus (60) is satisfied. a ab δQ[ηa,ηab]= K∧(2δe ηa − lδω ηab). (66) Going back to our equation for the charges we can ∂Σ Z see that they vanish for the black hole. Indeed, given a This formula for the global charge depends crucially on Killing vector η the charge associated, up to constants, AB the assumption that the topology of the manifold is Σ×< would be (or Σ × S1) and that Σ has a boundary. The black AB CD EF holes that we have been discussing do have this topol- δQ ∼ ABCDEF η R˜ δω , (62) ogy, however, the < (S1) factor does not represent time, Z 9 but rather, the angular coordinate. Indeed, the five di- We have just shown that the parameters r+ and b give mensional black hole has the topology B4 × S1 where B4 rise to conserved global charges. In the next section we 2 is a 4-ball whose boundary is S3 . As discussed before, shall compute the holonomies existing in the black hole the Euclidean black hole can be interpreted as thermal geometry and show that {r+,b} also represent gauge in- anti-de Sitter space provided one treats the angular co- variant quantities. ordinate as time. We shall now compute global charges for the black hole treating ϕ as the time coordinate. In the formula (66) ∂Σ thus represents the three-sphere pa- V. GROUP ELEMENT. HOLONOMIES rameterized by the coordinates t, χ, θ. The formula (66) depends on the 2-form K which was Since the topological black holes have constant space- not determined by the equations of motion. The only time curvature, their anti-de Sitter Yang-Mills curvature local conditions over K are dK =0(sinceK=db), and is equal to zero. Thus, they can be represented as a that it must have maximum rank [24]. In particular, K mapping g from the manifold to the gauge group G.This must be different from zero everywhere. It turns out that mapping is not trivial due to the identifications needed to global considerations suggest a natural choice for the pull produce the black hole. Here we shall exhibit the explicit back of K into ∂Σ=S3.SinceKis a two form, its dual form of the map g and extract relevant information from is a vector field. We then face the problem of defining a it such as the value of the holonomies, the gauge invariant vector field on S3, different from zero everywhere. As it quantities, the possibility of finding Killing spinors, and is well known there are not too many possibilities. In the the temperature of the black hole. In three dimensions coordinates t, χ, θ that parameterized the sphere in the the group element was calculated in [29]. metric (41) with the change (52) two natural choices for A word of caution concerning global issues is necessary. ∗ ∗ the dual of K are: Kt = ∂t or Kχ = ∂χ. It is direct to As we have pointed out above, the rotating black hole ∗ see that Kχ does not give rise to any conserved charges. is known only in the Schwarzschild coordinates which, ∗ 2 We shall then take K =(k/π )∂t with k an arbitrary in Minkowskian signature, do not cover the full exterior but fixed constant. [The normalization factor is included manifold. This problem is solved when J = 0 passing for convenience π2 = dθdχ.] to Kruskal coordinates. Here we shall be interested in The formula for the charge thus becomes the holonomies generated by going around the angular R k coordinate ϕ, and for this purpose the Schwarzschild co- δQ[η ,η ]= (2η δea − lη δωab)dS. (67) ordinates are good enough. a ab π2 a t ab t ZS3 We shall make use of the spinorial representation of The black hole geometries described above have three the so(4,2) algebra, given by the matrices Jab =Γab = 1 [Γ , Γ ]andJ = 1 Γ ,whereΓ aretheDiracmatri- commuting Killing vectors, ∂t, ∂χ and ∂ϕ.Thesemetric 4 a b a,6 2 a a Killing vectors translate into connection Killing vectors ces in five dimensions. Here A = {a, 6} with a =1,···,5. µ We use the following representation of Dirac matrices: via ξ Aµ. Thus, for example, invariance