Squashed, magnetized black holes in $D=5$ minimal gauged supergravity

We construct a new class of black hole solutions in five-dimensional Einstein-Maxwell-Chern-Simons theory with a negative cosmological constant. These configurations are cohomogeneity-1, with two equal-magnitude angular momenta. In the generic case, they possess a non-vanishing magnetic potential at infinity with a boundary metric which is the product of time and a squashed three-dimensional sphere. Both extremal and non-extremal black holes are studied. The non-extremal black holes satisfying a certain relation between electric charge, angular momenta and magnitude of the magnetic potential at infinity do not trivialize in the limit of vanishing event horizon size, becoming particle-like (non-topological) solitonic configurations. Among the extremal black holes, we show the existence of a new one-parameter family of supersymmetric solutions, which bifurcate from a critical Gutowski-Reall configuration.


Introduction and motivation
The study of black objects in gravity models with a negative cosmological constant has attracted recently considerable interest, being fueled by studies of the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence [1], [2]. This conjecture basically proposes a 'dictionary' between classical AdS bulk gravitational solutions (in D−dimensions) and field theory states at strong coupling (in (D − 1)−dimensions).
Of particular interest in this context are the solutions of five dimensional N = 4 SO(6) gauged supergravity, which is thought to be a consistent truncation of type IIB supergravity on AdS 5 × S 5 [3], [4]. In its minimal version, the bosonic sector of this model is just Einstein-Maxwell (EM) theory with a negative cosmological constant and a Chern-Simons (CS) U(1) term (with a fixed value of the coupling constant). Despite its (apparent) simplicity, this theory possesses a variety of interesting solutions which have been investigated extensively over the last two decades.
Restricting to configurations possessing an event horizon, one remarks that most of the studies in the literature concentrate on two different classes of solutions. First, there are the black holes (BHs) with a spherical horizon topology 1 in a globally AdS 5 spacetime background, in which case the dual theory is formulated in a D = 4 Einstein universe. The Schwarzschild-AdS BH is the simplest example, while the most general such EMCS solutions rotate in two planes and possess four global charges: the mass, the electric charge, and two angular momenta [5]. A considerable simplification is obtained for an Ansatz with two equalmagnitude angular momenta, an assumption which factorizes the angular dependence of the problem. These BH solutions have been found in closed form by Cvetič, Lü and Pope (CLP) in [6] (see also [7]). Remarkably, their extremal limit contains a subset of solutions that preserves some amount of supersymmetry [8]. An extension of the CLP BHs which possesses an extra parameter Φ m associated with a non-zero magnitude of the magnetic potential at infinity has been reported in the recent work [9].
Second, there are the black branes, which approach at infinity the Poincaré patch of the AdS spacetime. These solutions have a Ricci flat horizon, while their dual field theory states reside in a D = 4 Minkowski spacetime. The most general such configurations appear to be those reported in [10], [11]; in addition to the mass and electric charge, they possess an extra parameter corresponding to the magnitude of the magnetic field at infinity.
However, it is worth remarking that the AdS/CFT correspondence does not constrain the way of approaching the boundary of spacetime, asymptotically locally AdS (AlAdS) solutions being also relevant. An interesting class of such configurations are the D = 5 AdS black strings 2 originally found by Copsey and Horowitz in [12]. These are natural AdS 1 Black rings with an S 2 × S 1 event horizon topology exist as well, approaching at infinity a globally AdS5 background. Such solutions have been constructed in [13] using approximate methods, and fully nonperturbatively in [14]. 2 These solutions have been generalized to higher dimensions and a more general topology of the event horizon in [15]. The mass-angular momentum-electric charge diagram is shown for the supersymmetric solitons in [16], the Gutowski-Reall BHs [8] and the new supersymmetric BHs in this work.
counterparts of the (better known) uniform black strings in a D = 5 Kaluza-Klein theory, the horizon topology being S 2 × S 1 . Also, the conformal boundary, where the dual theory lives, is the product of time and S 2 × S 1 .
The main purpose of this work is to report the existence of a new class of solutions of the D = 5 minimal gauged supergravity model. These solutions possess a squashed sphere in the boundary metric and can be viewed as interpolating between (some versions of) the three classes of black objects mentioned above.
Moreover, we find that a particular set of these configurations has special properties, forming a new one-parameter family of supersymmetric BHs.
A discussion of the basic properties of these solutions was given in the recent work [17], in a slightly different context.

Summary of results
In a convenient set of coordinates, the conformal boundary metric of the solutions in this work reads  Figure 2: (a) The electric charge-angular momentum diagram is shown for the supersymmetric solitons in [16], the Gutowski-Reall BHs [8]  with v a control parameter, dΩ 2 (v) the metric on a squashed S 3 sphere and 0 ≤ θ < π, 0 ≤ φ < 2π, 0 ≤ψ < 4πv. The presence of a squashed sphere in the boundary geometry of some asymptotically locally AdS configurations has been found before in the literature, the D = 4 nutty instantons (reviewed in Appendix A) being perhaps the best known case.
Clearly the sphere in (1.1) becomes a round one for v = 1, in which case the solutions approach a globally AdS background. Another case of interest is v = 0, the bulk solutions becoming AdS black strings and vortices, with a boundary which is the product of time and S 2 × S 1 (with S 1 parametrized byψ, whose periodicity is arbitrary in this limit). Finally, for large values of v, one can show that, after a proper rescaling, the boundary geometry (1.1) is the product of time and a twisted R 3 part.
In this work we provide evidence for the existence of a family of black objects with a conformal boundary given by (1.1). The solutions are constructed numerically within the framework of the D = 5 minimal gauged supergravity model, and possess a gauge potential with both a magnetic and an electric part; they also rotate in the bulk, with equal-magnitude angular momenta.
The main properties of the generic nonextremal solutions can be summarized as follows: (i) They possess an event horizon of spherical topology and are regular on and outside the horizon. Also, they do not present other pathologies (such as closed timelike curves (CTCs)).
(ii) In addition to the mass M , the electric charge Q and the angular momenta J, the new solutions possess an extra parameter c m associated with a non-zero magnitude of the magnetic potential at infinity.
(iii) A particular set of BHs with c m = 0 does not trivialize as the horizon size shrinks to zero, a limit which describes a one-parameter family of squashed spinning charged solitons. The angular momentum J and the electric charge Q of these solutions are determined by the magnetic flux at infinity Φ m through the base space S 2 of the S 1 -fibration, with J = Φ m Q.
(iv) The generic BH solutions possess an extremal limit, with a nonzero event horizon area. Moreover, supersymmetric BHs exist as well, forming a one-parameter family. These BHs bifurcate from a critical Gutowski-Reall [8] configuration, their mass, angular momenta and electric charge having relatively simple expressions in terms of the squashing parameter v only.
We note that supersymmetric solitons exist as well within the same framework, being investigated in the interesting work [16]. However, they do not correspond to a limit of the supersymmetric BHs.
The mass − angular momentum − electric charge diagrams summarizing the picture for these three different types of supersymmetric solutions are shown in Figure 1 and in Figure  2.

