aa r X i v : . [ h e p - t h ] M a r Preprint typeset in JHEP style - HYPER VERSION
Near the horizon of D black rings Farhang Loran, Hesam Soltanpanahi
Department of Physics, Isfahan University of Technology,Isfahan 84156-83111, IranE-mail: [email protected], h [email protected]
Abstract:
For the five dimensional N = 2 black rings, we study the supersymmetryenhancement and identify the global supergroup of the near horizon geometry. Weshow that the global part of the supergroup is OSp (4 ∗ | × U (1) which is similarto the small black string. We show that results obtained by applying the entropyfunction formalism, the c-extremization approach and the Brown-Henneaux methodto the black ring solution are in agreement with the microscopic entropy calculation. Keywords:
Black Holes in String Theory, Supergravity Models . ontents
1. Introduction and Summary 12. N = 2 5 D black rings 4
3. Enhancement of supersymmetry 9
4. Near horizon physics 14
A. Killing vectors of AdS × S geometry 18
1. Introduction and Summary
Five dimensional supergravity is interesting from several points of view. Such a su-pergravity can be constructed by compactifying the eleven dimensional supergravity,on some six dimensional manifolds e.g. CY or T . There are several supersymmetricsolutions for the five dimensional supergravity that preserve either one half or all ofthe supersymmetry [1]. These solutions contain three kinds of black objects whichare half-BPS known as black holes, black strings and black rings which their nearhorizon geometries are AdS × S , AdS × S and AdS × S × S respectively. Eachof these solutions has a specific charge configuration. A black hole has only elec-tric charges, a black string has only magnetic charges, while a black ring has bothelectric and magnetic charges. A nice review on these black solutions of N = 2 fivedimensional supergravity is [2].Black ring is the first example of a black object with a non-spherical horizontopology and asymptotically flat geometry which carries angular momentum alongthe S direction [3]. Furthermore, the existence of this solution implies that theblack hole uniqueness theorems can not be extended to five dimensions, except in the– 1 –tatic case [4]. The generalization of the uniqueness of black holes to five dimensionsis studied in [5], where it is shown that the dipole charge appears in the first law ofthermodynamics in the same manner as a global charge. Therefore there exist blackobjects with the same global charges but with different horizon topologies. Someother developments are listed in [6]-[11]. For a good review on black ring see [12].In this paper we study some features of the supersymmetric large black rings inthe five dimensional N = 2 supergravity which have the non-zero classical horizonarea. Large black rings are half-BPS and in the near horizon limit they exhibitsupersymmetry enhancement [13]. We want to investigate the symmetry of the nearhorizon geometry of the supersymmetric black rings. For this purpose, we note that N = 2 n supergravity in five dimensions with 8 n real supercharges has an Sp (2 n ) R-symmetry group with the supersymmetry parameter ε i , i = 1 , · · · , n , transformingas representation. Using this fact, one can solve for the supersymmetry spinorand calculate the global part of the superalgebra. Doing so, we show that the globalpart of the superalgebra is OSp (4 ∗ | × U (1), which is similar to the small blackstring obtained in [14].The most important reason for investigating the supergroup of the near horizongeometry of the black objects is the AdS/CFT correspondence. AdS /CFT corre-spondence is not well-defined yet in