On timelike supersymmetric solutions of Abelian gauged 5-dimensional supergravity

We consider 5-dimensional gauged supergravity coupled to Abelian vector multiplets, and we look for supersymmetric solutions for which the 4-dimensional K\"ahler base space admits a holomorphic isometry. Taking advantage of this isometry, we are able to find several supersymmetric solutions for the ST$[2,n_v+1]$ special geometric model with arbitrarily many vector multiplets. Among these there are three families of solutions with $n_v+2$ independent parameters, which for one of the families can be seen to correspond to $n_v+1$ electric charges and one angular momentum. These solutions generalize the ones recently found for minimal gauged supergravity in JHEP 1704 (2017) 017 and include in particular the general supersymmetric asymptotically-AdS$_5$ black holes of Gutowski and Reall, analogous black hole solutions with non-compact horizon, the three near horizon geometries themselves, and the singular static solutions of Behrndt, Chamseddine and Sabra.


Introduction
Exact solutions of supergravity theories have been and continue to be instrumental in gaining new insights into string theory and related areas of research. In particular asymptotically anti-de Sitter solutions, which occur naturally in gauged supergravity, are interesting from the point of view of the AdS/CFT correspondence, since in that context they can be viewed as gravitational duals of strongly coupled quantum systems living on the AdS boundary.
Symmetry has always been one of the main tools in the search for exact solutions of gravity theories, since requiring the invariance of the solution under some symmetry transformation can dramatically simplify the usually formidable task of solving the equations of motion.
In the supergravity setting it is natural to look for solutions with some unbroken supersymmetry. This implies that the bosonic equations of motion are related through the Killing Spinor Identities [1], reducing the problem of solving them to that of solving just a small subset plus the first order supersymmetry equations.
However, while assuming unbroken supersymmetry makes the problem more tractable, it is usually not enough to find explicit solutions, and one has to make some additional assumptions or to impose a specific ansatz in order to solve the equations. 1 An approach that has proven to be very successful in ungauged 5-dimensional supergravity, with or without vector multiplets, is to assume that the 4-dimensional base space, which for that theory has to be hyperKähler, admits one triholomorphic isometry. In this case the base space has a Gibbons-Hawking metric [3,4], and it turns out that the solutions can be completely characterized in terms of a small number of building blocks, namely harmonic functions on 3-dimensional flat space [5,6]. The same ansatz has also been effective for N = 1, d = 5 supergravity with vector multiplets and non-Abelian gaugings [7] , but without Fayet-Iliopoulos terms, in which case the base space is again a 4-dimensional hyperKähler space.
Recently [8] a similar ansatz was applied to the case of minimal d = 5 gauged supergravity, where a U(1) subgroup of the SU(2) R-symmetry group is gauged by adding a Fayet-Iliopoulos term to the bosonic action. In this case the base space is just Kähler, instead of hyperKähler, and the ansatz consists in assuming that it admits a holomorphic isometry. The metric of the base space can then be written in terms of two functions [9] in a form that generalizes the Gibbons-Hawking metrics, and the problem of finding supersymmetric solutions is reduced to that of solving a system of fourth order differential equations for these two functions plus a third one.
The aim of this paper is to apply the same ansatz in the case of N = 1, d = 5 supergravity with vector multiplets and Abelian Fayet-Iliopoulos gaugings, where a U(1) subgroup of the SU(2) R-symmetry group is gauged with a linear combination of the vector fields of the theory, in which case the base space is again Kähler.