The model and Ansatz
The action for D = 5 minimal gauged supergravity is given by where R is the curvature scalar, L is the AdS length scale (which is fixed by the cosmological constant Λ = −6/L 2 ) and A µ is the gauge potential with the field strength tensor F µν = ∂ µ A ν − ∂ ν A µ . Also λ is the CS coupling constant, with λ = 1 in the minimal gauged supergravity case. Since a number of basic results do not depend on the precise value of λ, we shall keep it general in all relations below, such that (2.1) will describe a generic Einstein-Maxwell-Chern-Simons (EMCS) model. However, the numerical results will cover the SUGRA case only. Finally, K is the trace of the extrinsic curvature for the boundary ∂M and h is the induced metric of the boundary. The field equations of this model consist of the Einstein equations together with the Maxwell-Chern-Simons (MCS) equations A general parametrization of the metric Ansatz which covers both the generic and the supersymmetric configurations possesses a local SU (2) × U (1) × U (1) symmetry and reads with σ i the left-invariant one-forms on S 3 , σ 1 = cos ψdθ + sin ψ sin θdφ, σ 2 = − sin ψdθ + cos ψ sin θdφ, σ 3 = dψ + cos dθdφ, the coordinates θ, φ, ψ being the Euler angles on S 3 , with the usual range (in particular, a periodicity 4π for ψ). Also, note the existence of gauge freedom degree in the line element (2.4)), which will be fixed by convenience. A gauge field Ansatz compatible with the symmetries of (2.4) contains an electric potential and a magnetic one, with The general configurations satisfy the following set of equations which follow from (2.2), (2.3): where a prime denotes a derivative w.r.t. the radial coordinate r. Also, we notice the existence of the scaling symmetry with p an arbitrary nonzero constant. We remark that one cannot take a 0 = 0, unless the magnetic potential also vanishes, a k = 0. Also, the equations of motion possess two first integrals 3 with c t , c W two constants of integration. The CLP BHs are a solution of the above equations, the corresponding expression of (F i , W ) and (a 0 , a k ) being given e.g. in the Appendix A of Ref. [18]. In practice, the nonsupersymmetric solutions are found for a reparametrization of (2.4) which fixes the metric gauge and enforces the far behaviour, with (2.10) The supersymmetric solutions are found for a more complicated parametrization of (2.4), which is discussed in Section 4.

The solutions in the far field
The far field expression of the solutions is found assuming that (i) they approach at infinity a locally AdS spacetime, with a conformal boundary metric given by 4 (1.1), and, (ii) they possess a boundary magnetic field. As such, as r → ∞, the metric functions 1/F 1 (r) and F 0 (r) behave as r 2 /L 2 , F 2 (r) and F 3 (r)/v 2 as r 2 , while W (r) vanishes. Also we assume that 3 The origin of these first integrals can be traced back to the fact that the Einstein equation E t ψ and the MCS equations possess a total derivative structure. 4 In fact, the form (1.1) is found only after a suitable rescaling, see the discussion in Section 3.2.
a k (r) → c m in the same limit, (c m , v) being input parameters. This implies the existence of a nonvanishing asymptotic magnetic field, F θφ → − 1 2 c m sin θ, such that the parameter c m can be identified 5 with the magnetic flux at infinity through the base space S 2 of the S 1 fibration [9], One should remark that the assumptions (i) and (ii) above are not related. There exist 'magnetized' solutions possessing a round sphere at infinity [9], and also vacuum BHs with a conformal boundary geometry (1.1). However, as we shall see in Section 4, the existence of a Killing spinor imposes that both (i) and (ii) should hold, i.e. c m = 0 and v = 1, with a special relation between these two constants 6 .
An expression of the solution compatible with above assumptions can be constructed in a systematic way, being shared by both (extremal and nonextremal) BHs and solitons. The first terms in the large-r expansion read 7 containing, in addition to (v, c m ), the free parameters {α,β,ĵ, q, µ}. In principle, f (∞) = f ∞ is also a free parameter of the far field expansion, but we can always fix it to one by means of the scaling symmetry (2.8) We observe that the first integrals (2.9), evaluated for these asymptotics, imply the following relations: The CLP BHs (as well as their λ = 1 generalizations in [18]) have c m = 0, v = 1, in which case no log−terms are present in the far field asymptotics. 5 Static magnetized squashed BHs in D = 5 Kaluza-Klein theory were constructed in Ref. [19]. However, the properties of those solutions are very different as compared to the AlAdS case. 6 Here we exclude the supersymmetric Gutowski-Reall BHs, which have cm = 0 and v = 1, being recovered as a limit of the new solutions in this work. 7 The occurrence of log−terms in this asymptotic expansion makes the existence of an analytic solution unlikely. Moreover, this applies also in the supersymmetric case.

The near-horizon expansion
In this work we shall restrict our study of solutions to the region outside the (outer) BH horizon. For non-extremal solutions, this horizon resides at r = r H > 0, where the function f (r) vanishes. There the solutions possess the following expansion: (2.14) k } are free coefficients. In the quasi-isotropic coordinates we are using, the horizon of extremal black holes is located at r H = 0. As a result, the behavior of the functions near the horizon changes with respect to the non-extremal case, with the occurrence 8 of non-integer powers of r. The first terms in the near-horizon expression of the solutions are f (r) =f 4 r 2k +f 0 , etc...} being, however, very complicated. Let us also notice that the near-horizon expression of the solutions takes a simpler form when written in terms of a new radial coordinate x = r k . As such, the existence of squashed AdS 2 × S 3 solutions (described by the leading order terms in (2.15)) becomes transparent. They form a particular class of the EMCS-attractors discussed in a more general context in Ref. [18].