under transla- tions in ϕ is reproduce in the connection representations 0 −σm as invariance of the connection under gauge transforma- Γm = i , a ab σ¯m 0 tions with the parameter {eϕ,ωϕ }. From the formulas (54) and (55) one can compute the charge associated to with σ =(1,σ),σ ¯ =(1,σ), m =1, ,4and this invariance and obtain m − ··· Γ5=ıΓ1Γ2Γ3Γ4. For later reference we remind that in a ab 2 2 five dimensions there exists two irreducible independent δQ[eϕ,ωϕ ]=βδ(2kbr+)+Ωδ(k(r+ −b )). (68) representations for the Dirac matrices, namely Γm and The interpretation for (68) is straightforward: β and Ω −Γm. 2 2 are the conjugates of M =2kbr+ and J = k(r+ − b ). The gauge field A in the spinorial representation in Note that k, which acts as a coupling constant, is not terms of vielbein ea and spin connection ωab is defined an universal parameter in the action but the -fixed- value as, of the U(1) field strength K. This is similar to what ea 1 happens in string theory where the coupling constant is A = Γ + ωabΓ . (69) equal to the value of the dilaton field at infinity. 2l a 2 ab For notational simplicity, in this section we will set l =1. 2 In [14] Σ was taken to be S2 ×S1. This topology arises from A. Group element of 5D Minkowskian rotating black an error in the rank of the coordinate θ whose right value is hole 0 ≤ θ<π/2, as opposed to 0 ≤ θ<πas in [14]. Surprisingly, the actual value of the charges is insensitive to this problem To compute g for the rotating black hole it is conve- and the same value of M and J are obtained here. See Eq. nient to begin with the group element for the sector of (68) below. 10 anti-de Sitter space obtained from (41) with the change B. Group element for the extreme black hole t → it. The vielbein and spin connection for this metric are given in that section. As we have seen in previous sections, the extreme met- The group element g is related to gauge field A by ric cannot be obtained in a smooth way from the non- −1 A = g dg and A isgivenintermsofeand ω in (69). extreme one. It should then be not a surprise that the The equations that determines g thus are associated group elements cannot be deformed one into 1 the other. This fact reflects a deeper issue: The Killing ∂˜g = g( sinh ρ cos θΓ1 +coshρcos θΓ12 − sin θΓ14) , vectors that produce the non-extreme and extreme met- t 2 1 rics belongs to different classes and one cannot relate ∂ρg = gΓ2 , them by conjugation by an element of the anti-de Sitter 2 group. This was already the case in three dimensions 1 ∂ϕ˜g = g( cosh ρΓ3 − sinh ρΓ23) , [11], the novelty here is that, contrary to the 2+1 case, 2 the metrics cannot be deformed one into the other. 1 ∂θg = g( sinh ρΓ4 − cosh ρΓ24) , The metric for the extreme black hole was displayed in 2 (51). The vielbein can be chosen as 1 ∂ g = g( sinh ρ sin θΓ − cosh ρ sin θΓ − cos θΓ ) , χ 2 5 25 45 eρ e1 = − dt whose solution is F e2 = dρ 1 1 Γ3ϕ˜ Γ12t˜ Γ54χ Γ42θ Γ2ρ gadS = g0e 2 e e e e 2l , (70) eρ e3 = ∓ dt + Feρdϕ F with g0 an arbitrary constant group element. 4 ρ The black hole metric is obtained from (41) by making e = e dσ the transformation (46) and identifying points along the e5 = eρσdχ, (75) angular coordinate ϕ. Thus, we perform the same coordi- nate transformation in (70) obtaining the group element and the non zero components of the spin connection are for the rotating black hole 1 1 ρ 2 −ρ 1 ω 2 = − (e dt ∓ a e dϕ) (76) pϕϕ ptt −Γ45χ −Γ24θ Γ2ρ gBH = g0e e e e e 2 , (71) F 2 1 a −2ρ r2 r2 ω = ( e dρ σdσ) +− − 3 2 ∓ ± where pϕ = r+Γ3/2+r Γ12 and pt = Γ12. F − r+ The group element g encodes gauge invariant informa- a2σ ω1 =± dϕ tion through its holonomies. First note that 4 F ρ −1 2π(r Γ /2+r Γ ) 2 e 2 2 h := g g = e + 3 − 12 6=1. (72) ω 3 = (±dt − (1 + a σ ) dϕ) ϕ=2π ϕ=0 F 2 ρ We now define h = eW and compute the Casimirs asso- ω 4 = −e dσ AB 2 ρ ciated to W = W JAB, ω 5 = −e σdχ 2 2 2 2 2 3 a σ Tr W =4π (r−+r+), (73) ω 4 = dϕ 4 4 4 2 2 4 F Tr W =4π (r +6r r +r ). (74) 4 − − + + ω 5 = −dχ, (77) Odd powers of W vanish because W AB is antisymmet- with F = 1+a2(σ2 +e−2ρ)andρis related to radial ric. The fact that r+ and r− are related to the Casimirs coordinate r of (51) by ρ =log(r2−a2)1/2 . of the holonomy confirm that they are gauge invariant p quantities. The differential equation for g is now more compli- The only remaining independent Casimir allowed by cated but it can still be integrated. After a rather long the Cayley–Hamilton theorem is but direct calculation we find the group element for the extreme black hole AB CD EF C3 = ABCDEF W W W , 1 1 pϕϕ ptt −Γ45χ (Γ4/2−Γ24)σ Γ2ρ±log F Γ13 g = g0e 2 e e e e 2 , (78) which is zero for the black hole with two charges. In order 2 2 2 to have C3 6= 0 we would need to consider identifications where pϕ = ±a (Γ1/2−Γ12)+(a +2)Γ3/2+(a −2)Γ23, along a Killing vector with three parameters of the form pt = −Γ1/2 − Γ12 ∓ (Γ3/2 − Γ23), and g0 is a constant ξ = r1J01 + r2J23 + r3J45.NotethatJ01,J23 and J45 are matrix. commuting and therefore r1,r2 and r3 would represent We note that this group element cannot be derived observables of the associated spacetime. It is actually from the general case (71) by taking the limit r+ → r−. not difficult to produce a metric with three charges, but As stressed before, this reveals that the Killing vectors for our purposes here it is of no relevance so we skip it. used to construct these black holes belong to a distinct 11 classes in the isometry group. The value of the holonomy As in 2+1 dimensions [30], the five dimensional black in this case is (h = eπW ) hole has locally as many Killing spinors as anti-de Sitter space, but only some of them are compatible with the 2 Γ1 2 Γ3 2 identifications. The global Killing spinors will be those W = ±a ( − Γ12)+(a +2) +(a −2)Γ23 (79) 2 2 (anti-) periodic in ϕ or those independent of ϕ. We begin with the extreme massive case. Since that and the corresponding invariants are g in (78) is non-periodic in ϕ, it is necessary to choose Tr W 2 =8π2a2 and Tr W 4 =32π4a4. (80) 0 such that the angular dependence is eliminated. We find that there is only one possibility for each sign in the The group element associated to the massless black limit r+ →±r−,and0 must be proportional to hole can be obtained by letting a → 0 in the extreme g. One finds i ±1 1 . pϕϕ ptt −Γ45χ (Γ4/2−Γ24)σ 2 Γ2ρ ∓i g = g0e e e e e , (81) 1 where pϕ =Γ3/2−Γ23, pt = −Γ1/2−Γ12 ∓(Γ3/2−Γ23), As mentioned before, there exists other inequivalent and g0 is a constant matrix. spinorial representation of so(4,2). We note that in this new representation pϕ does not change as a function of C. Group element of 5D Euclidean black hole the Dirac matrices, but its explicit representation is mod- ified. We find that there is one Killing spinor for each sign provided that is proportional to We finally consider the group element associated to the 0 Euclidean black hole. The spinorial representation of the −i so(5,1) algebra can be obtained from that of so(4,2) by ∓1 . changing Γ1 by iΓ1 (and therefore Γ5 by iΓ5). ∓i To calculate the group element for Euclidean black 1 holes we proceed in the same way as in Minkowskian case. The analytic continuation does not modifies the The ϕ dependence of in the M = 0 black hole is fixed functional form of the group element for the portion of by anti-de Sitter space (41). The starting point is thus its group element (70) written in terms ofτ ˜ = it˜ instead of exp(−pϕϕ)=1−pϕϕ (86) t˜ 1 2 given that pϕ = Γ3 +Γ23 verifies p = 0. Clearly, the 1 1 2 ϕ Γ3ϕ˜ Γ12τ˜ −Γ45χ −Γ24θ Γ2ρ g = g0e 2 e e e e 2 . (82) global Killing spinors occurs only if 0 is a null eigenvec- tor of pϕ. As this matrix has two linearly independent The Euclidean rotating black hole is obtained making null eigenvector there are two global Killing spinors for the transformation the massless black hole. The same occurs in the other spinorial representation. τ˜ = Aτ + bϕ , ϕ˜ = r ϕ, (83) + As in 2+1 dimensions, the vacuum solution r+ = r− = 2 2 0 has the maximum number of supersymmetries: two for with A =(b +r+)/r+, which applied on (82) yields each representation of the γ matrices. 