The paper is organized as follows. Section 2 consists in a quick review of the theory and the conditions to impose on the fields in order to obtain (timelike) supersymmetric solutions. In Section 3 we adapt the supersymmetry equations to the assumption that the 4-dimensional Kähler base space of the solution admits a holomorphic isometry, after writing the general form for a metric of this kind. In Section 4, after making some additional assumptions, we find several supersymmetric solutions for the special geometric model ST[2, n v + 1] with an arbitrary number n V of vector multiplets. Among these are three general classes of superficially asymptotically-AdS solutions that can be seen as a generalization in the presence of vector multiplets of solutions found recently for pure gauged supergravity [8]. They are studied in some detail in Subsection 4.1, where the conserved charges are computed for one of the families, and it is shown that they include as particular cases black holes with compact or non-compact horizon, as well as static singular solutions. In Subsection 4.2 we give the explicit expression of the fields for supersymmetric black holes not included in the solutions of Subsection 4.1, despite being very similar to a subcase of them. We conclude in Section 5 with some final remarks.
2 Abelian gauged N = 1, d = 5 supergravity In this section we give a brief description of the bosonic sector of a general theory of N = 1, d = 5 supergravity coupled to n v vector multiplets in which a U(1) subgroup of the SU(2) R-symmetry group has been gauged by the addition of Fayet-Iliopoulos (FI) terms. The U(1) subgroup to be gauged and the gauge vector used in the gauging are determined by the tensor P I r , as we are going to explain. 2 Our conventions are those in Refs. [10,11] which are those of Ref. [12] with minor modifications.
The supergravity multiplet is constituted by the graviton e a µ , the gravitino ψ i µ and the graviphoton A µ . All the spinors are symplectic Majorana spinors and carry a fundamental SU(2) R-symmetry index. The n v vector multiplets, labeled by x = 1, ...., n v consist of a real vector field A x µ , a real scalar φ x and a gaugino λ i x . It is convenient to combine the matter vector fields A x µ with the graviphoton . It is also convenient to define a vector of functions of the scalars h I (φ). N = 1, d = 5 supersymmetry requires that these n v + 1 functions of the n v scalars satisfy a constraint of the form where the constant symmetric tensor C IJK completely characterizes the ungauged theory and the Special Real geometry of the scalar manifold. In particular, the kinetic matrix of the vector fields a IJ (φ) and the metric of the scalar manifold g xy (φ) can be derived from it as follows: first, we define and Then, a IJ is defined implicitly by the relations It can be checked that The metric of the scalar manifold g xy (φ), which we will use to raise and lower x, y indices is (proportional to) the pullback of a IJ We will use the completeness relation The FI gauging of any model of N = 1, d = 5 supergravity coupled to vector multiplets is completely determined by the choice of P I r , where r = 1, 2, 3 is a su(2) index. In the Abelian case, this tensor can be factorized as follows: where g is the gauge coupling constant, d r (which we can normalize d r d r = 1) chooses a direction in S 3 or, equivalently, a u(1) ⊂ su(2) to be gauged and c I (also normalized c I c I = 1) dictates which linear combination of the vector fields, c I A I µ , acts as gauge field. g I = gc I is a convenient combination of constants that we will use. We will not make any specific choices for the time being.
The bosonic action is given in terms of a IJ , g xy and C IJK and P I r where the Abelian vector field strengths are F I µν = 2∂ [µ A I ν] and the scalar potential V (φ) is given by The equations of motion for the bosonic fields are

Timelike supersymmetric solutions
The general form of the solutions of these theories admitting a timelike Killing spinor 3 was found in Refs. [13][14][15]. In what follows we are going to review it using the notation and results of Ref. [11] in which general non-Abelian gaugings were considered, 4 but restricting to Abelian FI gaugings. The building blocks of the timelike supersymmetric solutions are the scalar functionf , the 4-dimensional spatial metric h mn , 5 an antiselfdual almost hypercomplex structureΦ (r) mn , 6 a 1-formω m , the 1-form potentialsÂ I m and the scalars of the theory combined into the functions h I (φ). All these fields are defined on the 4dimensional spatial manifold usually called "base space". They are time-independent and must satisfy a number of conditions: 1. The antiselfdual almost hypercomplex structureΦ (r) mn , the 1-form potentialŝ A I m and the base space metric h mn (through its Levi-Civita connection) satisfy the equation∇ 3 A timelike (commuting) spinor i is, by definition, such that the real vector bilinear constructed from it iV µ ∼¯ i γ µ i is timelike. 4 Even more general gaugings were considered in [16] with the inclusion of tensor multiplets. 5 m, n, p = 1, · · · , 4 will be tangent space indices and m, n, p = 1, · · · , 4 will be curved indices. We are going to denote with hats all objects that naturally live in this 4-dimensional space. 6 That is: the 2-formsΦ (r) mn r, s, t = 1, 2, 3 satisfŷ 2. The selfdual part of the spatial vector field strengthsF I ≡ dÂ I must be related to the functionf , the 1-formω and the scalars of the theory by 3. while the antiselfdual part is related to the almost hypercomplex structure by 7 4. Finally, all the building blocks are related by the equation where the dots indicate standard contraction of all the indices of the tensors.
