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Black Holes and Rings of minimal 5-dim. Supergravity

In this talk, I will summarize recent developments in 5-dimensional supergravity. Apart from black ring solutions, we will discuss the way of obtaining regular (bubbling) solutions with the same charges as black holes. We outline the procedure for the solution in five and four dimensions. Finally we explore the close relationship between 4- and 5-dimensional supersymmetric stationary solutions.


Introduction
The aim of this talk is to address recent progress of 5-dimensional supergravity. We start with an issue related to the uniqueness theorem of black holes, which states that the most general 4-dim. black hole has to be of the Kerr-Newmann type, ie. is uniquely fixed by its conserved charges (mass, angular momentum, electric/magnetic charges) and the horizon topology is a 2-sphere [1,2]. It implies especially that any wormhole has to be hidden behind a spherical horizon (topological censorship) and hence appears as a black hole for an outside observer. It fueled moreover the idea that stable microscopic black holes can regarded as elementary particles [3], which however is not (yet) realistic for different reasons, eg. there are no stable spinning black holes in four dimensions and moreover the mass to charge ratio is either not realistic or moduli dependent, but one should re-address this issue once all moduli are fixed. On the other hand, one can also ask, what happens in higher dimensions? As it turns out, the topology can be different. In five dimensions for examples, black rings with the geometry S 1 × S 2 have been found [4,5] and for D > 5, the allowed topology might be even more rich. This is in agreement with the general statement [6], that the horizon manifold has to have positive scalar curvature if the asymptotic vacuum is flat, which forbids in four dimensions, for example, a T 2 topology of a black hole horizon.
The second issue that I want to address in my talk is related to the information loss due to black holes. Any information about states passing an event horizon seems to be lost -one can recover only thermal radiation and this violates obviously the unitary evolution of quantum mechanical processes. The corresponding calculation has been done so far in a semiclassical approximation and one expects that a consistent theory of quantum gravity will eventually resolve this problem. In string theory encouraging results are obtained, which however rely mostly on supersymmetry or assume a weakly coupled regime, where black holes can be approximated by an intersection of branes embedded in flat space time [7,8]. New light on this issue is shed by the conjecture of Lunin and Mathur, who propose to "resolve" black holes into a smooth geometry carrying the same conserved charges [9,10]. Following this conjecture, black holes are fuzzballs of extended string states and the appearance of an event horizon is only an artifact of the parameterization and there is always an equivalent bound state with a regular geometry. As we will see below, for a large class of supersymmetric black holes, this is in fact possible. But it is unclear in the moment whether something analogous exists also for the Schwarzschild solution, for example.
Finally, we comment on the mapping of 4-and 5-dim. supersymmetric black holes. This mapping is interesting because it opens the possibility to estimate unknown corrections to the 5-dim. supergravity from known corrections in four dimensions. These are, for example, higher derivative couplings (eg. higher curvature terms), which are understood in four [11,12] but not in five dimensions. Moreover, the entropy of supersymmetric 4-dim. black holes is the Legendre transform of the free energy, which can be calculated in topological string theory [13]. One may wonder about the analog statement in five dimensions, which will be related to the M-theory limit of the topological string theory. In fact, the one-to-one map relates any supersymmetric stationary solution of 4-d supergravity to a localized solution in five dimensions with electric charges and non-vanishing magnetic dipole charges and implies especially that the corresponding (Bekenstein-Hawking) entropies coincide.