1 1 ( r+Γ3+bΓ12)ϕ AΓ12τ −Γ45χ −Γ24θ Γ2ρ g = g0e 2 e e e e 2 . (84) E. Temperature from the group element. D. Killing spinors As a final application of the group elements we calcu- late here the temperature of the Euclidean black hole. In the Chern-Simons formulation there is, in principle, An immediate application for the group elements cal- no metric and one cannot impose the usual non-conical culated in the previous sections is to analyse the issue of singularity condition on the Euclidean spacetime to de- Killing spinors. Killing spinors are defined by termine the Euclidean period, or inverse temperature. However, there exists an analog condition which can be D d+A=0, (85) ≡ implemented with the knowledge of g and give the right where D is the covariant derivative in the spinorial repre- value for the temperature. −1 We can compute the temperature by imposing that g sentation. It is direct to see that = g 0,withd0 =0 ( constant spinor), is the general solution of (85) provided is single valued as one turns in the time direction, that g−1dg = A g(τ = β)g−1(τ =0)=1. (87) 12 In the case of rotating black hole this condition reads When these properties are fulfilled, for a given value of k, we will say that the metric (89) represents a black hole 2 2 k b + r+ with topology M × WD−k. Of course the Schwarzschild exp{ 2 βJ12} =1. (88) l r+ and topological black holes satisfy (i)-(iv) with k =2 and k = D − 1, respectively. The problem now is to find Notice that we now use the vectorial representation J12 solutions to Einstein equations (or its generalizations) instead of the spinorial representation Γ12,andlhas been satisfying (i)-(iv) with different values of k, and different reincorporated. Since J12 is a rotational generator we choices of W . 2 2 b +r+ find the condition 2 β =2πwhich gives the right We have investigated the conditions imposed by the l r+ value for β [14]. standard Einstein equations on the functions H and s, with and without cosmological constant. For k>2the equations become difficult to treat and we did not suc- VI. OTHER SOLUTIONS AND FINAL REMARKS ceed in producing other black holes solutions. If the cos- mological constant is zero, and W is a torus then we can show that there are no solutions satisfying (i)-(iv). In the last sections we have discussed black holes with Probably, this result can also be understood in terms of the topology M D−1 × S . These black holes have con- 1 the classical theorems relating the existence of horizons stant curvature and are constructed by identifying points and the topology of spacetime. For k = 2 we have found in anti-de Sitter space. A natural question now arises. black hole solutions with a standard Kruskal picture but Could we find other black hole solutions, not necessar- new topologies for W are allowed. These solutions are ily of constant curvature, with topologies of the form described below. M k W with W a compact manifold? × D−k We shall leave for future investigations the problem of solving the ansatz (89) in alternative theories of gravity k like Stringy Gravity or Chern-Simons gravity. A. The M × WD k black hole ansatz − The Einstein equations for a metric of the form (89) are consistent only if the Ricci tensor associated to gij Consider a D–dimensional space–time with a metric of satisfy the form i i 2 a b 2 2 Rj = ζδj , (91) ds = H(s)ηabdx dx + r (s)dΩD−k , (89) where ζ can be chosen, without lost of generality, to take a with ηab =diag(−1,1,...,1) in the k first coordinates x , a b 2 the values 0 or ±1. This means that, in general, the Eu- s = ηabx x ,anddΩ is the metric