contrast to the higher dimensional cases (see forexample [15]). Motivated by this phenomenon, the symmetry of the near horizongeometry of the small black hole solutions of N = 2 , et al in [14], study the symmetry of the near horizon geom-etry of small black string solutions to investigate AdS /CFT correspondence, whichin principle gives some information about AdS /CFT via dimensional reduction.Some recent results on the AdS /CFT correspondence and the small black stringscan be found in [17]. It seems that for studying AdS /CFT from AdS /CFT , theblack ring is a better starting point than the black string since the fibration of S over AdS is explicit. Thus our study of the near horizon physics of the black ringmight shed a new light on this subject.An important feature of supersymmetric black objects is the attractor mecha-nism. The attractor mechanism determines the value of the scalar fields near thehorizon independent of their asymptotic values, and also implies the enhancementof supersymmetry near the horizon [19]. Attractor mechanism as reformulated bySen, which is called the “entropy function formalism”, can be used to calculate theentropy of black holes with AdS × X near horizon geometry in diverse dimensions[20]. In [21, 22, 23] the entropy function formalism is applied to black rings. As men-tioned above, the near horizon geometry of the black ring solution is AdS × S × S ,where S is fibred nontrivially over AdS . This phenomenon as well as the Chern-Simons term in the five dimensional supergravity frustrates a direct application of The same property of black ring gives an opportunity to study the relation between 4 D blackholes and 5 D black rings [18]. – 2 –he entropy function formalism in this case. In fact there are two problems in ap-plying the entropy function method to black rings. First, in the Wald formula [25]there is a derivative of the Lagrangian density with respect to the Riemann tensorcomponents which for AdS × X near horizon geometry has only one independentcomponent. But in the case of the black ring near horizon geometry, the Riemanntensor has four independent components since S is fibred non-trivially over AdS . Second, the Chern-Simons term in the Lagrangian density is not gauge invariant, while in the entropy function formalism the gauge invariance of the Lagrangian den-sity is assumed. We study the entropy function formalism for the black ring andexplain how both of these problems can be resolved by dimensional reduction alongthe S . By such a dimensional reduction, the near horizon geometry reduces toAdS × S , which has only one relevant independent Riemann tensor component,and the Chern-Simons term becomes a sum of gauge invariant terms.In [28], Kraus and Larsen introduced the c-extremization approach for obtainingthe spacetime central charge of black objects with AdS × Y near horizon geometryin a simple way. Although, the c-extremization is introduced for black objects witha globally AdS component of the near horizon geometry, we show that by applyingthis method to the black ring which horizon geometry locally looks like AdS × S ,one obtains results which are in agreement with the outcome of the entropy functionformalism and microscopic calculations of the black ring entropy [29, 30, 11, 12].We recalculate the microscopic entropy by the Kerr/CFT correspondence [31],which is intrinsically a generalization of Brown-Henneaux approach [32] to AdS/CFTcorrespondence [33]. Choosing an appropriate boundary condition we show that theasymptotic symmetry group of the near horizon of supersymmetric black ring con-tains a Virasoro algebra. The corresponding central