Super throats with non trivial scalars

We find new BPS solutions in N=2, D=4 Fayet-Iliopoulos gauged supergravity with STU prepotential. These are stationary solutions carrying a Kerr-Newman throat spacetime geometry and are everywhere regular. One of the three scalar vector fields is non constant. Moreover, they carry non vanishing magnetic and electric fluxes, generating conduction currents, but not electric or magnetic charges.


Introduction
In the recent years there have been progresses in finding BPS, non BPS and thermal black holes solutions in N = 2 gauged supergravity in four dimensions, coupled with matter, see for example [1], [2], [3], [4] [5] and [6]. Such the solutions are very interesting for several reasons, among which a general understanding of the attractor mechanism, applications to AdS/CFT and its generalizations, and so on. A way of finding BPS solutions is to directly face the equations deduced in [4], where all timelike BPS equations are classified for N = 2, D = 4 gauged supergravity coupled to an arbitrary number of abelian vector multiplets. This is the strategy used here, even though a more general setting could be to follow the strategy of [3], where a very general structure for (non necessarily BPS) black hole solutions has been individuated and proposed as a possible general structure of the black hole solutions. Black hole solutions, however, are not the only interesting ones. Indeed, it is well known the role of Bertotti-Robinson solutions in supergravity, in particular, in relation to the attractor mechanism, [7]. Their typically represent the near horizon geometries of extremal static black holes. In the stationary case, the near horizon geometry may become more involved but, as shown in [8], it has several characteristics in common with the Bertotti-Robinson geometry. Indeed, these kind of geometries have been investigated also in different dimensions, see for example [9], [10], [11] and [12]. Here we will present a different class of BPS solutions, which are everywhere regular solutions with non constant scalar fields. More precisely, we consider the Fayet-Iliopoulos gauged supergravity coupled to three vector multiplets in the STU model with prepotential F (X 0 , X 1 , X 2 , X 3 ) = −2i √ X 0 X 1 X 2 X 3 . Our solutions looks like deformed Bertotti Robinson spacetimes: they are fibrations of AdS2 spaces over a genus zero Riemann surface with non constant scalar curvature. These are stationary solutions carrying both magnetic and electric fluxes. However, there are not electric or magnetic charges. In particular, the electric fluxes can be switched off by varying a parameter to zero, related to the imaginary part of the non constant scalar field. When the electric fluxes are present, then there appear also U (1) electric currents flowing along a Killing direction of the deformed S 2 sphere defining the base of the fibration. This looks to be a conduction current, since there is not any background charge density. In the same direction there is a steady flow of energy, so the energy remains constant in any spacelike compact region, but it is not related to the current flow. In particular, the currents are perpendicular to both the electric and the magnetic fields. When the electric flows are switched off the electric currents disappear and the AdS fibres become flat Minkowski spacetimes R 1,1 . We also find a static solution. Moreover, there are also other similar solutions with the same structure but with the base replaced by non compact spaces. We however did not studied them since we considered the compact ones more interesting. In any case, all these solutions, even if not black holes, are in the general form proposed in [3].

General conventions and equations of motion
We will follow the conventions in [1]. Let us consider N = 2, D = 4 gauged supergravity coupled to n V abelian vector multiplets. Its bosonic content is given by the vierbein e a µ , the U (1) gauge vectors A I µ , I = 0, . . . , nv and complex scalar fields z α , α = 1, . . . , n V , parameterising a special Kähler manifold, which is the base of a symplectic bundle having covariantly holomorphic sections Here D is the Kähler covariant derivative and K the Kähler potential, and ∂ᾱ is the partial derivative w.r.t.zᾱ. These sections are constrained by Usually one assumes the existence of a homogeneous function F of degree two, called the prepotential, such that From this we see that the Kähler potential can be computed as The scalars z α are coupled to the gauge fields via the period matrix N IJ defined by The bosonic part L b of the lagrangian density is where G I µν = ∂µA I ν − ∂ν A I µ are the field strengths and V is the scalar potential with g I = gξ I constant. In [4] it has been shown that if one looks for supersymmetric solutions admitting a timelike Killing spinor, then the most general supersymmetric background can be expressed in coordinates t, z, w,w as where b(z, w,w) is a complex function, Φ(z, w,w) a real function and σ = σwdw + σwdw + σzdz a one form. After setting the fields must satisfy the following system of coupled nonlinear equations Here ⋆ (3) is the Hodge star on the three-dimensional base with metric The fieldstrengths corresponding to a given solution b, Φ, σ and V, are: where Aµ is the gauge field of the Kahler U(1), By now we will refer to eq.(2.10) as the rotating equation, eq.(2.11) as the radial equation, eq.(2.12) as the Bianchi identity, eq.(2.13) as the Maxwell equations, and eq.(2.14) as the metric equation.