Once the building blocks that satisfy the above conditions have been found, the physical 5-dimensional fields can be built out of them 8 as follows: 1. The 5-dimensional (conformastationary) metric is given by (2.21) 2. The complete 5-dimensional vector fields are given by so that the spatial components are and the 5-dimensional field strength is In this equation the indices of C IJK have been raised using the inverse metric a IJ and one has the useful relations (2.18) 8 In the ungauged case the above conditions determine the quotients h I /f from whichf can be found by using the condition Eq. (2.1).

The scalar fields φ x can be obtained by inverting the functions h I (φ) or h I (φ).
A parametrization which is always available is As it has already been observed in Refs. [13,15] choosing d r = δ r 1 we see that Eq. (2.16) gives us additional information: it splits intô where we have definedP (2.29) The first equation means that the metric h mn is Kähler with respect to the complex structureĴ mn ≡Φ (1) mn . Taking this fact into account, 9 the integrability condition of the other two equations is 10 (2.34) This equation must be read as a constraint on the 1-form potentialsÂ I m posed by the choice of base space metric.
Eq. (2.19) takes a simpler form as well: 9 We use the integrability condition of Eq. (2.26) which leads to the relation between the Ricci and Riemann tensorŝ The Ricci 2-form, defined asR is, therefore, related to the Riemann tensor bŷ (2.33) 10 If P m vanishes (for instance, in the ungauged case), then we have a covariantly constant hyper-Kähler structure and, then, the base space is hyperKähler.
Tracing the first of these equations and Eq. (2.34) withĴ mn one finds a relation between the Ricci scalar of the base space metric, the scalar potential and the function f :R The last equation to be simplified by our choice is Eq. (2.20). Substituting in it Eq. (2.35) and using Eqs. (2.18) and the completeness relation Eq. (2.7) one findŝ In order to make progress one has to start making specific assumptions about the base space metric. In the ungauged [5,10] and the non-Abelian gauged cases [7] it has proven very useful to assume that the base space metric has an additional isometry because, then, it depends on a very small number of independent functions. Recently the same assumption was made for pure gauged supergravity [8], where the base space can be a general Kähler metric, allowing to reduce the problem of finding supersymmetric solutions to a system of fourth order differential equations for three functions. In what follows we are going to make the same assumption for the case at hand, in which vector multiplets are present, in the attempt to simplify the task of finding supersymmetric solutions. whose integrability condition is In a frame defined by the Vierbein

4)
the conserved complex structure is given by The Ricci tensor and Ricci scalar of the 4-dimensional metric can be expressed in terms of the functions H and W 2 in a compact form, where the 4-dimensional Laplacian acts on z-independent functions aŝ and ∇ 2 is the Laplacian operator associated with the 3-dimensional metric The expression for the Ricci scalar should be compared with Eq. (2.36). We will take the base space metric h mn dx m dx n to be of the form (3.1), and we will make the identificationΦ (1) =Ĵ. We can solve forP m in Eqs. (2.27) and (2.28) if we choose a particular form for the complex structuresΦ (2,3) . Without loss of generality they can be chosen to be where σ 2 is the second Pauli matrix Then we find that the flat components of P can be written in the compact form On the other hand, recalling the definition ofP m Eq. (2.29) we find for the gauge vector and its field strength Every (anti-)selfdual 2-form F ± on the four dimensional Kähler base space can be written in terms of a 1-form living on the 3-dimensional space ϑ = ϑ i dx i as (3.16) The 2-forms we consider here are also