of a D−k–dimensional clidean space W can be any Euclidean, compact Einstein compact Euclidean space (W ), space with cosmological constant (D − k − 2)ζ/2. In the Schwarzschild case, for example, this space is a sphere dΩ2 = g (y)dyidyj i, j = k +1, ..., D. (90) ij (ζ = 1). Another example is provided by the new black For k = 1, the above metric represents a cosmological– holes found in [12], where ζ = −1andW is a Riemann like solution. For k = 2, the conformally flat 2– surface of genus g>1. dimensional space of coordinates xa can be seen, in some The Einstein equations for (89) are therefore, 2 cases, as a Kruskal diagram for a M × S black hole. 2 D−2 0=4(D−k)H r0[(k − 1)r +(D−k−1)sr0] (92) For a generic value of k we would have a M k W × D−k 0 0 black hole. However, for this affirmation to be true, the +2(k−1)HH r[(k − 2)r +2(D−k)sr ] following properties must be satisfied: + 2Λr2 − (D − k)ζ H3 +(k−1)(k − 2)sr2H02 r(k−2) (i) The functions H(s)andr(s) must be well behaved 0= 3H02 − 2HH00 +4H(H0r0 −Hr00). at the point s = 0, which will represent black hole D − k horizon. (ii) The function r must be positive at the horizon, B. Solutions with no cosmological constant r(s =0)=:r+ >0. For s>0, r must be a crescent function and go to infinite for s → s∞, where s∞ can be either infinity or some positive For Λ = 0 and ζ =0(D−ktorus), the above equations finite number. For s<0 the function r must to can be integrated once giving go to zero at some point s = s and the spacetime 0 2(D−k−1) k 02 k−2 geometry should be singular there. r s r H = α, (93) (iii) The function H must be positive for all values of s. where α is an integration constant. This implies that there is no regular solution in the horizon s =0forthe (iv) W must be a compact space. 13 variables r and H. Therefore, it is not possible to con- C. Solutions with cosmological constant struct a well behaved black hole geometry with topology k (D−k) M ×T , at least without the cosmological constant. Let us now discuss some solutions of equations (92) For k =2andζ6= 0 we have the standard 2– in presence of cosmological constant. We will again set dimensional Kruskal diagrams. The general solution, k =2,D>3 and proceed as in the preceding to obtain a with the regularity conditions explained before is given one–parameter family of solutions given, in Schwarzschild by coordinates by 0 4r+r N 2 (D − 3) H = , (94) ds2 = − dt2 + N −2dr2 + r2dΩ2, (101) ζ ζ ζ (D − 3) dr s = α exp r + , (95) r (r/r )D−3 − 1 with + Z + ω Λ˜ where α and r+ are real constants. N 2 =1− − r2, (102) The fact that H should be positive and r crescent force rd−3 ζ ζ and r to have the same sign. From (95) we learn that + ˜ 2Λ(D−3) they must be positive. The family of solutions obtained Λ= (D−1)(D−2) ,andωis a real integration constant can be written in Schwarzschild coordinates performing which is proportional to the mass. Note that we can the transformation of coordinates: always eliminate the factor 1/|ζ| in dt2 (but not the sign of ζ), redefining the time coordinate. Moreover, we can √ ζ(D − 3) redefine r and eliminate (D − 3)/|ζ| from dr2 (the price x0 = s cosh t (96) we pay here is the redefinition of dΩ2). 2r+ ! p For positive mass and negative cosmological constant √ ζ(D − 3) we have two possibilities depending on the sign of ζ. x1 = s sinh t , (97) 2r When ζ<0 we obtain the generalization of the black p + ! hole solutions found in [12,15] for any dimension. Again, that brings the metric into the form when D>4, the metric dΩ2 can represent any surface i i with Rj = ζδj, which has more solutions than a surface (D − 3) dr2 of genus g.InD= 6, for example this could have the ds2 = −N 2dt2 + + r2dΩ2, (98) ζ N 2 topology T 2 × T 2. When ζ>0 we obtain the generalization of with Schwarzschild–Anti–de Sitter solutions to higher dimen- D 3 sions. Again we can extend this solutions to exotic r+ − N 2 =1− . (99) topologies. r For positive cosmological constant and ζ>0wehave √ h i√ [For s<0 we replace s with −s. In the above formu- the generalization of the Schwarzschild–de Sitter solu- las s is the function of r given by (95).] tions, which have an event horizon and a cosmological If we choose dΩ2 to be a (D − 2)–sphere of radius 1, horizon. When ζ<0 we have again the same kind of so- it is easy to show that ζ = D − 3 and we obtain the lution, but with surfaces of constant radius of hyperbolic Schwarzschild solution. geometry. It is possible, nevertheless, to find some new solutions. In D = 6, for example, we can construct a black hole with topology M 2 × S2 × S2,whichreads ACKNOWLEDGMENTS ds2 = −N 2dt2 + N −2dr2 (100) During this work we have benefited from many discus- 1 sions and useful comments raised by Marc Henneaux, + r2 dθ2 +sin2θ dϕ2 + dθ2 +sin2θ dϕ2 . 3 1 1 1 2 2 2 Miguel Ortiz, Peter Peld´an, Claudio Teitelboim and Jorge Zanelli. This work was partially supported by Note that for r+ = 0 this metric does not represent a the grants Nos. 1970150, 3960008 and 3970004 from flat spacetime and it has a naked curvature singularity FONDECYT (Chile). The institutional support to the at r =0. Centro de Estudios Cient´ıficos de Santiago of Fuerza Similar solutions can be constructed in any dimension, A´erea de Chile and a group of Chilean companies (Em- where the topology of the horizon can be any product presas CMPC, CGE, Codelco, Copec, Minera Collahuasi, m m m of spheres S 1 × S 2 ×···×S M,wheremi ≥2and Minera Escondida, Novagas, Business Design Associates, imi=D−2. Xerox Chile) is also recognized. P 14 [1] L. Abbot and S. Deser, Nucl. Phys. B195, 76 (1982). [2]C.Frondsal,Rev.Mod.Phys.37, 221 (1965). [3] G. Gibbons, in “Proceedings of Supersymmetry, super- gravity and related topics”, del Aguila, de Azcarraga and Iba˜nez eds, World Scientific 1995. [4] J.M. Maldacena, hep-th/9711200. [5] E. Witten, hep-th/9802150 [6] E. Witten, Commun. Math. Phys. 121, 351 (1989). [7] S. Carlip, Phys. Rev. D51, 632-637 (1995) [8] A.Strominger, hep-th/9712251 [9] M. Ba˜nados, T.Brotz and M. Ortiz, hep-th/9802076 [10] M. Ba˜nados, C. Teitelboim and J.Zanelli, Phys. Rev. Lett. 69, 1849 (1992). [11] M. Ba˜nados, M. Henneaux, C. Teitelboim and J.Zanelli, Phys. Rev. D48, 1506 (1993). [12] S Aminneborg, I Bengtsson, S Holst and P Peld´an, Class. Quant. Grav. 13, 2707 (1996). [13] M. Ba˜nados, C. Teitelboim and J.Zanelli, “New topology and signature in higher dimensions”, unpublished. [14] M. Ba˜nados, Phys. Rev. D57, 1068 (1998). [15] R. B. Mann, gr-qc/9705007; W. L. Smith and R. B. Mann, gr-qc/9703007; J. Creighton and R. B. Mann, gr-qc/9710042; D. Brill, J. Louko and P. Peld´an, Phys. Rev. D56, 3600 (1997); L. Vanzo, Phys.Rev. D56, 6475 (1997); R. B. Mann, gr-qc/9709039. [16] S. Holst and P. Peld´an, Class. Quantum Grav. 14, 3433 (1997). [17] S. Deser, R. Jackiw and G. t’ Hooft, Ann. Phys. (NY) 152, 220 (1984). [18] J. Maldacena and A. Strominger, hep-th/9804085 [19] A.H. Chamseddine, Nucl. Phys. B346, 213 (1990). [20] A. Ach´ucarro and P.K. Townsend, Phys. Lett. B180,89 (1986). [21] G. Moore and N. Seiberg, Phys. Lett. B220, 422 (1989); S. Elitzur, G. Moore, A. Schwimmer and N. Seiberg Nucl. Phys. B326, 108 (1989). [22] See A. P. Balachandran, G. Bimonti, K.S. Gupta, A. Stern, Int. Jour. Mod. Phys. A7, 4655 (1992). [23] M. Ba˜nados, Phys. Rev. D52, 5816 (1995). [24] M. Ba˜nados, L.J. Garay and M. Henneaux, Nucl. Phys. B476, 611 (1996); Phys. Rev. D53, R593 (1996). [25] V.P. Nair and J. Schiff, Phys. Lett. B246, 423 (1990); Nucl. Phys. B371, 329 (1992). [26] M. Ba˜nados, R. Troncoso and J.Zanelli, Phys.Rev. D54, 2605 (1996). [27] M. Ba˜nados, C. Teitelboim and J.Zanelli, Phys. Rev. D49, 975 (1994). [28] R. Troncoso and J. Zanelli, hep-th/9710180. [29] D. Cangemi, M. Leblanc and R.B. Mann, Phys. Rev. D48, 3606 (1993). [30] O. Coussaert and M. Henneaux, Phys. Rev. Lett. 72, 183 (1994). 15