ST U model
We are interested in looking for stationary solutions with at least one non constant scalar, and characterised by a nontrivial σ. To this end, we consider the ST U model with prepotential The symplectic section can be parametrised in terms of three complex scalar fields τ1, τ2 e τ3 by choosing Z 0 = 1, τα andτᾱ are the complex coordinates on the scalar manifold. The Kähler potential and the non vanishing components of the metric on the scalar manifold are respectively In particular, we notice the relations between the prepotential and the period matrix. Inspired by [1] and [6], in order to solve the equations, we propose the ansätze Compatibly with this choice, we also make the ansätzē Replacing in equations (2.11) and (2.14), we get with the condition g B > 0. Moreover, we take g A < 0 because of the requests Reτ > 0 and Im(N IJ ) < 0. Bianchi's equations become, for I = A and for I = B ∂z h(η respectively. These are solved by setting 1 1 with the factor 16 included for convenience. This is not the most general solution, but we will discuss the most general case in the Conclusions section where α and β are arbitrary constants. 2 The Maxwell equations become It follows immediately that s −ŝ is a real armonic function. So, in general we must havê where F is a holomorphic function. Moreover eq.(3.9) reduces to In this way our system of equations is reduced to eq. (2.10), (3.14), (3.16) and (3.17).

A Static solution
A simple static solution is easily obtained puttingŝ = −s = a, where a is a constant. It is then straightforward to prove that the general solution of eq.(3.17) is where c > 0 is an integration constant. 3 The full solution reads It is convenient to introduce the new variables θ and φ such that This way the metric takes the form For α different from zero this is a Bertotti-Robinson space-time, which is a AdS2 × S 2 , with scalar curvature − α 2 G a 2 for AdS2 and G 2 for S 2 . For α = 0, it is an R 1,1 × S 2 . The fields strenght are with duals (4.10) Therefore, the magnetic and electric fluxes are respectively. Since d ⋆ G I = 0, the electric four-current densities J I vanish. Since the solution is everywhere regular, this means that there are not charges but just free constant fields.

Stationary solutions
Much more interesting solutions to the system (3.14)-(3.17) can be found just assuming F constant, sayŝ = s + 2a where a is a constant. The resulting equations are stationary conditions for the action functional where s is a scalar function with potential ψ(s) = 2s(s + 2a)(s + a) and R is the scalar curvature of the two dimensional euclidean metric As noticed in [6], this action has an interpretation in terms of generalized dilaton theories. Following [13], it can be shown that the most general solution of the above system is given by where C is an integration constant and the coordinate x is defined by Different types of solutions depend on the sign of C. Here, we will concentrate on the case where C is negative and we set and assume 0 < k 2 < a 2 . (5.8) This choice allows us to obtain solutions with compact (w,w)-space. In order to accomplish this we have to choose for x the range (granting the positivity of ℓ) − √ 9) and to compactify the variable y = 1 2i (w −w). The two dimensional metric ds 2 reads now At the boundary of the range of x, ℓ(x) vanishes so that we have to be careful with y in order to avoid singularities. By symmetry, it is sufficient to investigate what happens at x ∼ √ After introducing the new coordinates we see that the metric is ds 2 ∝ dr 2 + r 2 dφ 2 (5.14) so that, in order to avoid conical singularities, we have to take φ periodic with period 2π. With this choice eq.(5.10) defines a smooth compact surface, and applying the Gauss-Bonnet theorem we easily find it is a surface of genus 0. Finally we can solve the rotation equation (2.10), that gives Using the coordinates t, z, x, φ we find the complete solution with α, β, a and k constant satisfying eq.(5.8), and the gauge fields are where A = 0, 1 and B = 2, 3.
In Appendix it is shown that the metric (5.16) is free of singularities. It has the structure of a fibration over a two dimensional space, with coordinates x, φ, of metric We know that ds 2 corresponds to a compact surface of genus zero, and since the conformal factor 2(a 2 − x 2 ) G is regular and strictly positive for x 2 ≤ a 2 − k 2 , the same is true for dΣ 2 . So, the base of the fibration has the topology of a sphere S 2 . The corresponding scalar curvature is which becomes singular in x = ±a for k = 0. This is why we keep k ≠ 0. As for what concerns the fibres, we see that fixing x = x0, φ = φ0, (5.16) reduces to which is an AdS2 with curvature (5.27) So, topologically, for α ≠ 0 this solution looks to be a deformed Bertotti-Robinson spacetime. However, it has not the same topology but it more resembles the Kerr throat geometry studied in [8]. To see this let us first notice that, after redefining the coordinate r = Gα k 2 (αz + β), (5.28) we can rewrite (5.16) as Then, we introduce a coordinate ψ such that A simple calculation shows that where sn(z, k) is the Jacobi elliptic function defined for k 2 < 1, by As usual, we will omit the elliptic modulus, writing for brevity Therefore, Finally, folowing [8], we also introduce the new coordinates τ, y and ϕ by means of the relations so that We recall that here u = √ a 2 +k 2 2α ψ, and the range of coordinates is where K(k) is the complete elliptic integral of the first kind. Exactly in the same way as in [8], we get that these coordinates cover the whole spacetime, which is geodetically complete. τ is the global time. The topology of this solution is not exactly the one of a Bertotti-Robinson spacetime, but is the same as the Kerr-Newman throat (4.2) of [8]. This allows us to the following conjecture. Conjecture: our solution is the near horizon limit of a rotating, possibly non BPS, extremal AdS black hole carrying both electric and magnetic charges and coupled to a complex scalar field that is non constant at the horizon. For α = 0 the fibres become two dimensional flat Minkowski spacetimes. In this case β ≠ 0 and can be rescaled to 1 by a coordinate redefinition. The metric then takes the form (5.43)