z-independent and so will the components of the corresponding 1-forms be. Thus, we introduce the z-independent 3-dimensional 1-forms Λ I , Σ I , Ω ± defined bŷ Requiring the closure of the 2-formsF I =F I+ +F I− one gets which means that, locally, for some functions K I . From the same condition, using Eq. (3.3) and the definition of the operator D 2 in that equation, one also gets (3.24) Using Eq. (3.15) and its full contraction withĴ one finds where an integration constant reflecting the possibility of adding to the solutions K I of eq. (3.24) solutions of the homogeneous equation has been set to zero without loss of generality, since from (3.23) it is clear that the K I 's are defined up to a constant times H. Using these relations, Eq. (3.24) contracted with g I is automatically satisfied, leaving n V independent equations. It is convenient to rewriteω aŝ in terms of which and, then, from Eq. (3.27) we find that Using either of the last two equations in Eq. (3.27) one gets an equation for ω: Before calculating its integrability condition it is convenient to make a change of variables (identical to the one made in the ungauged case) to (partially) "symplecticdiagonalize" the right-hand side. Thus, we define L I and M through (3.31) Substituting these two expressions into Eq. (3.30) and using the relation between the 1-form χ and the functions H and W , Eqs. (3.2), the equation for ω takes the form 11 11 We have left one ω z in order to get a more compact expression.
(3.32) and its integrability equation is just 12 On the other hand, we havê and, using all these partial results into Eq. (2.37), and (not everywhere, for the sake of simplicity) the new variables Eqs. (3.31), we arrive at (3.36) 12 One has 3 d 3 d = ∇ 2 .
We can now use the relation between the 3-dimensional Laplacian and the D 2 operator and the equations for the functions H and K I (3.3) and (3.24) and the equation for L I becomes   (3.25). This is still a very difficult problem, in particular because the constraint (3.20) involves the symmetric tensor C IJK with raised indices, which in general is not constant and cannot be written in a simple way in terms of, for instance, the functions h Î f . To simplify the task one could assume that C IJK is constant, as is the case for several interesting models, in which case (3.20) and (3.31) allow to write Σ I 2 in terms of H, K I and L I . One could then proceed as follows: first choose two functions H and W 2 solving equation (3.3), which amounts to choosing a base space, and subsequently solve the system of second order equations given by (3.24), (3.38) and (3.39) for K I , L I and M , subject to the algebraic constraints (3.25).
Once all these functions are known, eq. (3.31) gives h Î f and ω z , equations (3.2) and (3.30) can be integrated to give respectively χ and ω,ω is given by (3.26) andf can be obtained from the functions h Î f using the special geometric constraint We will also assume a = 0, in which case one can set a = 1 and b = 0 by shifting and rescaling the coordinate , so that Inspired by the pure supergravity case [8] we will take Ψ to be a third order polynomial in . In particular eq. (3.25), which implies such that Ψ = g I Ψ I and Eq. (3.24) can be integrated to give where α I are integration constants, which we will take to be independent of x 1 and x 3 . Eq. (3.25) implies then that Φ must be a solution of Liouville's equation with k given by 2k = g I α I . It is possible to choose without loss of generality k = 0, ±1 and (4.9) Equation (3.2) then determines χ up to a closed 1-form, (4.10) We now focus our attention on special geometric models for which the totally symmetric tensor with raised indices C IJK is constant. 