Classifying supersymmetric solutions of minimal supergravity
Before we discuss the relation of 4-and 5-dim. supergravity and the resolution of singularities, we should summarize what is known about supersymmetric solutions in five dimensions [14]. In contrast to four dimensions, 5-dim. black holes have in general two angular momenta J 1 and J 2 and the prime example of supersymmetric black holes in five dimensions is the BMPV solution [15], where both angular momenta are equal and which is charged under an U (1) gauge group. This solution has been generalized by coupling it to additional U (1) gauge groups and scalar fields [16] or to general N = 2 vector multiplets in [17,18]. It is straightforward to construct also a magnetic BPS string solution, which is infinitely extended and lifts to intersecting M5 branes in 11 dimensions. All these solutions rely however on certain assumptions for the geometry, the charges and/or the choice of harmonic functions, because they were not derived in complete generality. For minimal supergravity (ie. with 8 supercharges or N = 2), this analysis was done by Gauntlett etal. [14] by redoing basically the analysis of Tod in four dimensions [19]. The basic idea relies only on supersymmetry, which requires the existence of Killing spinors ǫ a (a = 1, 2) which can be combined into one scalar f = ε abǭ a ǫ b , one vector V α = ε abǭ a γ α ǫ b , and three 2-forms Φ ab αβ =ǭ a γ αβ ǫ b (symmetric in a, b). In the absence of sources, these spinors are globally well-defined and thus also the fermionic bi-linears are globally well-defined and, due to Fierz identities, the vector V is timelike with |V | 2 = −f 2 or null. The Killing spinor equations imply that the 2-forms are closed and have no components along V , which is a Killing vector. For the null case, it is pp-wave type solution (which we will not discuss here) and if V is timelike and we have stationary solutions, with hyper Kähler base space. Therefore the metric can be written as where h mn is the hyper Kähler metric on the base space. By employing the Killing spinor equations, one finds moreover [14] where G ± are (anti) selfdual forms on the base and dF = 0 implies that G + has to be closed. Now, a given solution is related to a choice of these forms and ω and f are solutions to these equations. For example, a static metric is related to G − = 0 or if the base space is compact and smooth, one obtains f = 1, G + = 0 whereas for the non-compact case f = 1 and G + = 0 yields the Gödel type solution.
The BMPV black hole corresponds to G + = 0 and a flat base space. In general, the base space can be any hyper Kähler space, but especially interesting is a Gibbons-Hawking space [20], which allows for a tri-holomorphic Killing vector and the corresponding dimensional reduction yields the known 4-dim.
(stationary) solutions. Denoting the Killing vector by ∂ ψ , this metric can be written as and thus N is 3-d harmonic function in x. For the simple case N = 1/| x| it is nothing but flat space and for N = 1 + n/| x| it is a Taub-NUT space with the NUT charge n. The general solution to the eqs. (2.2) becomes where (N, K, L, M ) are harmonic functions are their poles are related to the following conserved charges: In order to get the black ring solution with the horizon topology of S 1 × S 2 , one has to consider a 2-center solution. For example, a single black ring in a flat space at x = x 0 = (0, 0, −a) [5,21] requires with r = | x| and the three parameters p, q, a correspond to the magnetic dipole charge, the electric charge and the rotational parameter. Lifting this solution to 10 dimensions, yields the supertube solution of Emparan, Mateos and Townsend [22], which comprises a fundamental string, a D0-brane and the rotation generates a D2 dipole charge. We should also mention the non-extremal black ring solution has been found, see [23] for the most general example.

Bubbling solutions
Lunin and Mathur constructed a solution with the same conserved charges as a black hole, but without a singularity or horizon [9]. This "fuzzball" expands into a certain volume bounded by a surface whose area matches the Bekenstein-Hawking entropy. This surface grows with the degeneracy and the notion of an entropy is introduced by a coarse graining process, where one traces over the states within the volume. Moreover, they conjecture that string theory can always resolve (at least supersymmetric) black holes in fuzzballs of string states. In the strong version of this conjecture, also non-supersymmetric black holes as the Schwarzschild solution, for example, are expected to grow into fuzzballs. If this conjecture is true, there is no information paradox since all states remain visible.
But how is this possible? How can a horizon disappear?
To understand the answer, we have to recall that in compactified string/M-theory the horizon of a supersymmetric black hole is the location of (intersecting) branes. Especially 5-dim. black holes appear as intersecting M2-branes where the 4-form field strength solves the 11-dim. equation of motion After Calabi-Yau compactification this equation becomes where the selfdual 2-form G + is the one introduced in the last section. Expanding it in a set of harmonic 2-forms, the solution is given by (2.4) and the regular (bubbling) solution corresponds to the case, where the different sources of the harmonic functions cancel against each other. This means, that the M2-brane charge entering the function L is canceled by the dipole charges in K (related G + ) so that f −1 becomes constant near a given center. From the solution of f in (2.4), we see that this is only possible if N has also a pole at that center, ie. the NUT fiber has to degenerate at that point. Note, the dipole charges come from wrapped M5-branes and two intersecting M5-branes carry an effective membrane charge due to the F ∧ F term in (3.1). From the microscopic point of view, this membrane charge is due to a small instantons of the self-dual tensor field on the five brane world volume. In simple supergravity with only one U (1) gauge group this cancellation would imply that f becomes constant and the solution is trivial. We get only a non-trivial solution if the gauge group is [U (1)] n with n > 1 and in addition, if the solution has different centers so that at each center, a different charge is canceled. The modifications due couplings to additional vector multiplets have been derived in [24]. The selfdual 2-form G + has to be replaced by X A Θ A , where A = 1...n counts the different vector multiplets with the gauge fields Apart from gauge fields, each vector multiplet of 5-dim. minimal supergravity comes with a real scalar and the corresponding moduli space is defined by the cubic equation: V ≡ 1 6 C ABC X A X B X C = 1 with real X A and C ABC denoting the topological intersection numbers. The metric can then be written as

