Supersymmetric black holes and attractors in gauged supergravity with hypermultiplets
PPrepared for submission to JHEP
IFUM-1038-FT
Supersymmetric black holes and attractors in gauged supergravity with hypermultiplets
Samuele Chimento, Dietmar Klemm and Nicol`o Petri
Dipartimento di Fisica, Universit`a di Milano, andINFN, Sezione di Milano,Via Celoria 16, 20133 Milano, Italy.
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We consider four-dimensional N = 2 supergravity coupled to vector-and hypermultiplets, where abelian isometries of the quaternionic K¨ahler hypermul-tiplet scalar manifold are gauged. Using the recipe given by Meessen and Ort´ın inarXiv:1204.0493, we analytically construct a supersymmetric black hole solution forthe case of just one vector multiplet with prepotential F = − iχ χ , and the univer-sal hypermultiplet. This solution has a running dilaton, and it interpolates betweenAdS × H at the horizon and a hyperscaling-violating type geometry at infinity, con-formal to AdS × H . It carries two magnetic charges that are completely fixed in termsof the parameters that appear in the Killing vector used for the gauging.In the second part of the paper, we extend the work of Bellucci et al. on black holeattractors in gauged supergravity to the case where also hypermultiplets are present.The attractors are shown to be governed by an effective potential V eff , which is extrem-ized on the horizon by all the scalar fields of the theory. Moreover, the entropy is givenby the critical value of V eff . In the limit of vanishing scalar potential, V eff reduces (upto a prefactor) to the usual black hole potential. Keywords:
Black Holes, Supergravity Models, Black Holes in String Theory, Attrac-tor Mechanism. a r X i v : . [ h e p - t h ] A p r ontents N = 2 , d = 4 gauged supergravity 33 Supersymmetric solutions 54 A black hole solution 65 Attractor mechanism 126 Final remarks 17 Black holes in gauged supergravity theories provide an important testground to addressfundamental questions of gravity, both at the classical and quantum level. Among theseare for instance the problems of black hole microstates, the final state of black holeevolution, uniqueness- or no hair theorems, to mention only a few of them. In gaugedsupergravity, the solutions typically have AdS asymptotics, and one can then try tostudy these issues guided by the AdS/CFT correspondence. On the other hand, blackhole solutions to these theories are also relevant for a number of recent developmentsin high energy- and especially in condensed matter physics, since they provide thedual description of certain condensed matter systems at finite temperature, cf. [1] fora review. In particular, models that contain Einstein gravity coupled to U(1) gaugefields and neutral scalars have been instrumental to study transitions from Fermi-liquidto non-Fermi-liquid behaviour, cf. [2, 3] and references therein. In AdS/condensedmatter applications one is often interested in including a charged scalar operator in thedynamics, e.g. in the holographic modeling of strongly coupled superconductors [4].This is dual to a charged scalar field in the bulk, that typically appears in supergravitycoupled to gauged hypermultiplets. It would thus be desirable to dispose of analyticalblack hole solutions to such theories. In the first part of the present paper we will make The necessity of a bulk U(1) gauge field arises, because a basic ingredient of realistic condensedmatter systems is the presence of a finite density of charge carriers. – 1 – first step in this direction. Solving the corresponding second order equations of motionis generically quite involved, such that one is forced to resort to numerical techniques.For this reason we shall look here for BPS black holes, which satisfy first order equations,and make essential use of the results of [5], where all supersymmetric backgrounds of N = 2, d = 4 gauged supergravity coupled to both vector- and hypermultiplets wereclassified. This provides a systematic method to obtain BPS solutions, without thenecessity to guess some suitable ans¨atze. Let us mention here that black holes infour-dimensional gauged supergravity with hypers were also obtained numerically in[6]. Solutions that have ghost modes (i.e., with at least one negative eigenvalue of thespecial K¨ahler metric) were constructed in [7]. In five dimensions, a singular solution ofsupergravity with gauging of the axionic shift symmetry of the universal hypermultipletwas derived in [8]. Finally, ref. [9] analyzed the near-horizon geometries of static BPSblack holes in four-dimensional N = 2 supergravity with