Solitons: the small−r expansion
As a new feature in contrast to the CLP case (c m = 0, v = 1), the zero horizon size limit of the generic solutions is nontrivial. For a given v, this corresponds to a one-parameter family of spinning charged solitons with nonzero global charges. Such solutions possess no horizon, while the size of both parts of the S 3 -sector of the metric shrinks to zero 9 as r → 0.
A small-r approximate form of the solitonic solutions can be constructed as a power series in r, being compatible with the assumption of regularity at r = 0. The first terms in this expansion are with the free parameters {f 0 , m 0 ,m 2 , w 1 ; u, v 0 }. Finally, let us remark that after evaluating the first integrals (2.9) for the above asymptotics, one finds that the constants c W , c t vanish for solitons, (2.17)

Physical quantities 2.3.1 Event-horizon quantities
The horizon is a squashed S 3 sphere, with different sizes for the S 1 and the round S 2 parts of it. There the Killing vector ζ = ∂ t + 2Ω H ∂ ψ becomes null and orthogonal to the other Killing vectors on it. For non-extremal BHs, the induced horizon metric reads which leads us to define the horizon deformation parameter with m 2 , n 2 and f 2 the coefficients in (2.14). The area of the horizon A H , the Hawking temperature T H and the horizon angular velocity Ω H of these solutions are given by The horizon electrostatic potential Φ H as measured in a co-rotating frame on the horizon is In the extremal case, the induced horizon metric is while the horizon quantities are in terms of the constants which enter the near-horizon expansion (2.15).

Holographic renormalization and global charges
The global charges of the solutions are encoded in the constantsα,β,ĵ and q which enter the large−r expansion of the solutions (2.12). In computing them, we use the holographic renormalization of the D = 5 EMCS system as discussed e.g. in Ref. [21]. The first step is to rewrite the solution in the standard Graham-Fefferman coordinate system [22], by defining a new radial coordinate, This results in an equivalent asymptotic form of the line element ab log and of the gauge field The boundary metric tensor h ab is found by taking x = x 0 in (2.25), with x 0 sent to infinity in the final relations. Also, g (0) and A (0) are imposed as boundary conditions, providing the background metric and the external gauge potential for the four dimensional dual theory. For the solutions in this work one takes The terms g (2) , h (4) and B (2) are fixed by the equations of motion, while the terms g (4) and A (2) are not determined by the field equations. Their expression can easily be found from (2.12) together with (2.24), in practice they being extracted from the numerical output.
In the next step one defines a regularized total action I tot =I+I ct , which is the sum of (2.1) and a counterterm I ct , with [16], [21], [23] where R abcd , R ab , E ab denote the Riemann, Ricci and Einstein tensors, respectively, R is the Ricci scalar for the boundary metric h and F (0) = dA (0) is the boundary U(1) field. Note that in equation (2.28) we are only considering the counterterms that cancel the power-law and logarithmic divergences at the boundary, which provide finite expressions for the charges. However, additional counterterms can be added to the action [16] (see also the discussion in [24] for AlAdS solutions in supergravity). These additional counterterms in general introduce an ambiguity in the definition of the charges, but they can be useful in order to restore some lost symmetries at the boundary. As usual in AdS/CFT, one defines the expectation value of the stress tensor and current in the dual theory as Then provided that the boundary geometry has an isometry generated by a Killing vector ξ, a conserved charge can be associated with a closed surface Σ [25], with u a = δ t a a unit timelike vector (in general the expression (2.32) will contain a contribution from the flux < J a > [24], but this extra-term is not relevant for the particular solutions we are considering). The mass/energy M is the conserved charge associated with ξ = ∂/∂t; there is also an angular momentum J associated with the Killing vector ∂/∂ψ. A similar expression holds for the electric charge Q (2.33) It is straightforward to apply this formalism to the solutions with the asymptotics (2.12) (together with (2.24)). The nonvanishing components of the boundary stress tensor (2.31) are while the boundary current is After replacing these expressions in (2.32) and (2.33) one finds the global charges of the solutions Note also that the total derivative structure of the MCS equations implies the existence of a conserved Page charge The Page charge is proportional to the conserved charge c t of (2.13), with Q (P age) = −πc t .
In the standard c m = 0 case, the (holographic) electric charge Q and Q (P age) are the same. However, (2.37) and (2.38) do not coincide for solutions with a boundary magnetic field (and, in fact, the Page-charge vanishes for solitons, while Q = 0). One also notices that the stress tensor (2.34) is not traceless, its trace consisting of two parts. The first part is due to the conformal anomaly of the boundary CFT coming from the background curvature [26], [27] The part of (2.39) proportional to c 2 m results from the coupling of the CFT to a background gauge field [28] Moreover, one can verify that the following Ward identities are satisfied [21]

Numerical procedure
The equations (2.6) do not seem to possess closed-form solutions with v = 1 and/or c m = 0. Therefore all new configurations reported in this work are found numerically. The methods we have used are similar to those used in [29], [30], [31], [32] to find other numerical solutions with equal-magnitude angular momenta in D = 5 EM(CS) theory.
By making use of all the available symmetries, the set (2.6) of field equations can be reduced to a system of four second-order differential equations (ODEs) for the functions (f, m, n, a k ) together with two first-order ODEs for (ω, a 0 ). A relation between first-order derivatives of the functions f , n, m, ω and a k can be used as a constraint, which the numerical solutions must satisfy with a given precision.
In our numerical scheme, the input parameters are: i) the AdS length scale L, ii) the magnetic parameter c m , iii) the boundary squashing v, iv) the constantsĵ, q in the far field asymptotics, and, for non-extremal BHs, v) the event horizon radius r H . The event horizon data and the coefficients at infinity α, β and µ are read from the numerical output. In practice, we fix the AdS length scale L = 1 and construct families of solutions by varying the other input parameters.
The equations are solved by using a professional software package [33] which employs a collocation method for boundary-value ordinary differential equations and a damped Newton method of quasi-linearization. The number of mesh points used in our calculation was around 10 4 , distributed non-equidistantly on x, where x = 1−r H /r is a compactified radial coordinate employed in the BH non-extremal case; for solitons and extremal BH solutions one takes x = r/(1 + r) (with 0 ≤ x ≤ 1 in both cases). One should remark that the computation of global charges for these solutions is a nontrivial problem which requires a very good numerical accuracy, since the coefficientsα,β, appear as subleading terms in the asymptotic expansion (2.12). The typical relative accuracy of the solutions here is around 10 −10 .
Finally, let us mention that all solutions reported in this work are regular on the horizon or outside of it 10 . Also, since g tt = −f (r) < 0 for any r > r H , while the metric functions m(r) and n(r) remain strictly positive (in particular v 2 > 0), the solutions are also free of closed timelike (or null) curves, t being a time function (see the general discussion in [20], which covers also the framework here). Similar to the well known CLP case, the generic BHs possess, however, an ergoregion located between the horizon and the ergosurface r = r c (with g tt (r c ) = 0).