4)
The value of the function f and the scalar fields are now fixed by the algebraic equations and the solution is fixed by the set of harmonic functions M , N , L A and K A , ie. for each vector multiplet we have two functions or two charges (one electric charge and one magnetic dipole charge). The remaining two charges correspond to the graviphoton which enters the gravity multiplet.
The procedure to obtain a regular (bubbling) solution is now as follows [25,26]. One takes a multicenter solution (eg. for each U (1) a separate center) and cancels the poles in the harmonic function at each center so that There are two ways to understand these equations, either they fix the dipole charges (entering K A ) or one sees these equations as a fixing for moduli, which are related to the constant parts in the harmonic functions, see below. But these constraints are not yet enough for a well-defined supergravity solution.
One has also to proof that there are no closed time curves or Misner strings, which requires at each center. This will give a constraint on the position of the centers, which means that the moduli related to the positions in the external space are also lifted. We should also mention, that this procedure involves a subtle point. Namely, the cancellation requires some negative poles. For example in the simplest case one has to take as N function where the first term ensures that the base space is asymptotically flat. At the position x = x j is now a negative pole, which implies that the hyper Kähler base space changes its signature when N changes its sign and moreover it becomes singular at zeros of N . So, the base space is not well-defined everywhere, but the 5-dim. solution remains smooth! The simplest way to see this, is to investigate the corresponding 4-dim. solution, which is smooth at this point. But if the conditions (3.6) are satisfied, the 4-dim. solution is singular at the poles of N [see eq. (4.11)] and is regular only if N = const. To make this more clear, lets discuss an example.

Bubbling STU model
The STU-model is defined by V = ST U = 1 and we have An interesting question is, what is the least number of centers for which a bubbling solution exists? Before we discuss this, let us comment on the moduli. The complete solution is fixed by the set of harmonic functions and their constant parts as well as the different positions of the centers appear as moduli. If N = 0 (yielding a singular 4-dim. solution), we need only two centers and it is a straightforward exercise to distribute the harmonic functions so that the relations (3.6) are obeyed. For the STU model each center gives four equations and hence we fix 2×(3+1) constants, e.g. the constant parts in the harmonic functions. If we do not want to impose constraints on the charges, the integrability constraint in (4.13) yields another condition on the relative position; see [32,29]. Note, even if we want to set ω = 0, this constraint applies.
But how can we obtain a regular solution in four dimensions? This is the case, if we impose N = const., which in turn implies that A = consts in (2.3) and hence the hyper Kähler base becomes S 1 ⊗ R 3 . Now, to cancel the poles in ∆ A , we have to distribute the poles in K A at different centers. For the STU-model this is done if we assume three centers, so that each K i has only one center at r = r i ; i = 1, 2, 3. Thus, each ∆ i in (3.9) -(3.11) has only single poles, which can be canceled against each other. Eg. in ∆ 1 the poles from K 2 K 3 at r = r 2 and at r = r 3 are canceled if L 1 has two poles also r = r 2 and r = r 3 . The same can be done for ∆ 2 and ∆ 3 . In total these are six equations and in addition, there are three equations coming from the ω 5 constraint in (3.6), because this equation has poles at each center. So in total, we have now 9 constraints. In general, we do not expect to solve these constraints just by fixing the constant parts of the harmonic functions (there are too few of them) and therefore, one may have to impose constraints on the charges related to the separate centers. But note, the total charges of the bound state measured at infinity, can always be chosen freely. The positions are again fixed by solving the integrability constraint. It would be very interesting to work out this example in detail and see whether in fact all continuous parameters are fixed at the end. To make the story complete, one has also to make sure that there are no closed timelike curves, but also this issue can be addressed in four dimension; see below. Further interesting questions are then: How can one understand the entropy of this bound state? Can one break supersymmetry? Note, the naive non-extreme black hole solution does not allow a multicenter case, which is important in this setup.