gauging of abelian isometriesof the hypermultiplet scalar manifold, while the authors of [10] found nonrelativistic(Lifshitz and Schr¨odinger) solutions in the same theory for the canonical example of asingle vector- and a single hypermultiplet .Another point of interest addressed in this paper is the attractor mechanism [12–16], that has been the subject of extensive research in the asymptotically flat case, butfor which not very much has been done for black holes with more general asymptotics.First steps towards a systematic analysis of the attractor flow in gauged supergravitywere made in [17, 18] for the non-BPS and in [19–22] for the BPS case. Some interestingresults have been found, for instance the appearance of flat directions in the effectiveblack hole potential for BPS flows [20], a property that does not occur in ungauged N = 2, d = 4 supergravity [16], at least as long as the metric of the scalar manifold isstrictly positive definite.In the second part of our paper we extend the work of [18] to include also gaugedhypermultiplets. We shall construct an effective potential V eff that depends on boththe usual black hole potential and the potential for the scalar fields. V eff governs theattractors, in the sense that it is extremized on the horizon by all the scalar fields of thetheory, and the entropy is given by the critical value of V eff . As in [18], our analysis doesnot make use of supersymmetry, so our results are valid for any static extremal blackhole in four-dimensional N = 2 matter-coupled supergravity with gauging of abelianisometries of the hypermultiplet scalar manifold.The remainder of this paper is organized as follows: In the next section, we brieflyreview N = 2, d = 4 gauged supergravity coupled to vector- and hypermultiplets. For related work cf. [11], where Lifshitz solutions in general N = 2, d = 4 supergravity modelswere obtained by reducing d = 5 theories with AdS vacua. – 2 –ection 3 summarizes the general recipe to construct supersymmetric solutions providedin [5]. In 4, a simple model is considered that has just one vector multiplet with specialK¨ahler prepotential F = − iχ χ , and the universal hypermultiplet. In this setting,the equations of [5] are then solved and a genuine BPS black hole with running dilatonand two magnetic charges is constructed. Section 5 contains an extension of the resultsof [18] on black hole attractors in gauged supergravity to the case that includes alsohypermultiplets. Section 6 contains our conclusions and some final remarks. N = 2 , d = 4 gauged supergravity The gravity multiplet of N = 2, d = 4 supergravity can be coupled to a number n V of vector multiplets and to n H hypermultiplets. The bosonic sector then includes thevierbein e aµ , ¯ n ≡ n V + 1 vector fields A Λ µ with Λ = 0 , . . . n V (the graviphoton plus n V other fields from the vector multiplets), n V complex scalar fields Z i , i = 1 , . . . , n V , and4 n H real hyperscalars q u , u = 1 , . . . , n H .The complex scalars Z i of the vector multiplets parametrize an n V -dimensionalspecial K¨ahler manifold, i.e. a K¨ahler-Hodge manifold, with K¨ahler metric G i ¯ ( Z, ¯ Z ),which is the base of a symplectic bundle with the covariantly holomorphic sections . V = (cid:18) L Λ M Λ (cid:19) , D ¯ ı V ≡ ∂ ¯ ı V −
12 ( ∂ ¯ ı K ) V = 0 , (2.1)obeying the constraint (cid:10) V| ¯ V (cid:11) ≡ ¯ L Λ M Λ − L Λ ¯ M Λ = − i , (2.2)where K is the K¨ahler potential. Alternatively one can introduce the explicitly holo-morphic sections of a different symplectic bundle,Ω ≡ e −K / V ≡ (cid:18) χ Λ F Λ (cid:19) . (2.3)In appropriate symplectic frames it is possible to choose a homogeneous function ofsecond degree F ( χ ), called prepotential, such that F Λ = ∂ Λ F . In terms of the sectionsΩ the constraint (2.2) becomes (cid:10) Ω | ¯Ω (cid:11) ≡ ¯ χ Λ F Λ − χ Λ ¯ F Λ = − ie −K . (2.4)The couplings of the vector fields to the scalars are determined by the ¯ n × ¯ n periodmatrix N , defined by the relations M Λ = N ΛΣ L Σ , D ¯ ı ¯ M Λ = N ΛΣ D ¯ ı ¯ L Σ . (2.5) The conventions and notation used in this paper are those of refs. [5, 23] – 3 –f the theory is defined in a frame in which a prepotential exists, N can be obtainedfrom N ΛΣ = ¯ F ΛΣ + 2 i (cid:0) N ΛΓ χ Γ (cid:1) (cid:0) N Σ∆ χ ∆ (cid:1) χ Ω N ΩΨ χ Ψ , (2.6)where F ΛΣ = ∂ Λ ∂ Σ F and N ΛΣ ≡ Im ( F ΛΣ ).The 4 n H real hyperscalars q u parametrize a quaternionic K¨ahler manifold withmetric H uv ( q ). A quaternionic K¨ahler manifold is a 4 n -dimensional Riemannian mani-fold admitting a locally defined triplet (cid:126) K vu of almost complex structures satisfying thequaternion relation K K = K , (2.7)and whose Levi-Civita connection preserves (cid:126) K up to a rotation, ∇ w (cid:126) K vu + (cid:126) A w × (cid:126) K vu = 0 , (2.8)with SU(2) connection (cid:126) A ≡ (cid:126) A u ( q ) dq u . An important property is that the SU(2) curva-ture is proportional to the complex structures, F x ≡ d A x + 12 ε xyz A y ∧ A z = − K x . (2.9)We will only consider gaugings of abelian symmetries of the action. Under theaction of abelian symmetries, the complex scalars Z i transform trivially, so that we willbe effectively gauging abelian isometries of the quaternionic-K¨ahler metric H uv . Theseare generated by commuting Killing vectors k Λ u ( q ), [ k Λ , k Σ ] = 0, and the requirementthat the quaternionic-K¨ahler structure is preserved implies the existence of a triplet ofKilling prepotentials, or moment maps, P Λ x for each Killing vector such that P Λ x = 12 n H K xuv ∇ v k Λ u , D u P Λ x ≡ ∂ u P Λ x + ε xyz A yu P Λ z = − K xuv k Λ v . (2.10)The bosonic action reads S = (cid:90) d x (cid:112) | g | (cid:2) R + 2 G i ¯ ∂ µ Z i ∂ µ ¯ Z ¯ + 2 H uv D µ q u D µ q v +2 I ΛΣ F Λ µν F Σ µν − R ΛΣ F Λ µν (cid:63) F Σ µν − V ( Z, ¯ Z, q ) (cid:3) , (2.11)where the scalar potential has the form V ( Z, ¯ Z, q ) = g (cid:20) L Λ L Σ ( H uv k Λ u k Σ v − P Λ x P Σ x ) − I ΛΣ P Λ x P Σ x (cid:21) , (2.12)the covariant derivatives acting on the hyperscalars are D µ q u = ∂ µ q u + gA Λ µ k Λ u , (2.13)and I ΛΣ ≡ Im ( N ΛΣ ) , R ΛΣ ≡ Re ( N ΛΣ ) , I ΛΣ I ΣΓ = δ ΛΓ . (2.14)– 4 – Supersymmetric solutions
All the timelike supersymmetric solutions to N = 2 gauged supergravity in four dimen-sions were characterized by Meessen and Ort´ın in [5]. Here we summarize their results,restricted to the case of abelian gauging.The expressions and equations that follow are given in terms of bilinears con-structed out of the Killing spinors, X = 12 ε IJ ¯ (cid:15) I (cid:15) J , V a = i ¯ (cid:15) I γ a (cid:15) I , V xa = i ( σ x ) JI ¯ (cid:15) I γ a (cid:15) J , (3.1)and of the real symplectic sections of K¨ahler weight zero R ≡ Re ( V /X ) , I ≡ Im ( V /X ) . (3.2)The metric and vector fields take the form ds = 2 | X | ( dt + ω ) − | X | h mn dy m dy n , (3.3) A Λ = − R Λ V + ˜ A Λ m dy m , (3.4)where the 3-dimensional metric h mn must admit a dreibein V x satisfying the structureequation dV x + (cid:15) xyz (cid:16) A y − g ˜ A Λ P Λ y (cid:17) ∧ V z + g √ I Λ P Λ y V y ∧ V x = 0 . (3.5) | X | can be determined from R and I ,12 | X | = (cid:104)R|I(cid:105) , (3.6)the 1-form V is given by V = 2 √ | X | ( dt + ω ) , (3.7)and the spatial 1-form ω satisfies( dω ) xy = 2 ε xyz (cid:20) (cid:104)I| ∂ z I(cid:105) − g √ | X | R Λ P Λ z (cid:21) . (3.8) Eq. (3.5) corrects a typo in [5]; the terms containing the moment maps must have the oppositesign w.r.t. the one in [5]. – 5 –he complex scalars Z i , the sections R and I , the 1-form ω , the function X andthe hyperscalars q u are all time-independent.The complex scalars depend, in a way that depends on the chosen parametrizationof the special K¨ahler manifold, on the sections R and I . A common simple choice ofparametrization is χ = 1, χ i = Z i , in which case Z i = L i L = R i + i I i R + i I . (3.9)The effective 3-dimensional gauge connection ˜ A Λ must satisfy( d ˜ A Λ ) xy = ˜ F Λ xy = − √ ε xyz ( ∂ z I Λ + g B Λ z ) , (3.10)with B Λ x ≡ √ (cid:20) R Λ R Σ + 18 | X | I ΛΣ (cid:21) P Σ x , (3.11)from which follows the integrability condition˜ ∇ I Λ + g ˜ ∇ x B Λ x = 0 . (3.12)A similar condition holds for the I Λ ’s,˜ ∇ I Λ + g ˜ ∇ x B Λ x = g √ (cid:104)I| ∂ x I(cid:105) P Λ x + g | X | R Σ [ k Λ u k Σ u − P Λ x P Σ x ] , (3.13)where B Λ x ≡ √ (cid:20) R Λ R Σ + 18 | X | R ΛΓ I ΓΣ (cid:21) P Σ x . (3.14)Finally, the hyperscalars must satisfy the equation K x uv V x µ D µ q v + √ g | X | I Λ k Λ u = 0 . (3.15)For a given special geometric model the sections R can always, at least in principle,be determined in terms of the sections I , by solving the so-called stabilization equations .This means that to obtain a supersymmetric solution one needs to solve the aboveequations for I Λ , I Λ , ω , V x and q u . We now turn to the task of obtaining an explicit solution with non-trivial hyperscalars.To do so, we consider a simple theory with just one vector multiplet and one hyper-multiplet, n V = n H = 1. – 6 –ore specifically, let the hypermultiplet be the universal hypermultiplet [24]. Thescalar fields in this multiplet, denoted by ( φ, a, ξ , ξ ), parametrize the quaternionicspace SU(2 , / U(2), with metric H uv dq u dq v = dφ + 14 e φ (cid:18) da − (cid:104) ξ | dξ (cid:105) (cid:19) + 14 e φ [( dξ ) + ( dξ ) ] , (4.1)where (cid:104) ξ | dξ (cid:105) ≡ ξ dξ − ξ dξ , and the corresponding SU(2) connection has components A = e φ dξ , A = e φ dξ , A = e φ (cid:18) da − (cid:104) ξ | dξ (cid:105) (cid:19) . (4.2)As for the vector multiplet, we choose a special geometric model specified by the pre-potential F ( χ ) = − iχ χ , (4.3)with the parametrization χ = 1, χ = Z . Then it is easy to obtain from (2.4) theK¨ahler potential K = − log [4 Re ( Z )] and the K¨ahler metric G Z ¯ Z = ∂ Z ∂ ¯ Z K = 14 Re ( Z ) , (4.4)while the period matrix N ΛΣ , giving the scalar-vector couplings, is calculated fromeq. (2.6) to be N = − i (cid:18) Z Z (cid:19) . (4.5)Using the definition (3.2), the dependence of the R section on the I section forthis special geometric model is readily seen to be R = −I , R = −I , R = I , R = I , (4.6)so that the complex scalar is given by Z = R + i I R + i I = I − i I I − i I , (4.7)and 12 | X | = (cid:104)R|I(cid:105) = 2 (cid:0) I I + I I (cid:1) . (4.8)Since the theory includes two vector fields, we can choose to gauge up to two isometriesof the metric H uv . We choose to gauge the (commuting) isometries generated by theKilling vectors k Λ = k Λ ∂ a + δ c (cid:0) ξ ∂ ξ − ξ ∂ ξ (cid:1) , (4.9)– 7 –here k Λ and c are constants. This means that we are gauging the R group of thetranslations along a with the combination A Λ k Λ , and the U(1) group of rotations inthe ξ – ξ plane with the field A . (4.9) is a subcase of the Killing vector considered in[6], and corresponds to setting Q Λ A = Q Λ A = 0 , U = (cid:18) c − c (cid:19) (4.10)in eqs. (3.8) and (3.9) of [6]. The triholomorphic moment maps associated with theKilling vectors (4.9) can be obtained from (2.10), and are P Λ1 = − δ c ξ e φ , P Λ2 = δ c ξ e φ , (4.11) P Λ3 = δ c (cid:20) − e φ (cid:0) ( ξ ) + ( ξ ) (cid:1)(cid:21) + 12 k Λ e φ . With these choices the scalar potential (2.12) reads V = g (cid:26) Z + ¯ Z (cid:20) e φ (cid:104) k − c (cid:0) ( ξ ) + ( ξ ) (cid:1)(cid:105) − c − k c e φ (cid:21) + Z ¯ ZZ + ¯ Z e φ k − k c e φ (cid:27) . (4.12)For simplicity we will look for solutions with R = R = I = I = 0, which impliesfrom (4.7) that the scalar Z is real and from (3.4) that the gauge fields are in a purelymagnetic configuration. From eq. (3.8) follows that ω is a closed 1-form, and can bereabsorbed by a redefinition of the coordinate t , leading to static solutions. This choicealso implies that eq. (3.13) is trivially satisfied.We will also take the hyperscalar a to be constant and ξ = ξ = 0. Note thatthe scalar potential (4.12) has then a critical point at Z = − k /k and e φ = − c/k ,with V crit = 3 k g c / (8 k ). Since the absence of ghost modes requires Z >
0, one needs k /k < c/k <
0) to have a critical point of the potential. With thechoice ξ = ξ = 0, the moment maps (4.11) become P Λ1 = P Λ2 = 0 , P Λ3 = δ c + 12 k Λ e φ . (4.13)Eq. (3.5) implies then dV = 0, hence there exists a function r (that we will use as acoordinate) such that locally V = dr . (4.14)– 8 –e shall impose radial symmetry on the solution by requiring the scalar fields Z , φ and the sections I Λ to depend only on r .The φ , ξ and ξ components of equation (3.15) reduce then to the constraint A Λ x k Λ = 0 = ⇒ ˜ A Λ k Λ = 0 , (4.15)while the a component becomes φ (cid:48) = g √ e φ I Λ k Λ , (4.16)where the prime stands for a derivative with respect to r .If we now introduce the remaining coordinates ϑ and φ by choosing V = e W ( r ) dϑ , V = e W ( r ) f ( ϑ ) dϕ , (4.17)where at this stage f is an arbitrary function of ϑ , the remaining components of eq. (3.5)are satisfied provided that the following conditions are met W (cid:48) ( r ) = − g √ P Λ3 I Λ = − g √ (cid:18) c I + e φ I Λ k Λ (cid:19) , (4.18)˜ A = − f (cid:48) ( ϑ ) gc dϕ . (4.19)From (4.19) and the constraint (4.15) we also have˜ A = k k f (cid:48) ( ϑ ) gc dϕ . (4.20)Finally, (3.10) leads to the two equations (cid:20)(cid:0) I Λ k Λ (cid:1) (cid:48) − g √ (cid:0) I Λ (cid:1) k Λ P Λ3 (cid:21) e W ( r ) = ( − Λ √ k gc f (cid:48)(cid:48) ( ϑ ) f ( ϑ ) (no sum over Λ) , (4.21)while (3.12) is automatically satisfied since we obtained ˜ F Λ as the exterior derivativeof the effective connection ˜ A Λ .Equation (4.16) allows us to use the chain rule to trade the coordinate r for φ in(4.21), which after summing over Λ becomes12 ∂ φ (cid:104)(cid:0) I Λ k Λ (cid:1) (cid:105) − (cid:0) I Λ k Λ (cid:1) + 2 I k (cid:0) I k − I c e − φ (cid:1) = 0 . (4.22)If we impose the condition I k = I c e − φ , (4.23)– 9 –his equation is solved by I = αe φ k + c e − φ , I = ck αe − φ k + c e − φ , (4.24)where α is an integration constant. Substituting these expressions back in (4.21) forΛ = 0 or Λ = 1, we obtain an expression for the function W ( r ), e W ( r ) = (cid:20) αgc (cid:0) k + c e − φ (cid:1) e − φ (cid:21) f (cid:48)(cid:48) ( ϑ ) f ( ϑ ) . (4.25)The expression (4.25) is also a solution of equation (4.18), which is non-trivial, provingthe constraint (4.23) to be consistent with all the equations. From (4.25) we alsoconclude that f (cid:48)(cid:48) ( ϑ ) /f ( ϑ ) should be a positive constant, therefore f ( ϑ ) in general takesthe form f ( ϑ ) = γ sinh ( δϑ + ρ ) , (4.26)where γ , δ and ρ are constants. We can now go back to the coordinate r by solvingequation (4.16) to obtain the dependence of φ on r , obtaining φ = −