Black holes
In the generic case, given (v, c m ), the solutions possess three independent charges M, J and Q. Therefore finding their domain of existence is a considerable task which is beyond the purposes of this paper. Instead, we shall analyze several particular classes of solutions, hoping that they capture a part of the general pattern.

v = 1: a globally AdS background
Let us start with the simplest case of solutions possessing a round S 3 -part in the boundary metric. For c m = 0, these are the Cvetič, Lü and Pope (CLP) BHs [6]. However, as found in the recent work [9], they possess a generalization with a nonvanishing magnetic field in the far field, which can also be described within the framework in Section 2.
The results in [9] show that the qualitative behaviour of the BHs with small |c m | resembles that of the unmagnetized CLP case. However, a different picture is found for large enough values of c m , with a monotonic behaviour of mass and horizon area as a function of temperature (also the solutions do not appear to possess an upper bound on |c m |). In contrast to the CLP case, one finds BHs which have J = 0 but still rotate in the bulk, with a nonvanishing angular momentum density, T t ψ = 0. Extremal BHs with c m = 0 exist as well, possessing generically a nonzero horizon area. Moreover, the BH solutions satisfying a certain relation between J, Q and c m do not trivialize as r H → 0, becoming solitonic deformations of the AdS background.

v = 1: static, vacuum configurations
As expected, the v = 1 solutions in [9] possess generalizations with a squashed sphere at infinity, and new qualitative features occur as well. To simplify the problem, let us consider first the static, vacuum configurations, in which case it is possible to perform a systematic study of the solutions together with their relevant limits. These BHs are found within a consistent truncation of the general Ansatz (2.4), (2.5) with W = 0, a k = a 0 = 0. Our numerical results clearly indicate the existence of v = 1 (static, vacuum) BH solutions of the equations (2.6), smoothly interpolating between the asymptotics (2.14) and (2.12).
These solutions are most naturally interpreted as squashed 11 BHs, being in some sense the AdS counterparts of the (Λ = 0) Kaluza-Klein solutions in [34]. Such configurations exist for an arbitrary value of the horizon size, without an extremal limit. In fact, their thermodynamics is similar to the one of the Schwarzschild-AdS BHs [35]. For any v, their temperature is bounded from below, and one finds two branches consisting of small (unstable) and large (stable) BHs. As r H → 0, a singularity-free solitonic configuration is approached, the size of both parts of the S 3 -sector of the metric shrinking to zero. The properties of these solutions are discussed in the next subsection.
Let us now explore 12 the behaviour of the squashed BHs as a function of v. For a given value of r H , the parameter v can take arbitrary values. Apparently, as v → 0, the size in the far field of the U(1) fiber over S 2 shrinks to zero, such that this limit does not seem to be well defined. However, this is not the case. Following the discussion in the Appendix A, we consider an equivalent form of (2.4) which absorbs the v 2 factor in the asymptotic value of n(r) via a redefinition of ψ, with Also, the numerics shows that for small v, the size of the S 1 -circle in the horizon metric (2.18) becomes proportional to v [17]. As such, the limit v → 0 is smooth for theψ-parametrization, and describes the AdS 5 black strings and vortices, originally found in [12] for a different metric Ansatz. The black strings' horizon metric reads the horizon topology being S 2 × S 1 , while the conformal boundary metric is the product of time and a line element of the form (3.2). Also, the solutions possess a nontrivial r H → 0 limit describing AdS vortices. We note that for both black strings and vortices, the range of ψ (usually denoted as z-coordinate in the literature) is not fixed a priori 13 . No upper bound seems to exist for the value of v, although the numerical integration becomes more difficult as we increase this parameter, with the mass M diverging as v → ∞. However, a careful analysis of this limit reveals the existence of a different solution of the field equations. Following again the discussion in the nutty instanton case (see Appendix A), 11 Their basic properties have been discussed in a different context in [35], [36]. 12 In understanding the limiting-v behaviour, some useful hints are provided by the nutty-instanton toy model in Appendix A. 13 However, for black strings, the Gregory-Laflamme instability [37] implies the existence of a critical periodicity of theψ-coordinate for a given value of the mass [38].
we define the scaled coordinates Then as λ → ∞ the line element (2.4) (with W = 0) becomes which corresponds to a 'twisted' black brane configuration. The horizon is located again at somer =r H , with an induced horizon geometry In the absence of an analytical solution, the expression of F i (r) is found numerically 14 . We also remark that they do not possess a solitonic limit. Their conformal boundary metric is 15 (with 0 ≤ Θ < ∞, 0 ≤ φ < 2π and an arbitrary periodicity for Ψ). Although a t = const. surface is topologically a direct product of Ψ and the (Θ, φ)-plane, the product is "twisted" (or warped), and the boundary is not flat (its Ricci scalar is proportional to N 2 ). More details of the limiting solutions can be found in Appendix B.