Symmetries and conservation laws. If
Tµν is the energy-momentum tensor, we can construct the tensor density Tµν , which satisfies where repeated latin index run from 1 to 3, while the greek ones from 0 to 3. If S is a compact 3-dimensional spatial region with boundary ∂S with normal directions nj and area element dσ, we can rewrite it in integral form For λ = t, y the last term vanishes identically and the l.h.s. of the above equation individuate conserved quantities. 4 The energy-momentum tensor reads so, the full tensor density T = √ γT is 5 We then see that for λ = y the conservation equation is just an identity that gives just zero. The only interested conserved quantity is associated to T t t . For where ∆z = z2 − z1. This energy is conserved in any compact region, in a stationary way, since there is a constant flow of energy in the φ direction, due to the T φ t term. Let us give a better look to the Killing vector fields. Looking at (5.34) we see that beyond translation in t and φ, the metric is invariant under the rescaling r ↦ λr, t ↦ t λ .
which, as before, do not represent actual electric charges. Indeed, the solution being regular everywhere, included the fieldstrenghts and their duals, there is not any natural candidate as source of charges (apart at most at infinite in z). Further hints come from the electromagnetic currents J I , related to the Maxwell fields by d ⋆ G I = 4π ⋆ J I . Taking into account that ⋆ ⋆ J = −J, the corresponding current vector fields are (5.63) Therefore, there is not even any electric charge density, but only a conduction current along the φ direction. Notice that there are not electric fields along this direction. Nevertheless, these currents go to zero with α, exactly like the electric field, while the magnetic field survives.

Conclusions and perspectives
We have found new interesting supersymmetric backgrounds in N = 2, D = 4 gauged supergravity coupled to vector multiplets for the STU model with prepotential F (X 0 , . . . , These are everywhere regular solutions with one nonconstant complex scalar field and both magnetic and electric fluxes. A parameter α can be set to zero to switch off the electric fluxes. In this limit the topology of the solutions is the one of a fibration R 1,1 over a surface of genus 0, while for α ≠ 0 the solutions have the same topology as the Ker-Newman throat of [8]: a fibration of AdS2 fibres on the base of conformal spheres with non constant scalar curvature. These are stationary solutions, with a rotation parameter along a Killing spacelike direction on the conformal spheres. When the electric fluxes are switched on, current densities flow in the same direction of the rotation, without any electric charge density, so they look more as conduction currents than convective currents. It is worth to mention that, however, these currents are perpendicular to both the electric and magnetic fields. It is interesting to notice that these solutions have the same topology of the near horizon geometry of extremal black holes. We indeed conjecture that these solutions are indeed the near horizon limit of some rotating extremal black hole configuration coupled to electric and magnetic fields. It would be interesting to understand if they carry thermodynamical properties. Another possible generalization of the present paper is the following. In order to solve the equations we assumed ∂ 2 z h = 0, equivalent to the ansatz (3.12), (3.13). However, equations (3.10) and (3.11) require a weaker assumption, which is This system of equations is much more difficult to solve than the one considered in the present paper. It would be interesting to find nontrivial solutions of this system and to study the corresponding spacetime solutions. These and other questions will be considered in a forthcoming paper.