13 log (cid:18) − αg √ r + β (cid:19) , (4.27)where β is yet another integration constant. Note that all the integration constantscan be reabsorbed by the coordinate change( t , r , ϑ , ϕ ) −→ (cid:32) √ αgk c t , − √ αg (cid:0) r − β (cid:1) , ϑ − ρδ , ϕδγ (cid:33) , (4.28)that allows to write the complete solution as ds = 16 r g k c (cid:34)(cid:18) k c r (cid:19) r dt − (cid:18) k c r (cid:19) − dr r − (cid:0) dϑ + sinh ϑ dϕ (cid:1)(cid:35) , (4.29) A = − cosh ϑgc dϕ , A = k k cosh ϑgc dϕ , (4.30) φ = − log r , Z = ck r . (4.31)We start the analysis of the solution by noting that it has no free parameters, sinceall the constants appearing in (4.29)–(4.31) are completely determined by the choice– 10 –f gauging. Observe also that in order to maintain the correct signature and to have Z >
0, which is required to have a real K¨ahler potential, we have to impose k c > r = 0 and, if k c <
0, also in r = (cid:112) − k /c .The singularity in r = r S ≡ r = r H ≡ (cid:112) − k /c is not and corresponds instead to a Killing horizon, always covering thecurvature singularity.With the metric written in the form (4.29), it is immediate to see that in theasymptotic limit r → + ∞ it reduces to ds = 16 r g k c (cid:20) r dt − dr r − (cid:0) dϑ + sinh ϑ dϕ (cid:1)(cid:21) , (4.32)which is manifestly conformally equivalent to AdS × H . Note that (4.32) is very similarto hyperscaling violating geometries, which in d dimensions have the form ds = r − θd − (cid:18) r z dt − dr r − r ( dx i ) (cid:19) , (4.33)where i = 1 , . . . , d −
2. Here, z is the dynamical critical exponent and θ the so-calledhyperscaling violation exponent. Under the scaling r → r/λ , x i → λx i , t → λ z t , (4.33)is not invariant, but transforms covariantly, ds → λ θ/ ( d − ds . Geometries of the form(4.33) have been instrumental in recent applications of AdS/CFT to condensed matterphysics, cf. e.g. [25]. (4.32) exhibits actually a scaling behaviour similar to that of(4.33). To see this, introduce new coordinates x , y on H according to x + iy = e iϕ tanh ϑ + 1 e iϕ tanh ϑ − , (4.34)which casts (4.32) into the form ds = 16 r g k c (cid:20) r dt − dr r − dx + dy x (cid:21) . (4.35)Under the scaling r → rλ , t → λt , x → λx , y → λy , (4.36)(4.35) transforms as ds → ds/λ .In the near-horizon limit, r → r H , after the coordinate change t → t/
4, the metrictakes the form ds = − g c k k (cid:20) r dt − dr r − (cid:0) dϑ + sinh ϑ dϕ (cid:1)(cid:21) , (4.37)– 11 –hich is AdS × H , while the scalar fields take the values φ = −
12 log (cid:18) − k c (cid:19) , Z = − k k . (4.38)The magnetic charges are given by P Λ = 14 π (cid:90) F Λ = p Λ V , V = (cid:90) sinh ϑ dϑ ∧ dϕ , (4.39)yielding for the magnetic charge densities p = − πgc p = k k πgc . (4.40)The Bekenstein-Hawking entropy density can then be written as s = S V = − k k g c = 32 π p p . (4.41) In [18] the authors presented a generalization of the well-known black hole attractormechanism [12–16] to extremal static black holes in N = 2, d = 4 gauged supergravitycoupled to abelian vector multiplets. In this section we closely follow their argument,generalizing it by taking into account the presence of gauged hypermultiplets. As in[18], we make no assumption on the form of the scalar potential, of the vectors’ kineticmatrix N or on the scalar manifolds, so that our results are valid not only for N = 2supergravity, but for any theory described by an action of the form (2.11).The equations of motion obtained from the variation of (2.11) are R µν + T µν + 2 G i ¯ ∂ ( µ Z i ∂ ν ) ¯ Z ¯ + 2 H uv D µ q u D ν q v − g µν V = 0 , (5.1) ∇ ν ( (cid:63)F Λ νµ ) + g k Λ u D µ q u = 0 , (5.2) D Z i + ∂ i F Λ µν (cid:63) F Λ µν + 12 ∂ i V = 0 , (5.3) D q u + 14 ∂ u V = 0 , (5.4)– 12 –here T µν ≡ I ΛΣ (cid:16) F Λ ρµ F Σ νρ − g µν F Λ ρσ F Σ ρσ (cid:17) , (5.5)the dual field strengths are given by F Λ µν ≡ − (cid:112) | g | δSδ (cid:63) F Λ µν = R ΛΣ F Σ µν + I ΛΣ (cid:63) F Σ µν , (5.6)and the second covariant derivatives on the