v = 1: the generic case
Increasing the complexity of the solutions, we first notice that the (vacuum, static) BHs of the previous subsection possess spinning generalizations. They are interpreted as squashed rotating BHs, their thermodynamics being similar to that of the (v = 1) Myers-Perry-AdS BHs with two equal angular momenta. Static, electrically charged BHs with v = 1 exist as well. These solutions are generalizations of the Reissner-Nordström-AdS BHs, with a squashed horizon (and a squashed sphere at infinity) and possess similar thermal properties. What J = 0 or Q = 0 brings new is the absence of a smooth particle-like solitonic limit 16 (we recall c m = 0). Instead, one notices the existence of extremal BHs with a nonzero horizon area, which are smooth on and outside the horizon.
Moreover, as expected, spinning v = 1 solutions with Q = 0 and c m = 0 exist as well. They can be interpreted as squashed counterparts of the CLP BHs and appear to share all their basic properties. Again, these unmagnetized solutions do not possess a smooth solitonic limit. Also, their behaviour in terms of the squashing parameter v is similar to the one in the vacuum static case. In particular, the limit v = 0 describes a generalization of the AdS black strings in [12] with a nonzero electromagnetic field and a momentum along the ψ-direction.
However, these limits are in some sense less interesting, since, as we shall see in the next Section, they do not allow for supersymmetric solutions. Therefore let us now consider the general case of spinning, magnetized solutions with a squashed sphere at infinity. As a general remark, our numerical results show that they share a number of basic properties of the v = 1 solutions with c m = 0 in [9]. For example, for large enough J, Q, the magnetic field induces subleading effects only, and we recover the general pattern found for the CLP BHs. Also, these generic solutions do not allow for a smooth black string limit as v → 0. However, the limit v → ∞ is well defined, describing (after a suitable rescaling) a new family of twisted charged black branes. Although the asymptotics of these solutions is very similar to (2.14), (2.12) they possess a number of distinct properties, see the discussion in Appendix B.
The typical profiles of the metric and gauge functions of squashed magnetized black holes are presented in Fig. 3(a), for the a typical magnetized spinning BH (note the absence of nodes in the profile of the magnetic potential, a feature which holds for all configurations in this work, including the solitonic ones).  Let us now discuss some thermodynamical properties of these solutions, as shown in One can see that in general, the BHs possess an extremal limit, T H = 0. However, the situation is different for BHs with a specific relation between J and Q as given by (3.9) (shown with the blue line in Fig. 4). For these solutions, the temperature diverges as the horizon size decreases, a smooth solitonic configuration being approached as the horizon size shrinks to for solitons. Then the first integrals (2.9), imply that W = 0 and also a0 = 0. zero. Anticipating the discussion in Section 4, we mention that in general, the extremal BHs are not supersymmetric. However, for particular values of the charges and the squashing, the limiting solutions satisfy the susy equations. This is the case of the extremal black hole marked with a red point (and also the solitonic limit of the blue line).
In Fig. 4(a) we show the event horizon area A H vs. the temperature T H . Depending on the values of Q and J, the area can be a monotonic function of T H (red, cyan and orange lines); a more complicated picture is also allowed, with regions where A H decreases with increasing temperature (purple and blue lines). In the solitonic limit of the blue line, the area vanishes as T H → ∞, although the limiting solitons are regular everywhere. Also note that this family of BHs is special, with a finite minimum temperature.
In Fig. 4(b) we present a similar figure for the mass M vs the temperature T H . In general, the behaviour of M is similar to the area. However, in the solitonic limit the mass does not vanish, reaching a finite value, which, as we shall see, depends on the soliton charges and squashing.
The horizon angular velocity Ω H is shown in Fig. 4(c), again as a function of T H . As an interesting feature, we note that the configurations can present a counter-rotating horizon, depending on the specific combination of the charges. In the solitonic limit, the angular velocity vanishes since there is no horizon.
In Fig. 4(d) we present the horizon deformation (as given by rel. (2.19)) as a function of T H . Although the squashing v of these particular sets is fixed to v = 1.56, the horizon  deforms depending on the black hole electric charge Q, and angular momentum J (and in general, on the magnetization parameter, which in these Figures is also fixed). Although ≥ 1 for the solutions there, the horizon deformation can also be smaller than one for a different choice of the input parameters. As a result, one can see from (2.9) that the solutions' electric charge and angular momentum possess a dependence on the squashing v as given in the corresponding relations in (4.35). Therefore, for a given v, they form a one parameter family of solutions which are constructed by varying the value of the horizon radius r H (note that other quantities of interest of the solutions (e.g. mass, horizon area and temperature) are unconstrained, being read from the numerical output).
The reason for this special choice of Q, J and c m is that the extremal limit of these solutions possesses some special properties, being supersymmetric, as we will see in the next Section. Here they are constructed directly, as solutions of the second-order equations of motion.
In Fig. 5 we show the mass M as a function of the squashing parameter v and the Hawking temperature T H . We can see that M strongly increases as v decreases, with the existence of a minimal value for a given v. In Fig. 6 we show the event horizon area A H as a function of the horizon deformation and the Hawking temperature T H . Note that in the extremal limit, all solutions converge to a single point (in red), meaning that all the extremal BHs possess the same horizon properties. An explanation of this feature will be provided in the next Section when studying the susy squashed BHs Further properties of BHs are shown in Figs. 7, 8. There the choice of the magnetization parameter is the same as above, c m = L(1 − v 2 )/ √ 3, and c t = c W = 0. This implies that Q and J possess a different dependence on v as given in relation (4.4) below, other quantities being determined by the value of r H . As such, these configurations provide a different cut in the parameter space of solutions, the set of susy solitons being recovered in the vanishing horizon size limit (see the discussion below).
We mention that we have also studied families of solutions with < T a a > = 0 (which thus do not possess a supersymmetric limit). The picture we found here is similar to the generic case in Figure 4 (red, cyan and orange lines), with the existence of an extremal limit possessing a nonzero horizon area.

Solitons
We start the discussion of the solitonic solutions of the model with the vacuum static case. These configurations naturally emerge as the zero horizon size limit of the corresponding families of BHs discussed in the Section 3.2.2, representing deformations of the globally AdS 5 spacetime. As such, their most natural interpretation is as providing a background for models with given geometric parameter v. As expected their mass is nonvanishing, being shown in Figure 9 as a function of v. These configurations can also be viewed as the zero horizon size limit of families of charged and/or spinning BHs with c m = 0. However, as already discussed above, both J and Q vanish as r H → 0.
One feature a nonzero boundary magnetic field brings new is the existence of a nontrivial limit of the solutions which describes spinning electrically charged solitons. Similar to the vacuum case, they possess no horizon, while the size of both parts of the S 3 -sector of the metric shrinks to zero as r → 0.
An interesting property of the solitons is that their electric charge and angular momentum are proportional. To prove it, we notice that since c t = c W = 0, they satisfy the simple relations 17 Then, after expressingĵ, c m and q and in terms of the angular momentum J (as given by rel. (2.36)), the magnetic flux at infinity Φ m (rel. (2.11)), and the electric charge Q (rel. (2.37)) one finds These relations are universal, being satisfied for any value of the CS coupling constant λ = 0. As we have already seen in Fig. 4 the solitons appear as the r H → 0 limit of particular black hole solutions which satisfy (3.9). Further insight on the properties of soliton solutions 17 These relations can also be viewed as a consequence of the vanishing of the Page charge for solitons. In Fig. 7 we represent the mass M of BHs as a function of the squashing parameter v and the temperature T H for the same solutions as in Fig. 8. The solitons are approached in the limit of vanishing area and diverging temperature. This results in a family of solutions with finite mass and varying squashing parameter v (we recall that the magnetic parameter c m is determined by v such that the boundary stress tensor (2.39) is traceless). We have verified that these solutions correspond in fact to the supersymmetric configurations in [16] (this provides another useful test of our numerical results). More details on these special solutions is presented in the next Section.
Finally, let use remark that the considered choice c 2 m = L 2 (1 − v 2 ) 2 /3 is not a necessary condition for the existence of solitons (while (3.9) is mandatory). In fact, solitonic solutions in a globally AdS background (v = 1) have been already reported in Ref. [9], most of their properties being recovered for other values of v.