Relating four-and five-dimensional BPS solutions
The whole class of solutions that we just summarize, can be mapped to the known stationary solutions in four dimensions [27]. This is a one-to-one map, without any smearing or any other approximation and different aspects of this relation are explored in [24,28,29]. It is an important relationship, because it opens the possibility to understand corrections to (classical) 5-dim. supergravity. These are for example, higher curvature or general higher derivative corrections or, since the (asymptotic) 5-dim. space time has a compact circle (ie. R 4 × S 1 ) one can expect even instanton corrections (as the one discussed in [30]) to 5-dim. solution, which are of course suppressed by the radius of the S 1 .
It is important to realize that this mapping relates localized solutions, there is no smearing procedure along the 5th coordinate. To understand it, consider the single center Reissner-Nordstrom black hole, which reads in 5-dimensions and becomes after the transformation ρ 2 = 4R r The reduction along x 5 = Rψ (σ 3 = dψ + cos 2 θdϕ) yields If one identifies N and H, this is exactly the 4-dim. Reissner-Nordstrom black hole and note, the 5-dim. black hole was localized in all four spacial directions! A crucial point for the mapping is that the entropy of both black holes should be equal. In fact, one finds where A 3/2 are the areas of the corresponding 3-and 2-dim. black hole horizons. There is straightforward generalization: N → 1 + pR r so that S 5 = S 4 = 2π pq 3 and if the NUT charge is p = 1, the asymptotic space is effectively 4-dimensional (S 1 × R 3 ), but as r → 0 the geometry is equivalent to the 5-dimensional metric (4.2) and therefore, by varying R, the geometry interpolates between the 4-and 5-dimensional case [28].
It is now straightforward to include additional vector multiplets in five dimensions and the resulting 4-dimensional geometry is given by the stationary solutions discussed in [27]. All couplings and fields of the 4-dim. Lagrangians can be expressed in terms of the symplectic section (Y I , , which has been rescaled by the central charge [31]. In order to solve the gauge field equation and Bianchi identities one introduces a set of harmonic functions and supersymmetry requires that the the section obeys the attractor equations The solution can then be written as where the variables x A are solutions of the set quadratic equations The 4-dim. metric can then be written as 4.12) and the integrability of the last equation becomes [32] There is now the following dictionary between the 4-and 5-dim. quantities [29] (4.14) It is straightforward to verify that the 4-dim. solution is regular at N = 0 (all terms ∼ 1/N or ∼ 1/N 2 cancel) and therefore also the 5-dim. solution has to be regular. The effective angular momentum in five dimensions is related to the combination of harmonic functions and the condition e −2U > 0 sets an upper bound on the angular momentum beyond which the 5-dim. solution develops closed timelike curves whereas the 4-dim. becomes singular. For the multi-center case, the constraint (4.13) can be interpreted that the charge vectors (p I , q I ) of the different centers as well as the symplectic vector (H I , H I ) r=∞ have to be mutual local.
This was the general solution, but there are a number of interesting examples: (i) The simplest one are the well-known single-center black holes with H I = h I + p I R r , H I = h I + qI R r , which describes in five dimensions a rotating black hole in a Taub-NUT space (the angular momentum as well as the NUT charge become an electric and magnetic charge in four dimensions). (ii) Black rings are mapped on a 2-center solution in four dimensions; one center is regular and has no NUT charge, but the other carries the NUT charge and is singular. (iii) The Gödel type solution is obtained if M = gz (ie. a linear function in one coordinate, g = const.), N = R/r, L = const. and K = 0. Since M is linearly growing function, the solution in four dimensions develops a curvature singularity at finite radius: r = r c where e −2U = 0. As for the over-rotating BMPV black hole, this singularity is related the vanishing circle of the 5th direction. We should also mention, that the 4-dim. solution can develop closed timelike curves if det(−e 2U ω i ω j + e −2U δ ij ) < 0 (i, j = 1, 2, 3). This constraints as well as the absence of Dirac-Misner strings have to be ensured for consistency. (iv) It is straightforward to combine the different solutions by adding the appropriate harmonic functions. Black holes in a Gödel space time corresponds to adding a linear function to at least one of the harmonic functions [33]. Although this correction drops out on the black hole horizon, it will always yield a singularity at some finite radius. Within supergravity, the only way to avoid this singularity is to cut-off the region in space time by introducing a domain wall by gz → g(1 − |z − z 0 |), where z 0 should be chosen appropriately. But as always in such scenarios, one has to address the question, what happens if the wall moves close to the singularity? (v) We should also note, that the reduction of the 5-dim. solution over the S 2 instead of the non-trivial S 1 circle yields the BTZ black hole in three dimension and again the entropy agrees with the one in five and four dimensions.