scalars act as D Z i = ∇ µ ∂ µ Z i + Γ ijk ∂ µ Z j ∂ µ Z k , (5.7) D q u = ∇ µ D µ q u + Γ uvw D µ q v D µ q w + gA Λ µ ∂ v k Λ u D µ q v . (5.8)The metric for the most general static extremal black hole background with flat, spher-ical or hyperbolic horizon can be written in the form ds = e U ( r ) dt − e − U ( r ) (cid:2) dr + e W ( r ) (cid:0) dϑ + f κ ( ϑ ) dϕ (cid:1)(cid:3) , (5.9)with f κ ( ϑ ) = sin ϑ , κ = 1 ,ϑ , κ = 0 , sinh ϑ , κ = − . (5.10)We require that all the fields are invariant under the symmetries of the metric, namelythe time translation isometry generated by ∂ t and the spatial isometries generated bythe Killing vectors ∂ ϕ , cos ϕ ∂ ϑ − f (cid:48) κ f κ sin ϕ ∂ ϕ , sin ϕ ∂ ϑ + f (cid:48) κ f κ cos ϕ ∂ ϕ . (5.11)The scalar fields can then only depend on the radial coordinate r , and the request ofinvariance of the field strength 2-forms F Λ leads to F Λ = 12 F Λ µν ( x ) dx µ dx ν = F Λ tr ( r ) dt ∧ dr + F Λ ϑϕ ( r, ϑ ) dϑ ∧ dϕ , (5.12)with F Λ ϑϕ ( r, ϑ ) = 4 πp Λ ( r ) f κ ( ϑ ) , (5.13)where p Λ ( r ) is a generic function of r . The Bianchi identities ∇ ν (cid:0) (cid:63)F Λ νµ (cid:1) = 0 ⇐⇒ ∂ [ µ F Λ νρ ] = 0 (5.14)– 13 –mply that p Λ must be constant. With field strengths of this form, it is always possibleto choose a gauge in which the gauge potential 1-forms can be written as A Λ = A Λ t ( r ) dt + A Λ ϕ ( ϑ ) dϕ . (5.15)The r -component of the Maxwell equations (5.2) reduces then to the condition k Λ u ( q ) ∂ r q u = 0 , (5.16)while the ϑ -component is automatically satisfied and the ϕ -component gives A Σ ϕ k Σ u k Λ u = 0 (5.17)for every value of Λ, or equivalently k Λ u ( q ) p Λ = 0 . (5.18)Finally if we define a function e Λ ( r ) such that F Λ tr ( r ) = 4 πI ΛΣ (cid:0) e Σ ( r ) − R ΣΓ p Γ (cid:1) e U − W ) , (5.19)we have F Λ ϑϕ = 4 πe Λ ( r ) f κ ( ϑ ) and the t -component of the Maxwell equations becomes4 πe U − W ) ∂ r e Λ = g e − U A Σ t k Σ u k Λ u . (5.20)The quantities p Λ and e Λ ( r ) are the magnetic and electric charge densities inside the2-surfaces S r of constant r and t , p Λ = 14 π V (cid:90) S r F Λ , e Λ ( r ) = 14 π V (cid:90) S r F Λ , V = (cid:90) S r f κ ( ϑ ) dϑ ∧ dϕ . (5.21)The non-vanishing components of T µν are given by T tt = T rr = − T θθ = − T ϕϕ = (8 π ) e U − W ) ˜ V BH , (5.22)where ˜ V BH is the so-called black hole potential,˜ V BH = − (cid:0) p Λ , e Λ ( r ) (cid:1) (cid:18) I ΛΣ + R ΛΓ I ΓΩ R ΩΣ − R ΛΓ I ΓΣ − I ΛΓ R ΓΣ I ΛΣ (cid:19) (cid:18) p Σ e Σ ( r ) (cid:19) , (5.23)which however, unlike the usual definition, has an explicit dependence on r throughthe varying electric charges e Λ . It is also straightforward, using the expressions (5.13),(5.19) and the definition (5.6), to verify that ∂ i F Λ µν (cid:63) F Λ µν = (8 π ) e U − W ) ∂ i ˜ V BH , (5.24)– 14 –here on the left-hand side only the dual field strengths F Λ are taken to depend onthe complex scalars Z i and only through the matrices R ΛΣ and I ΛΣ appearing in (5.6),while on the right-hand side the charges e Λ ( r ) are considered to be independent of the Z i . Equations (5.1), (5.3) and (5.4) then reduce to e U (2 U (cid:48) W (cid:48) + U (cid:48)(cid:48) ) − (8 π ) e U − W ) ˜ V BH − g e − U A Λ t k Λ u A Σ t k Σ u + V , (5.25) e U (cid:0) U (cid:48) + W (cid:48) + W (cid:48)(cid:48) (cid:1) − (8 π ) e U − W ) ˜ V BH + e U G i ¯ Z i (cid:48) ¯ Z ¯ (cid:48) + e U H uv q u (cid:48) q v (cid:48) − g e − U A Λ t k Λ u A Σ t k Σ u + V , (5.26) e U (cid:0) − κe − W + 2 W (cid:48) + W (cid:48)(cid:48) (cid:1) − g e − U A Λ t k Λ u A Σ t k Σ u + V = 0 , (5.27) e U (cid:0) Z i (cid:48)(cid:48) + 2 W (cid:48) Z i (cid:48) + G i ¯ ∂ l G k ¯ Z l (cid:48) Z k (cid:48) (cid:1) − (8 π ) e U − W ) ∂ i ˜ V BH − ∂ i V = 0 , (5.28) e U ( q u (cid:48)(cid:48) + 2 W (cid:48) q u (cid:48) + Γ uvz q v (cid:48) q z (cid:48) ) − g e − U A Λ t k Λ v A Σ t ∇ v k Σ u − ∂ u V = 0 , (5.29)where a prime denotes a derivative with respect to r .In the near horizon limit ( r →