Supersymmetric solutions
The only known supersymmetric BHs within the framework in Section 2 are those found by Gutowski and Reall in Ref. [8]. They possess a round sphere at infinity (v = 1) and no boundary magnetic field (c m = 0), being a special limit of the CLP solution. These solutions contain a single parameter, α > 1/2, the global charges being given by while the horizon area, horizon angular momentum, electrostatic potential and horizon angular velocity are It is natural to inquire if the more general squashed magnetized solutions in this work also possess a supersymmetric limit. A hint in this direction comes from the observation [40] that the two contributions in the trace < T a a > of the stress tensor (2.39) exactly cancel for In the extremal BH case, this requirement leads to a two parameter family of solutions (the parameters can be taken as the squashing v and the electric charge Q). However, the situation changes for solitons, the above condition together with the charge-angular momentum relation (3.9) leading to a single family of solutions which can be parametrized in terms of v. We have found numerically that these solutions correspond 18 in fact to the supersymmetric solitons in [16]. Although they cannot be written in closed form, the supersymmetry allows for an almost complete description of the solution in terms of the squashing parameter v. The mass, angular momentum and electric charge of the supersymmetric solitons are given by [16] M = πL 2 5 288 We have already shown in the previous section that these supersymmetric solitons can be connected with squashed magnetized black holes in the limit of vanishing size of the horizon.
In what follows, we show that, in addition to these solitons, the EMCS equations possess as well a one-parameter family of supersymmetric BHs.

The formalism
In constructing the supersymmetric solutions which fit the framework in Section 2, the most convenient approach is to use the general formalism proposed in [8]. Then such configurations have the following line element 19 : withσ 1 = − sinψdθ + cosψ sin θdφ,σ 2 = cosψdθ + sinψ sin θdφ,σ 3 = dψ + cos θdφ, and the gauge potential The function a is the solution of a sixth order equation Any solution to this equation (together with (4.7), (4.6)) corresponds to a configuration which preserves at least one quarter of the supersymmetry. Let us also remark that the a−equation with λ > 0. The only (known) closed form solution of the eq. (4.8) which describes a BH has been found by Gutowski and Reall (GR) and has a(ρ) = αL sinh( ρ L ), (4.10) 19 To agree with the standard notation in the literature, in this Section we use ρ for the radial coordinate, instead of r as in (4.5).
Also, one notes that the line element (4.5) can be put into the form (2.4) by taking (4.12) The reason we introduce the constant u in the above relations is that the supersymmetric solutions are found in a frame which rotates at infinity, with Then the transformation (4.12) brings the solution to a static frame at infinity, such that the solutions become a particular limit of the general case in Sections 2, 3. For example, the corresponding expression of the U(1) potential (2.5) reads

The large−ρ expansion
Despite the absence of an exact solution in the general squashed magnetized case, it is still possible to find an approximate solution at the limits of the domain of integration. Keeping the notation of Ref. [16], the first terms in the far field expansion of a(ρ) read 20 The corresponding expression for supersymmetric solutions can be found in the Appendix A of Ref. [16].
Since the generic solitons and BHs possess the same far field expansion, the expressions there are valid also for the solutions in this work.
such that the far field expansion resembles (2.12) 1 + 4a 2 + 4c + 4c log r L + . . . , while the asymptotic form of the gauge potentials is One can see that W → 0 as r → ∞, such that the solution is indeed written in a nonrotating frame at infinity 21 . Then, from (4.18), one finds that the constant c in the far field expansion corresponds to the magnetic flux parameter Also, one can see that the solution necessarily possesses a squashed sphere at infinity 22 , with 20) 21 Here we have used u = limρ→∞ f 3 Ψ b 2 −f 3 Ψ 2 = 2 4c−1 , and replaced it in (4.12). 22 Note that given the asymptotics (4.15), (4.17), the conformal boundary metric is ds 2 (bdry) = 4a 2 0 L 2 (σ 2 0 v 2 dt 2 , which is slightly different from (1.1). However, one can always set a0 = 1/2 by rescaling the radial coordinate (which implies a redefinition of other constants in (4.15)). Also, the scaling (2.8) can be used to dispose of the 1/v 2 factor in the above expression of gtt, such that the standard form (1.1) is recovered. Moreover, let us remark that although the a−equation (4.8) is invariant under the transformation (4.9), that symmetry is fixed by imposing the far field asymptotics (4.15). the zero-trace condition (4.3) being satisfied. The squashing parameter v takes arbitrary values, the solutions with v 2 < 0 possessing CTCs.
An important observation here is that after evaluating the first integrals (2.9), for the asymptotic form (4.17), the constants a 4 and a 6 can be expressed in terms of c t , c W together with a 2 , c: (4.22)

The near-horizon expansion
Without any loss of generality, the horizon is located at ρ = 0. There one assumes the existence of a power series expansion for the function a(ρ), with Then, after replacing the above expression in the sixth-order equation (4.8) and solving order by order in k, one finds that the problem possesses (at least) two independent solutions describing the near-horizon of a BH. The argument goes as follows. First, the existence of a horizon requires α 0 = 0. Then, to lowest order the eq. which admits two independent solutions. The first one has α 4 = 0, and leads to an expression for a(ρ) containing odd powers of ρ only, with in terms of two coefficients α 1 , α 3 . However, after using the scaling symmetry (4.9) (with λ = α 1 6α 3 ), one finds that this corresponds in fact to the small-ρ expansion of the Gutowski-Reall solution (4.10) (where α = α 1 ).
The second choice to satisfy the Eq. (4.25) is α 1 = 2 2 11 , which leads to a second consistent small-ρ expansion of a(ρ) different from (4.26). One should remark that this possibility has been noticed in Ref. [16], where the near-horizon expression of f (ρ) has been already displayed.
The first few terms in this alternative expression of a(ρ) are a(ρ) L = 2 2 11 This solution translates into the following near-horizon expansion in terms of the metric Ansatz (2.4): while for the gauge potential one finds Also, this near-horizon expansion implies the following expressions for the constants c t , c W , which enter the first integrals (2.9): Then, after replacing in (4.21), one finds an expression of the far field coefficients a 4 and a 6 in terms of v and a 2 .