14 Comparing the expression for Σ I in (4.6) with the one in (3.20) it seems a natural choice to introduce n v + 1 first order polynomials in , Q I , such that with eq. (4.6) implying the constraints (4.12) One can then, after computing the functions L I from the definition (3.31), use equation (3.38) to obtain an expression for ∂ M . Since the expression must be the same for each of the n v +1 equations (one for each value of I), the following proportionality conditions must be met: where x, y = 1, . . . , n v , η xy is the Minkowski n v -dimensional metric, and the other components of C IJK vanish. This model reduces to pure supergravity for n v = 1 and h 1 = h 0 , and includes as a special case the STU model for n v = 2. In what follows x-type indices will be raised and lowered with η xy and their contraction will be denoted by a dot (e.g. g·c 1 ≡ g x c 1 x ). The constraints (4.12) become The conditions (4.13) and equation (3.39) are satisfied for an arbitrary choice of gauging constants g I only if one of the following sets of conditions is met: If g·g = 0: If g 0 = 0: If g x = 0 ∀x: The functionf can be computed from (4.11) using the special geometric constraint (2.1), givinĝ (q 00 + q 10 ) q 0 ·q 0 + 2q 0 ·q 1 + q 1 ·q 1 2 1/3 . (4.16) We are interested in particular in asymptotically anti-de Sitter solutions. Given that the line element of AdS 5 (with radius ) can be written in standard supersymmetric form as [8] one expects that for such solutions as → ∞f tends to a constant and Ψ diverges like 3 . These conditions translate to q 10 q 1 ·q 1 = 0 and g I c 3 excluding all the solutions above except the first six for arbitrary gauging. Out of these, however, only the first two are actually asymptotically AdS, at least locally, since in the other cases ω z does not present the correct behavior, being proportional to −1 (one can also check that their scalar curvature does not tend to a constant as → ∞). In the following we will analyze some properties of these two cases.

Case 1
We will now analyze in detail the solutions with parameters satisfying the conditions The functionsf and Ψ becomê while ω z can be obtained from eq. (3.31) after integrating ∂ M , where d is an arbitrary constant, and ω from eq. (3.30) ω = 3 64 Since ω is of the formωχ withω constant, it is always possible to reabsorb ω in ω z with a shift in the t coordinate, t → t +ωz, leading to ω = 0 and The full solution is invariant under the rescaling t → t/α, → α , q 10 → q 10 /α, c 1 I → αc 1 I , c 0 I → α 2 c 0 I . Since we are assuming q 10 = 0 we can use this freedom to set where we introduced for convenience the constant defined by 15 3 g 0 g·g = 2 , (4.26) 15 The solutions presented here are superficially asymptotically AdS 5 , with AdS radius | |.
The line element is then (4.28) Using the parametrization (2.25) the physical scalars are given by The full gauge potentials are given, according to eq. (2.22), by where the 4-dimensional partÂ I can be obtained from (3.17), (3.18), (3.23),

33)
A x = 2g 0 g x + 3 16 Pure supergravity is recovered by choosing g x = g 0 δ 1 x , q 0x = q 00 δ 1 x and q 1x = q 10 δ 1 x . With this choice one recovers the class of asymptotically AdS solutions of minimal gauged N = 1, d = 5 supergravity found in [8].
For each value of k the solutions are determined by n v + 2 parameters, q 0x , c 1 0 and g I c 0 I . The metric however only depends on the q 0x 's through the combinations g ·q 0 and q 0 ·q 0 , so it is always determined by four parameters, independently of the number of vector multiplets n v .

Supersymmetric black holes
If an event horizon exists, it must be situated in = 0, wheref = 0 and the supersymmetric Killing vector ∂ t becomes null. Sincef , H and ω z only depend on , it is possible to perform a coordinate change such that after which the metric takes the form (4.38) The combination (f −1 H −1 −f 2 ω z 2 ) tends to a constant in the limit → 0, so the hypersurface = 0 is null, and is thus a Killing horizon, iff 2 ω z goes to zero. The only possibility to satisfy this condition without giving rise to singularities is to take the scaling limit 39) 16 Note that here h x = η xy h y .