0) one has U ∼ log rr AdS , W ∼ log (cid:18) r H r AdS r (cid:19) , (5.30)where r AdS is the AdS curvature radius. We require all the fields, their derivatives, thescalar potential and the couplings to be regular on the horizon. Then we can choose agauge such that A Λ t (cid:12)(cid:12) r =0 = 0 = ⇒ A Λ t r → ∼ F Λ rt (cid:12)(cid:12) r =0 r . (5.31)It is also reasonable to assume that the derivative of the electric charges ∂ r e Λ remainsfinite on the horizon. In this case, eq. (5.20) implies that in the near-horizon limit thequantity A Σ t k Σ u k Λ u is at least of order r . If we expand in powers of r , in the gauge(5.31) the order zero term automatically vanishes, while for the order one term we have0 = ∂ r (cid:0) A Σ t k Σ u k Λ u (cid:1)(cid:12)(cid:12) r =0 = − F Σ tr k Σ u k Λ u (cid:12)(cid:12) r =0 = ⇒ F Λ tr k Λ u (cid:12)(cid:12) r =0 = 0 . (5.32)Using (5.31) and (5.32) one can see that the terms with A Λ t in the equations of motion, e − U A Λ t k Λ u A Σ t k Σ u and e − U A Λ t k Λ v A Σ t ∇ v k Σ u , go to zero in the near-horizon limit. In– 15 –his limit the equations of motion (5.25)–(5.29) thus reduce to1 r = (8 π ) V BH r H − V , (5.33) κr H = 1 r + V , (5.34) ∂ i (cid:20) (8 π ) V BH r H + V (cid:21) = 0 , (5.35) ∂ u V = 0 , (5.36)where V BH ≡ ˜ V BH | e Λ ( r ) → e Λ (0) . Solving the first two equations for r H and r one gets r H = κ ± (cid:112) κ − π ) V BH VV (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r =0 , (5.37) r = ∓ r H (cid:112) κ − π ) V BH V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r =0 , (5.38)and since of course r > r H >
0, which means that flat or hyperbolic geometries, κ = 0 , −
1, are only possible ifthe scalar potential takes negative values on the horizon, V | r =0 <
0. Spherical geometry( κ = 1), on the other hand, is compatible with both positive or negative values of V on the horizon, but for V | r =0 > V BH V | r =0 < π ) , since V BH is always positive.We can introduce an effective potential as a function of the scalars, V eff ( Z, ¯ Z, q ) ≡ κ − (cid:112) κ − π ) V BH VV , (5.39)defined for V BH V < π ) , and write r H = V eff | Z H ,q H , (5.40) r = V eff (cid:112) κ − π ) V BH V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z H ,q H , (5.41)– 16 –ith Z iH ≡ lim r → Z i , q uH ≡ lim r → q u . Because of equations (5.35)–(5.36), V eff isextremized on the horizon by all the scalar fields of the theory, ∂ i V eff | Z H ,q H = 0 , ∂ u V eff | Z H ,q H = 0 . (5.42)The values Z iH , q uH of the scalars on the horizon are then determined by the ex-tremization conditions (5.42), and the Bekenstein–Hawking entropy density is given bythe critical value of V eff , s = S V = A V = r H V eff ( Z H , ¯ Z H , q H )4 . (5.43)For a given theory this critical value, and thus also the entropy, depend only on thecharges (on the horizon) p Λ and e Λ (0), so that the attractor mechanism still works. Onthe other hand Z iH and q uH may not be uniquely determined, since in general V eff mayhave flat directions.The limit for V → V eff only exists for κ = 1, in which case V eff → (8 π ) V BH and one recovers the attractor mechanism for ungauged supergravity. The fact thatthis limit does not exist for κ = 0 , − V eff . In particular one has on the horizon ∂ q V = ∂ Z V = ∂ Z V BH = 0 . (5.44) In this paper, we considered N = 2 supergravity in four dimensions, coupled to vector-and hypermultiplets, where abelian isometries of the quaternionic K¨ahler manifold aregauged. In the first part, we analytically constructed a magnetically charged super-symmetric black hole solution of this theory for the case of just one vector multipletwith prepotential F = − iχ χ , and the universal hypermultiplet. This black hole has arunning dilaton, and interpolates between AdS × H at the horizon and a hyperscaling-violating type geometry at infinity, which is conformal to AdS × H . To the best ofour knowledge, this represents the first example of an analytic genuine BPS black holein gauged supergravity with nontrivial hyperscalars; previously known solutions of thistype were only constructed numerically [6].– 17 –iverging scalars fields of the form (4.31) are common in two and three dimen-sions, but are sometimes regarded as a sign of pathology in four or higher dimensions.However, similar to the linear dilaton black holes of [26], our solutions have finite en-tropy, magnetic charges and curvature at large r , in spite of the diverging scalars, andshould thus be regarded as physically meaningful . In any case, it may be interesting toconsider more general models and gaugings, and to look for asymptotically AdS blackholes with running hyperscalars, that might be more relevant for gauge/gravity dualityapplications. Unfortunately the equations of [5] become immediately quite involvedonce the complexity of the model increases, but perhaps our solution may serve as astarting point that helps solving analytically the equations of [5] in a more complicatedsetting. We hope to come back to this point in a future publication.In the second part of the paper, we extended the work of [18] on black hole attrac-tors in gauged supergravity to the case where also hypermultiplets are present. Theattractors were shown to be governed by an effective potential V eff , which is extremizedon the horizon by all the scalar