Numerical approach
We have not succeeded in solving analytically 23 the sixth-order Eq.(4.8), to find a solution connecting the asymptotics (4.27) and (4.15). However, its numerical integration is straighforward. In our approach, we reformulate the problem in terms of a new function a(ρ) = e − ρ L a(ρ), which remains finite as ρ → ∞. Similar to the treatment of the nonsupersymmetric case, a new radial compactified radial coordinate x is introduced, with 0 ≤ x ≤ 1 and ρ = x 1−x . The resulting differential equation forâ(x) is written as a set of six first-order equations, which are solved with the following boundary conditionŝ and The first five algebraic relations above result directly from (4.27), while the condition at infinity is compatible with the asymptotics (4.15). Also note that the parameter α 3 = d 3â is free, resulting from the numerical approach. Hence the boundary conditions present a single free parameter α 4 = ξ, which we use as a control parameter to generate non-trivial solutions.
As an initial solution, we takeâ = 1 and change the value of ξ in small steps. This setting works well, and the numerical iteration converges quickly (note that we have used the same solver [33] as in the generic case). The solutions have around 2000 points in the mesh, with a numerical error of 10 −8 or lower for a(x) and its derivatives.
Once the profile of a(ρ) is known, the full solution is reconstructed from (4.7) together with (4.11), (4.13). The coefficients {a 0 , c, a 2 , a 4 , a 6 } in the far field expansion (4.15) are extracted from the numerical output.
In Fig. 10(a) we present several profiles of the functionâ(x). The corresponding profiles of the functions of the susy Ansatz (multiplied with suitable factors) are shown in Figure  10(b), 11 and 12(a). The results in these plots are found for several values of ξ.
The relation between the squashing parameter v 2 of the far-field expansion and the parameter ξ of the near-horizon behaviour is presented in Figure 12(b). Note that the solutions are not invariant to changes in the sign of ξ. For example, the squashing v 2 grows for negative ξ, while for positive values it decreases, with v → 0 for ξ 0.03. For larger values of ξ, a 23 However, one cannot exclude the existence of a (partial) analytical solution. For example, a closed form (approximate) expression of the supersymmetric solitons has been reported in Ref. [16]. The solution there has been found to first order in a perturbative expansion around the AdS background in terms of cm. So far we did not succeed in finding a similar expression in the BH case. small branch of solutions with CTCs is found (dashed green line). Also note that the solution with ξ = 0 (red point) is a particular Gutowski-Reall black hole. The relation of the remaining coefficients {a 0 , a 2 , a 4 , a 6 } in the far field expansion (4.15) with the squashing parameter v is presented in Figures 13 and 14, where we mark again the Gutowski-Real solution with a red point, and the sector with CTCs in green.
Finally, we also show the dependence of the parameter α 3 as a function of the squashing v 2 in Fig. 14(b). We mark again the Gutowski-Real solution with a red point, and the sector with CTCs in green.

General properties
The computation of all quantities of interest of the solution is a direct application of the which depend on the squashing parameter v, only. while on the squashed BHs side, this corresponds to the limit 24 v → 1 (i.e. a round S 3 sphere at infinity and no boundary magnetic field).
Also, note the solutions with v 3.61691 possess a vanishing total angular momentum, although they still rotate in the bulk (T t ψ = 0 and Ω H = 0), while their mass and electric charge are positive.
The limit v = 0 (i.e. c = 1/4 in the far field expansion (4.15)) is special. While the horizon geometry does not change, the asymptotics are different in this case, the conformal boundary structure being lost. One finds e.g.
with q 0 , q 3 and q w possessing a complicated dependence on a 0 , a 2 , a 4 and a 6 . Thus, similar to the extremal case in Section 3, the v → 0 limit does not result in a black string configuration. Moreover, this limiting solution is not asymptotically (locally) AdS (despite the absence of (obvious) pathologies). We hope to return elsewhere with a detailed study of this interesting limiting solution.
The infinite squashing limit v → ∞ is taken again together with the rescaling (3.3), (3.4). This results in a BH solution with a different topology (e.g. the horizon geometry is of the form (3.6)), whose spacetime asymptotics are again non-standard, the AdS conformal boundary structure being lost. This limit is described by an exact solution, which is discussed in Appendix B.
Finally, let us mention that we have found numerical evidence for the existence of solutions with v 2 < 0. However, such configurations possess closed timelike curves (in the bulk and on the boundary) and are less interesting. Non-supersymmetric solutions with this behaviour exist as well.

Further remarks. Conclusions
The solutions of the D = 5 gauged supergravity models play a central role in the AdS/CFT correspondence, providing a dual description of strongly-coupled CFT on the four-dimensional AdS boundary. The main purpose of this work was to report the existence of a new class of BH solutions of the minimal gauged supergravity model and to provide a discussion of their basic properties (see also [17]) . They are built within the same general framework as the well-known Cvetič-Lü-Pope BHs, sharing some of their basic properties; for example, the horizon has a spherical topology and the two angular momenta have equal magnitude. However, the conformal boundary of the new BHs in this work possesses a squashed sphere, 24 The supersymmetric solitons in [16] approach instead the globally AdS background as v → 1.
such that the solutions could become in certain limits black strings or black branes. Moreover, new features occur as one allows for a nonvanishing value of the magnetic field at infinity. For example, as discussed in Section 3.3, this supports the existence of smooth particle-like solitons, which satisfy a universal relation between angular momentum and electric charge. Moreover, a particular set of extremal configurations corresponds to a new one-parameter family of supersymmetric black holes, which were reported in Section 4. They satisfy a certain relation between the squashing parameter and the magnetic parameter and bifurcate from a critical Gutowski-Reall configuration.
There are many open questions and avenues for future investigation. For example, the general framework in this paper provides a ground for further study of the properties of these solutions, such as a systematic investigation of their domain of existence, thermodynamics and stability. Moreover, various limits of the solutions briefly mentioned in Section 3 certainly deserve a systematic study. Also, our results in the generic non-susy case were found for the supersymmetric value of the CS coupling, λ = 1. However, it would be interesting to consider other values as well. Here we remark that the results in [18] (valid EMCS BHs with v = 1 and no boundary magnetic field) show the existence of new qualitative features (e.g. the existence of excited solutions) once the CS coupling constant exceeds a critical value. As yet another possible direction, we note that it is straightforward to extend the framework introduced in Section 2 to other (odd) spacetime dimensions D > 5 and the same matter content. Therefore we predict the existence of AlAdS solutions also in that case, which would share some basic properties of the solutions in this work.
Another interesting question is the relevance of such configurations in an AdS/CFT context. Here one remarks that the supersymmetric solitons have been interpreted in [16] as providing the gravity supersymmetric dual of an N = 1 supersymmetric gauge theory on a squashed Einstein universe background, with a nontrivial background gauge field coupling to the R-symmetry current. We expect that the interpretation of the supersymmetric BHs in this work would be similar, describing different phases of the same D = 4 model.
Finally, let us remark that the presence of a nontrivial magnetic field on the boundary can be viewed in a larger context as a consequence of the 'box-type' behaviour of the AdS spacetime. As realized in [41], [42], [43], for the D = 4 case, the AdS asymptotics supports the existence of everywhere regular Maxwell-field multipole solutions, which survive when including the backreaction. That is, the U(1) field mode, which in the (asymptotically) flat case is divergent at infinity, gets regularized, leading to new families of Einstein-Maxwell solutions (which include both BHs and solitons). Although further study is necessary, this feature appears to be universal, some partial results being reported in [44] for D = 2k + 1 ≥ 5 dimensions (see also [45], [46]).
This observation leads us to predict the existence of a variety of other A(l)AdS solutions of the D = 5 minimal gauged supergravity model. First, we remark that for the solutions in this work, the asymptotics of the electric potential are standard, while the magnetic part can be interpreted as an AdS dipolar field. However, this is just the simplest type of non-standard boundary conditions for the U(1) field (supported by AdS asymptotics), which has the advantage to lead to a codimension-1 numerical problem. More general solutions should exist as well. For example, our preliminary numerical results indicate the existence of generalizations of the D = 5 Reissner-Nordström BHs with a vanishing magnetic field and an asymptotic value of the electric potential a 0 = c e cos(2θ) (with c e a control parameter). These configurations are static and not spherically symmetric, without being possible to factorize the θ-dependence on both metric and gauge sectors. Therefore the numerical treatment of this problem is much more complicated, the solutions being found by solving partial differential equations. In the absence of an electric charge, they possess a nontrivial solitonic limit and can be interpreted as AdS electric dipoles. More general solutions of the D = 5 minimal gauged supergravity model possessing higher-order multipoles for both the electric and magnetic potentials should also exist. We hope to return elsewhere with a systematic discussion of these aspects.