in which case the functions that determine the metric becomê (4.40) (2g 0 g·q 0 + q 00 g·g) + 1 2 (g 0 q 0 ·q 0 + 2q 00 g·q 0 ) . For k = 1 these are the supersymmetric black holes of [14] with the choice (4.14), while for k = 0 and k = −1 one gets a generalization of the black holes with noncompact horizon found in [8] for pure gauged supergravity. For them to be regular, any curvature singularity should lie behind the horizon = 0. Since the curvature scalars diverge whenf −3 vanishes, then the zeroes of (4.40) must be negative, which translates to the conditions and either (g·q 0 ) 2 < q 0 ·q 0 g·g , (4.44) in which case there is only one real root, or (g·q 0 ) 2 ≥ q 0 ·q 0 g·g and g·q 0 g 0 > 0 , (4.45) in which case all roots are negative. Further constraints on the parameters come from the requirementf that also implies H > 0. The near horizon geometries of these black holes are themselves supersymmetric solutions and are included in the class of solutions we presented. They can be obtained from equations (4.20), (4.21) and (4.24) by taking the limit (4.39) and choosing q 10 = 0. They are analogous to the three near horizon geometries obtained in [17] for pure supergravity, in particular one can easily see from (4.38) that dimensional reduction along v gives the geometries AdS 2 × S 2 , AdS 2 × H 2 or AdS 2 × E 2 , and that the horizon geometry is given by a homogeneous Riemannian metric on the group manifolds SU(2) (in which case the metric is that of a squashed S 3 ), SL (2, R) or N il respectively for k = 1, −1 or 0. The entropy for the compact k = 1 case was computed in [14].
For k = 0 it is not possible to eliminate the cross term in a simple way, and the metric is In the pure supergravity case this reduces to a metric without free parameters and having constant curvature scalars [8]. Here this is not true in general, and only happens if in which case the metric is the same as in the pure supergravity case, but it is still possible to have independent vector fields and non-trivial scalar fields.

Case 2
The solutions with are almost identical to the black hole limit of the ones in Subsection 4.1, given in equations (4.40), (4.41) and (4.42), with the additional constraint g·q 0 = 0. However there is an additional term in the 4-dimensional gauge potentialsÂ x proportional to the constants c 1 x , which were zero in the aforementioned limit. These constants are not completely arbitrary, being constrained by the relations g·c 1 = q 0 ·c 1 = c 1 ·c 1 = 0.
After the rescaling (4.25) the functions determining the metric arê and (4.80) For k = 1, the mass, angular momenta and electric charges are Keeping into account the constraints to which the constants q 0x and c 1 x are subject, it is easy to check that the relation (4.56) is satisfied.

Conclusions
In this paper we have adapted the equations that determine the timelike supersymmetric solutions of N = 1, d = 5 Abelian gauged supergravity coupled to vector multiplets to the assumption that the Kähler base space admits a holomorphic isometry. While the resulting system of equations is much more involved than in the pure supergravity case, we were able, thanks in part to the experience gained in this latter case, to obtain several supersymmetric solutions. Of these, the more interesting ones are three classes (for k = 0, ±1) of superficially asymptotically-AdS (globally asymptotically-AdS for k = 1) solutions, which are a direct generalization of the similar solutions found for pure supergravity in [8], and which include various already known solutions.
It is worth noting that the special geometric model ST[2, n v + 1] considered here admits as a special case the so-called U(1) 3 model, which is just the STU model with equal gauging parameters g I . This means that in this particular subcase our solutions can be oxidized to type-IIB supergravity as described in [20].
The solutions constructed here only have one independent angular momentum, however there are in the literature examples of supersymmetric black holes with two independent angular momenta in N = 1, d = 5 Abelian gauged supergravity, both without and with vector multiplets [21,22]. It would be interesting to study whether less restrictive assumptions than those made in this paper could lead to solutions generalizing these black holes. Another possible extension of our work would be to consider more general gaugings, for instance a combination of the Abelian Fayet-Iliopoulos gauging considered here and non-Abelian gaugings of the scalar manifold isometries. Work along these lines is in progress [23].