fields of the theory. Moreover, the entropy is given bythe critical value of V eff , and in the limit of vanishing scalar potential, V eff reduces (up toa prefactor) to the usual black hole potential. The resulting attractor equations (5.42)do not make use of supersymmetry; they are valid for any static extremal black hole.It would be interesting to analyze them for some specific models, for instance the onesworked out in [27] and considered also in [6] that arise from M-theory compactifications. R , R µν R µν and R µνρσ R µνρσ decay for large r like r − , r − and r − respectively. – 18 – eferences [1] S. A. Hartnoll, “Lectures on holographic methods for condensed matter physics,” Class.Quant. Grav. (2009) 224002 [arXiv:0903.3246 [hep-th]].[2] C. Charmousis, B. Gouteraux, B. S. Kim, E. Kiritsis and R. Meyer, “Effectiveholographic theories for low-temperature condensed matter systems,” JHEP (2010) 151 [arXiv:1005.4690 [hep-th]].[3] N. Iizuka, N. Kundu, P. Narayan and S. P. Trivedi, “Holographic Fermi and non-Fermiliquids with transitions in dilaton gravity,” JHEP (2012) 094 [arXiv:1105.1162[hep-th]].[4] S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, “Building a holographicsuperconductor,” Phys. Rev. Lett. (2008) 031601 [arXiv:0803.3295 [hep-th]].[5] P. Meessen and T. Ort´ın, “Supersymmetric solutions to gauged N = 2 d = 4 sugra: thefull timelike shebang,” Nucl. Phys. B (2012) 65 [arXiv:1204.0493 [hep-th]].[6] N. Halmagyi, M. Petrini and A. Zaffaroni, “BPS black holes in AdS from M-theory,”JHEP (2013) 124 [arXiv:1305.0730 [hep-th]].[7] K. Hristov, H. Looyestijn and S. Vandoren, “BPS black holes in N = 2, d = 4 gaugedsupergravities,” JHEP (2010) 103 [arXiv:1005.3650 [hep-th]].[8] M. Gutperle and W. A. Sabra, “A supersymmetric solution in N = 2 gaugedsupergravity with the universal hypermultiplet,” Phys. Lett. B (2001) 311[hep-th/0104044].[9] H. Erbin and N. Halmagyi, “Abelian hypermultiplet gaugings and BPS vacua in N = 2supergravity,” arXiv:1409.6310 [hep-th].[10] N. Halmagyi, M. Petrini and A. Zaffaroni, “Non-relativistic solutions of N = 2 gaugedsupergravity,” JHEP (2011) 041 [arXiv:1102.5740 [hep-th]].[11] D. Cassani and A. F. Faedo, “Constructing Lifshitz solutions from AdS,” JHEP (2011) 013 [arXiv:1102.5344 [hep-th]].[12] S. Ferrara, R. Kallosh and A. Strominger, “ N = 2 extremal black holes,” Phys. Rev. D (1995) 5412 [hep-th/9508072].[13] A. Strominger, “Macroscopic entropy of N = 2 extremal black holes,” Phys. Lett. B (1996) 39 [hep-th/9602111].[14] S. Ferrara and R. Kallosh, “Supersymmetry and attractors,” Phys. Rev. D (1996)1514 [hep-th/9602136].[15] S. Ferrara and R. Kallosh, “Universality of supersymmetric attractors,” Phys. Rev. D (1996) 1525 [hep-th/9603090]. – 19 –
16] S. Ferrara, G. W. Gibbons and R. Kallosh, “Black holes and critical points in modulispace,” Nucl. Phys. B (1997) 75 [hep-th/9702103].[17] J. F. Morales and H. Samtleben, “Entropy function and attractors for AdS blackholes,” JHEP (2006) 074 [hep-th/0608044].[18] S. Bellucci, S. Ferrara, A. Marrani and A. Yeranyan, “ d = 4 black hole attractors in N = 2 supergravity with Fayet-Iliopoulos terms,” Phys. Rev. D (2008) 085027[arXiv:0802.0141 [hep-th]].[19] M. H¨ubscher, P. Meessen, T. Ort´ın and S. Vaul`a, “Supersymmetric N = 2Einstein-Yang-Mills monopoles and covariant attractors,” Phys. Rev. D (2008)065031 [arXiv:0712.1530 [hep-th]].[20] S. L. Cacciatori and D. Klemm, “Supersymmetric AdS black holes and attractors,”JHEP (2010) 085 [arXiv:0911.4926 [hep-th]].[21] G. Dall’Agata and A. Gnecchi, “Flow equations and attractors for black holes in N = 2U(1) gauged supergravity,” JHEP (2011) 037 [arXiv:1012.3756 [hep-th]].[22] S. Kachru, R. Kallosh and M. Shmakova, “Generalized attractor points in gaugedsupergravity,” Phys. Rev. D (2011) 046003 [arXiv:1104.2884 [hep-th]].[23] J. Bellorin and T. Ort´ın, “All the supersymmetric configurations of N = 4, d = 4supergravity,” Nucl. Phys. B (2005) 171 [hep-th/0506056].[24] S. Cecotti, S. Ferrara and L. Girardello, “Geometry of type II superstrings and themoduli of superconformal field theories,” Int. J. Mod. Phys. A (1989) 2475.[25] L. Huijse, S. Sachdev and B. Swingle, “Hidden Fermi surfaces in compressible states ofgauge-gravity duality,” Phys. Rev. B (2012) 035121 [arXiv:1112.0573[cond-mat.str-el]].[26] K. C. K. Chan, J. H. Horne and R. B. Mann, “Charged dilaton black holes withunusual asymptotics,” Nucl. Phys. B (1995) 441 [gr-qc/9502042].[27] D. Cassani, P. Koerber and O. Varela, “All homogeneous N = 2 M-theory truncationswith supersymmetric AdS vacua,” JHEP (2012) 173 [arXiv:1208.1262 [hep-th]].(2012) 173 [arXiv:1208.1262 [hep-th]].