Acknowledgement
We gratefully acknowledge support by the DFG Research Training Group 1620 "Models of Gravity". E. R. acknowledges funding from the FCT-IF programme. This work was also partially supported by the H2020

A. A squashed S 3 boundary: limiting behaviour of nutty instantons in AdS 4
A simple model, which helps to understand the limiting behaviour of the solutions in this work in terms of the squashing parameter v is provided by the AdS 4 nut-charged instantons. These solutions solve the vacuum Einstein equations on the Euclidean section and can be written in a form resembling (2.4), with (σ i being the one-forms on S 3 as given by (2.5)), where This line element can be put into the standard form given in the literature via the coordinate transformation which results in .

(A.4)
The nutty instantons possess two constants: M , which is a mass parameter, and n-the nut parameter. Also, r is a radial coordinate, whileψ parameterizes a circle S 1 , which is fibred over the two sphere S 2 , with coordinates θ and φ. As a result, the metric (A.4) is only locally asymptotically AdS, while its boundary is a squashed three sphere as r → ∞ . As discussed in [47], this becomes a round S 3 for M = 0, n = L/2, such that (A.4) corresponds to AdS 4 . In the generic case, the absence of a conical singularity imposes some constraints on the value of M , the solutions possessing a variety of interesting features. However, these aspects are of no interest in the context of this work (for a detailed analysis, we refer the reader to Refs. [47], [48], [49], [50], [51], [52]). Instead we shall simply only consider their small/large n limits. For n = 0, one recovers the Schwarzschild-AdS Euclideanized solution, with a S 2 × S 1 topology of an r = const. surface. Note that, for the parametrization (A.1), the limit n → 0 should be taken with a rescaled ψ-coordinate, which, however has a period fixed by M, L.
Another case of interest is n → ∞, in which a different solution is recovered. To understand this limit, one starts again with the line element (A.4) and defines the scaled coordinates which is the planar version of the Taub-NUT-AdS spacetime [47]. This limit possesses a number of interesting properties; here we mention only that an r = const. surface has an R 3 topology, with a warped product R × R 2 . Also, they possess no Misner string singularity (i.e. the periodicity of Ψ is arbitrary) with a breakdown of the entropy/area relationship [50], [51].

B.1 The generic case
The D = 5 Schwarzschild-AdS BH possesses a well-known generalization with a planar horizon topology, which approaches a Poincaré patch at infinity. Adding extra charges (and also a boundary magnetic field) results in generalizations of the Schwarzschild black brane, which possess a variety of interesting properties (see e.g. [10], [11]). We have found that the infinite squashing limit of the BHs in this work leads to a new set of solutions which can be viewed as a generalization of the configurations in [10], [11]. Their horizon (and the spacelike part of the boundary metric) is R 3 , being topologically a direct product of a flat direction and the R 2 -plane. However, this product is 'twisted' (or warped), and the induced geometry is not flat.
These solutions have a line element m(r)L 2 dr 2 r 2 + Again, we assume that the configurations are AlAdS, with a conformal boundary metric given by (3.7), while a k → c m as r → ∞. Then a far field solution can be constructed in a systematic way, the leading order terms in the asymptotics being f (r) = 1 − induced metric on the horizon with a form very similar to (3.6). The horizon area density and Hawking temperature are

B.2 A supersymmetric solution
An interesting question here concerns the possible existence of a supersymmetric limit of the black brane solutions discussed above. To address it, we use a slightly modified version of the framework employed for black holes with a spherical horizon topology. These solutions are a particular member of the timelike case in [53] (see also [54] for a related study). Then we consider the usual framework with with a Kähler metric ds 2 B on a four-dimensional base space (characterized by a one-form X 1 , while P is the Ricci one-form potential) and w a transverse one-form. The solutions of interest are found for the following choice of the base space with the one forms σ 1 = dx, σ 2 = dy, σ 3 = dz + 1 2 (xdy − ydx), x, y, z possessing the usual range. Then a similar reasoning as in the case above of a Bergmann manifold as the base space, leads to a formulation of the problem in terms of a single function a(ρ), with w = Ψ(ρ)σ 3 , P = p(ρ)σ 3 , X = −a 2 (ρ)σ 1 ∧ σ 2 + b(ρ)σ 2 ∧ dρ , (B.8) and the gauge potential The function a(ρ) satisfies again a sixth-order equation which can formally be written in the compact form (4.8). However, the expressions for p, f and g are (slightly) different, with some terms which are absent here: dψ + Θ 2 2 dφ 2 (B.14) This corresponds, in fact, to the solution obtained by Gutowski and Reall in Ref. [8], as the large size limit of the configuration (4.10). It describes a black brane, the above line element possessing a horizon at r = L, whose properties are functions of the parameter α (e.g. the event horizon area density is A H = α 4 L 3 /(3 √ 3)). However, as r → ∞, a non-asymptotically AdS geometry is approached, which corresponds to a plane-fronted wave with a vanishing magnetic field [8].
Based on the results in Section 4, one may expect the existence of other solutions of the sixth-order a-equation, different from (B.13). However, we have failed to find any, the freedom in the choice of the boundary conditions at ρ = 0 (as implied by (4.25)) being absent in this case. Thus one finds a single possible form of the solutions at ρ → 0, corresponding to (4.26). Then the numerical integration of the a-equation leads always to the solution (B.11).