charge equals the c-extremizationresult. By defining the Frolov-Thorne temperature [35] and using the Cardy formulawe calculate the CFT entropy and show that it equals the Bekenstein-Hawking en-tropy.The main results of this work are that in five dimensional N = 2 supergravity theglobal part of the near horizon supergroup of the large black ring is OSp (4 ∗ | × U (1).At the leading order, the entropy function, c-extremization and Brown-Henneaux ap-proaches are in agreement with each other and with the microscopic results obtainedin [29, 30, 11, 12]. The difficulty in incorporating the Chern-Simons term into the entropy function formalism isstudied in [24]. Interestingly, for the D1-D5-P black holes, where a similar problem is encountered, a general-ization of the entropy function formalism is found in [26], which can not be applied in the blackring problem. One encounters a Similar problem in the case of four dimensional spinning black holes. Applyingentropy function formalism for spinning black holes is studied in [27] The Chern-Simons action is gauge invariant up to a boundary term. – 3 –he paper is organized as follows. In section 2 we review black ring solution offive dimensional N = 2 supergravity and its near horizon geometry. In section 3 weshow the supersymmetry enhancement near the horizon of black ring and determinethe global part of the superalgebra. In section 4 we apply the entropy function, c-extremization and Brown-Henneaux formalisms for large black rings where we showthat the macroscopic and microscopic entropies of black rings are equal to each other.In appendix A Killing vectors of AdS × S component of the black ring near horizongeometry, used in section 3 are given. N = 2 5 D black rings In this section we briefly review the N = 2 5 D black ring solution in superconfor-mal formalism. In this approach the symmetry group of supergravity is enlargedto superconformal group which can be reduced to the initial model by imposing asuitable gauge fixing condition. The supersymmetry variations of field content areindependent of the Lagrangian and one can consequently apply these variations atany level of corrections. The field content of superconformal gravity are arranged in Weyl, vector and hy-permultiplets. The bosonic fields of Weyl multiplet are the vielbein e aµ , an auxiliary2-form field v ab and an auxiliary scalar field D . The bosonic part of each vectormultiplet contains a 1-form gauge field A I and a scalar field X I , where I = 1 , · · · , n v labels the gauge group. The hypermultiplet contains scalar fields A iα , where i = 1 , SU (2) doublet index and α = 1 , · · · , n refers to USp(2n) group.In the off-shell formalism the bosonic part of the action of N = 2 supergravityin five dimensions at the leading order is [36] I = 116 πG Z d x p | g |L , (2.1)in which L = ∂ a A iα ∂ a A αi + (2 ν + A ) D ν − A ) R ν − A ) v ν I F Iab v ab + 14 ν IJ ( F Iab F J ab + 2 ∂ a X I ∂ a X J ) + e − C IJK ǫ abcde A Ia F Jbc F Kde . (2.2) A = A iα A αi , v = v ab v ab and ν = 16 C IJK X I X J X K , ν I = 12 C IJK X J X K , ν IJ = C IJK X K , (2.3)where C IJK are the intersection numbers of the internal space. The fermion fieldsare the gravitino ψ iµ and the auxiliary Majorana spinor χ i which are in the Weyl– 4 –ultiplet, the gaugino Ω Ii in the vector multiplet and hyperino ζ α in the hypermul-tiplet.As we are interested in supersymmetric bosonic solutions, in which fermion fieldsare set to zero and the solution is invariant under supersymmetry variations, we con-centrate on the study the bosonic terms of the supersymmetry variations of fermionswhich are given as follows δψ iµ = D µ ε i + v ab γ µab ε i − γ µ η i ,δχ i = Dε i − γ c γ ab ˆ D a v bc ε i + γ ab ˆ R ab ( V ) ij ε i − γ a ε i ǫ abcde v bc v de + 4 γ ab v ab η i ,δ Ω Ii = − γ ab F Iab ε i − γ a ∂ a X I ε i − X I η i ,δζ α = γ a ∂ a A αi − γ ab v ab ε i A αi + 3 A αi η i , (2.4)where δ ≡ ¯ ǫ i Q i + ¯ η i S i + ξ aK K a and the covariant derivatives are defined by D µ ε i = (cid:18) ∂ µ + 14 ω abµ + 12 b µ (cid:19) − V iµ j ε j , (2.5)ˆ D µ v ab = ( D µ − b µ ) v ab = ∂ µ v ab + 2 ω c [ a v b ] c − b µ v ab , (2.6)in which b µ is a real boson in the Weyl multiplet and is SU (2) singlet [36].There is a well-known gauge to fix the conformal invariance of the off-shell for-malism and reduce the superconformal symmetry to the standard symmetries of fivedimensional N = 2 supergravity, A = − , b µ = 0 , V ijµ = 0 . (2.7)In this gauge the last equation of (2.4) gives η i in terms of ε i as, η i = 13 γ ab v ab ε i . (2.8)In the gauge (2.7) and also after solving the equation of motion of the auxiliary fields D and v ab , the Lagrangian density (2.2) reduces to the standard form of the bosonicpart of N = 2 supergravity in five dimensions, L = R − G IJ F Iab F Jab − G IJ ∂ a X I ∂ a X J + e − C IJK A Ia F Jbc F Kde ǫ abcde , (2.9) Here γ a a ··· a m = m ! γ [ a γ a · · · γ a m ] which is antisymmetric in all indices. Also the covariantcurvature ˆ R ijµν is defined by ˆ R ijµν = 2 ∂ [ µ V ijν ] − V i [ µ k V kjν ] + fermionic terms , where V ijµ is a boson inthe Weyl multiplet which is in of the SU (2). For the solution we are going to consider, this termvanishes. Q i is the generator of N = 2 supersymmetry, S i is the generator of conformal supersymmetryand K a are special conformal boost generators of superconformal algebra [36]. – 5 –here G IJ = − ∂ I ∂ J (ln ν ) = 12 ( ν I ν J − ν IJ ) , (2.10)and the supersymmetry variations (2.4) simplify as δψ iµ = (cid:0) D µ + v ab γ µab − γ µ γ ab v ab (cid:1) ε i ,δχ i = (cid:0) D − γ c γ ab D a v bc − γ a ǫ abcde v bc v de + ( γ ab v ab ) (cid:1) ε i ,δ Ω Ii = (cid:0) − γ ab F Iab − γ a ∂ a X I − X I γ ab v ab (cid:1) ε i , (2.11)where we have used (2.8). In § The five dimensional N = 2 supergravity have several half-BPS black hole, blackstring and black ring solutions. Here we review the large black ring solutions following[7]. These solutions have both electric Q I and magnetic p I charges. From elevendimensional supergravity point of view, these charges correspond to the M M U (1) solution which is the most symmetric solution. The M-theoryconfiguration corresponding to this solution consists of three M2-branes and threeM5-branes oriented as [7] Q M − − − − − Q M − − − − − Q M − − − − − p M − − ψp M − − ψp M − − ψ (2.12)where directions z i , i = 1 · · ·
6, span the internal 6-torus and ψ is the ring directionof black ring.The 11 D supergravity solution takes the form ds = ds + X ( dz + dz ) + X ( dz + dz ) + X ( dz + dz ) , A = A ∧ dz ∧ dz + A ∧ dz ∧ dz + A ∧ dz ∧ dz , (2.13)where A is the three-form potential with four-form field strength F = d A .The five dimensional solution is specified by a metric ds , three scalars X I , andthree one-forms A I , with field strengths F I = dA I . In ring coordinates the solution All the fields are independent from internal space and exterior derivative d on A I is defined infive dimensional space. – 6 –s written as follows ds = ( H H H ) − / ( dt + ω ) − ( H H H ) / d x ,d x = R ( x − y ) h ( y − dψ + dy y − + dx − x + (1 − x ) dφ i ,A I = H − I ( dt + ω ) + p I [(1 + y ) dψ + (1 + x ) dφ ] ,X I = H − I ( H H H ) / . (2.14)In these coordinates, y = −∞ corresponds to the location of the ring, and Q I and p I are the electric and magnetic charges respectively. The harmonic functions H I aredefined by H = 1 + Q − p p R ( x − y ) − p p R ( x − y ) , (2.15)and the same for H and H with cyclic permutation. For simplicity we choose Q = Q = Q = Q, p = p = p = p. (2.16)The one-form ω which is related to the angular momentum of the solution is ω = ω ψ dψ + ω ϕ dϕ with ω ψ = p R ( y − Q − p (3 + x + y )] − p (1 + y ) ,ω ϕ = p R (1 − x )[3 Q − p (3 + x + y )] . (2.17)The ADM charges of this solution are given by M = π G Q,J ψ = π G p [6 R + 3 Q − p ] , J ϕ = π G p (3 Q − p ) . (2.18)The coordinate ranges are −∞ ≤ y ≤ , − ≤ x ≤ , ≤ ψ ≤ π, ≤ ϕ ≤ π. (2.19)To make the above solution free of closed causal curves for y ≥ −∞ , one requiresthat, 2 p L ≡ X i 1; 1) = Osp (4 ∗ | Osp (4 ∗ | × U (1).It is interesting to note that this supergroup is the same as the small blackstring near horizon supergroup [14]. Of course, in [14] the superalgebra of smallblack string in N = 4 five dimensional supergravity is calculated by embedding thesolution of N = 2 supergravity. For this solution, the supergroup of near horizon is The symmetries of the near horizon geometry of the extremal black ring and four dimensionalspinning black holes are studied for example in [40] and [41] respectively. – 13 – sp (4 ∗ | × U (1), where the Osp (4 ∗ | 4) part of superalgebra is { G Ir , G Js } = − δ IJ L r + s + ( r − s )( t α ) IJ J α + ( r − s )( ρ A ) IJ R A , [ L m , L n ] = ( m − n ) L m + n , [ L m , G Ir ] = (cid:16) m − r (cid:17) G Im + r , [ J α , G Ir ] = ( t α ) IJ G Jr , [ R A , G Ir ] = ( ρ A ) IJ G Jr , (3.31)in which t α and ρ A are the representation matrices for SU (2) and Sp (4) respectivelyand R A are the generators of Sp (4). In [16], it was shown that this global part ofsuperalgebra in N = 4 supergravity is reduced to Osp (4 ∗ | 2) in N = 2. This resultshows that in AdS/CFT analysis black ring solution behaves like a small black string.It is straightforward to repeat the above calculations when higher order correc-tions are considered, as higher order corrections only modify p and L in the metric(2.24) [43]-[47]. Therefore, after adding e.g. the supersymmetric correction [36],the supersymmetry is still enhanced near the horizon and the superalgebra does notchange. 4. Near horizon physics A special feature of the supersymmetric black ring is the geometry of the near horizonof this solution such that an AdS × S is locally AdS (2.23)-(2.24). This specialnear horizon topology allows one to apply both the entropy function [20] and the c-extremization [28] formalisms on black ring. We also use Brown-Henneaux approach[32] to calculate the CFT entropy of extremal black ring. In this section we briefly review the entropy function formalism applied to the blackring solution to calculate the corresponding macroscopic entropy [21], [22], [23].In the entropy function formalism the entropy can be found from the extremumof the entropy function, S = 2 π ( e I q I − f ) , (4.1)in which, f = Z dx H p | g H |L , (4.2) It is interesting to note that Osp (4 ∗ | 2) factor is also present in the small black string [14],the small black hole [16] and the large black ring solutions (3.29) of N = 2 supergravity in fivedimensions. – 14 –nd q I is defined by q I = ∂f∂e I . To apply the entropy function for the near horizongeometry of the black ring one uses the ansatz [21] ds = v ( − r dt + dr r ) + v ( dθ + sin θdφ ) + w ( dψ + e rdt ) ,F I rt = e I + a I e , F I θφ = p I sin θ, X I = M I , I = 1 , , , (4.3)where e is conjugate to the angular momentum of the ring. Extremizing the entropyfunction (4.1) with respect to the v , v , w, M I and N I gives, v = v = p , w = p e , e I + e N I = 0 , M I = p I ( C IJK p I p J p K ) / . (4.4)Using (4.4) and (4.1) one obtains, S mac = 2 πp L. (4.5)The same result is obtained in the off-shell formalism in [22]. In [28] Kraus and Larsen showed that for D -dimensional black objects with AdS × S D − near horizon geometry one can define the c-function as c ( l A , l S ) = 3Ω Ω D − πG D l A l D − S L , (4.6)which extremization with respect to the radii of AdS and S , gives the average ofthe left and right central charges of CFT dual of AdS . As we have discussed in § component(2.23). Thus one can expect that c-extremization formalism [28] can also be appliedfor black ring solution.We consider the following ansatz, ds = l A d Ω AdS + l S d Ω S , X I = mp I ,F Irt = e I + e a I , F Iθφ = p I sin θ, v rt = v v θφ = v sin θ. (4.7)After solving the equations of motion of D, v ab , m and a I one finds D = 12 p , m = p − , v = − p, v = 0 , e I + e a I = 0 , (4.8) In this section following the usual conventions in both the entropy function and the c-extremization formalisms we use ( − + + + +) signature and choose G = π/ It was discussed in [28] that only the bulk part of the action contributes in this definition. These results can also be derived from the supersymmetry variations of fermions (2.11) [22]. – 15 –here p ≡ ( C IJK p I p J p K ) / . By extremizing the c-function one obtains, l A = 2 l S = p, c = 6 p . (4.9)In the semiclassical regime c ≫ c ∼ c L ∼ c R , in which c L ( R ) is the left (right)central charge of the CFT. In this limit, ( c L − c R ) is negligible as it is given by higherorder corrections, and c = ( c L + c R ) is given by the c-extremization method (4.9).Since the black ring solution in N = 2 five dimensional supergravity corresponds tothe (0 , S mic = 2 π r c ˆ q , (4.10)where [12] ˆ q = p p p Q Q p + Q Q p + Q Q p ) (4.11)+ 14 p p p [( p Q ) + ( p Q ) + ( p Q ) ] − J ψ = pL . (4.12)Thus, at the leading order, the result of the c-extremization formalism (4.9) andmicroscopic description of the entropy of the black ring are in agreement with theentropy calculated by the macroscopic entropy function formalism (4.5), S mac = S mic = 2 πLp . (4.13) In this section we recalculate the microscopic entropy of supersymmetric black ringsfrom another viewpoint by using the Kerr/CFT formalism [31]. In this method, whichis intrinsically a generalization of the Brown-Henneaux approach [32], the Virasorogenerators of the CFT dual are related to the asymptotic symmetry group (ASG) ofthe near horizon metric. The asymptotic symmetry group (ASG) of a near horizonmetric is the group of allowed symmetries modulo trivial symmetries. By definition,an allowed symmetry transformation obeys the specified boundary conditions [31].A possible boundary condition for the fluctuations around the geometry (2.24) is, h µν ∼ O r /r /r r /r /r /r /r /r /r /r /r , (4.14) In U (1) supergravity which is the subject of our study in this paper, all p I are equal to eachother and denote them by p . We are using the conventions of [29] for left and right moving central charges. – 16 –n the basis ( t, r, θ, φ, ψ ). It is easy to show that the general diffeomorphism preserv-ing the boundary conditions (4.14) is given by, ζ = (cid:20) C + O ( 1 r ) (cid:21) ∂ t + [ rǫ ′ ( ψ ) + O (1)] ∂ r + O ( 1 r ) ∂ θ + O ( 1 r ) ∂ φ + (cid:20) ǫ ( ψ ) + O ( 1 r ) (cid:21) ∂ ψ , (4.15)where C is an arbitrary constant and ǫ ( ψ ) is the arbitrary smooth periodic functionsof ψ . By using the basis ǫ n ( ψ ) = − e − inψ for the function ǫ ( ψ ), it is easy to showthat the ASG generates contains a Virasoro algebra generated by ζ n = − e − inψ ∂ ψ − in r e − inψ ∂ r , (4.16)which satisfy [ ζ m , ζ n ] = − i ( m − n ) ζ n + n .The generator of a diffeomorphism has a conserved charge. The charges associ-ated to the diffeomorphisms (4.16) are defined by [34], Q ζ = 18 π Z ∂ Σ k ζ [ h, g ] , (4.17)where ∂ Σ is spatial surface at infinity and k ζ [ h, g ] = 12 [ ζ ν ∇ µ h − ζ ν ∇ σ h σµ + ζ σ ∇ ν h σµ + h ∇ ν ζ µ − h σν ∇ σ ζ µ + 12 h νσ ( ∇ µ ζ σ + ∇ σ ζ µ )] ∗ ( dx µ ∧ dx ν ) , (4.18)in which ∗ denotes the Hodge dual in 5D. In the Brown-Henneaux approach [32] thecentral charge is given by18 π Z ∂ Σ k ζ m [ L ζ n , g ] = − i c ( m − m ) δ m + n, . (4.19)Plugging the metric (2.24) and diffeomorphisms (4.16) in (4.19) one obtains, c = 6 p , (4.20)which is in agreement with the c-extremization result (4.9).The Frolov-Thorne temperature can be determined by identifying quantum num-bers of a matter field in the near horizon geometry with those in original geometry.For the chiral CFT given by (4.16) a matter field can be expanded in eigen modesof the asymptotic energy ω and angular momentum m asΦ = X ω,m,l ϕ ωml e − i ( ωt − mψ ) f l ( r, θ, φ ) , (4.21)– 17 –imilar to [31] one can show that, here, the Frolov-Thorne temperature is T FT = 12 πe . (4.22)The Cardy formula gives the microscopic entropy of chiral CFT (4.16) as follows, S CFT = π c T FT = 2 πLp , (4.23)which is in precise agreement with the result obtained by utilizing the entropy func-tion and the c-extremization methods (4.13). Acknowledgement . We would like to thank M. Alishahiha, H. Ebrahim andR. Fareghbal for useful comments and discussions. A. Killing vectors of AdS × S geometry In this appendix we derive the Killing vectors of AdS × S geometry which is ap-peared in the near horizon of black ring solution of N = 2 five dimensional super-gravity. The metric of this part is (2.23), ds = − p ( dr r + L p dψ + Lrp dψdt ) , (A.1)and Killing equation is X ρ ∂ ρ g µν + ∂ µ X ρ g ρν + ∂ ν X ρ g µρ = 0 . (A.2)The components of Killing equation are ∂ t X ψ = 0 , (A.3) ∂ t X r + 2 Lr p ∂ r X ψ = 0 , (A.4) X r + r∂ t X t + r∂ ψ X ψ = 0 , (A.5) X r − r∂ r X r = 0 , (A.6) ∂ ψ X r + 2 Lr p ∂ r X t + 4 L r p ∂ r X ψ = 0 , (A.7) ∂ ψ X t + 2 Lpr ∂ ψ X ψ = 0 . (A.8)Equations (A.3) and (A.6) show that, X ψ = f ( r, ψ ) , X r = rg ( t, ψ ) . (A.9)– 18 –o we can simplify (A.3)-(A.8) to obtain, ∂ t g ( t, ψ ) + 2 Lr p ∂ r f ( r, ψ ) = 0 , (A.10) g ( t, ψ ) + ∂ t X t + ∂ ψ f ( r, ψ ) = 0 , (A.11) ∂ ψ g ( t, ψ ) + 2 Lr p ∂ r X t + 4 L rp ∂ r f ( r, ψ ) = 0 , (A.12) ∂ ψ X t + 2 Lpr ∂ ψ f ( r, ψ ) = 0 . (A.13)In (A.10) the first term is a function of t and ψ and the second term is a function of r and ψ . Therefore, each term only is a function of ψ , ∂ t g ( t, ψ ) = − Lr p ∂ r f ( r, ψ ) = h ( ψ ) , (A.14)and consequently, g ( t, ψ ) = h ( ψ ) t + g ( ψ ) , f ( r, ψ ) = p Lr h ( ψ ) + f ( ψ ) . (A.15)Now we can simplify (A.11)-(A.13) as, h ( ψ ) t + g ( ψ ) + ∂ t X t + p Lr ∂ ψ h ( ψ ) + ∂ ψ f ( ψ ) = 0 , (A.16) ∂ ψ h ( ψ ) t + ∂ ψ g ( ψ ) + 2 Lr p ∂ r X t − Lpr h ( ψ ) = 0 , (A.17) ∂ ψ X t + 1 r ∂ ψ h ( ψ ) + 2 Lpr ∂ ψ f ( ψ ) = 0 . (A.18)From (A.16) one finds X t = − (cid:18) h ( ψ ) t + g ( ψ ) t + p Lr ∂ ψ h ( ψ ) t + ∂ ψ f ( ψ ) t (cid:19) + I ( r, ψ ) . (A.19)(A.19) and (A.17) give2 ∂ ψ h ( ψ ) t + ∂ ψ g ( ψ ) + 2 Lr p ∂ r I ( r, ψ ) − Lpr h ( ψ ) = 0 . (A.20)This implies that, ∂ ψ h ( ψ ) = 0 ⇒ h ( ψ ) = c . (A.21)Thus (A.15) simplifies as g ( t, ψ ) = c t + g ( ψ ) , f ( r, ψ ) = p Lr c + f ( ψ ) , (A.22)– 19 –nd (A.16)-(A.19) become c t + g ( ψ ) + ∂ t X t + ∂ ψ f ( ψ ) = 0 , (A.23) ∂ ψ g ( ψ ) + 2 Lr p ∂ r X t − Lpr c = 0 , (A.24) ∂ ψ X t + 2 Lpr ∂ ψ f ( ψ ) = 0 , (A.25) X t = − (cid:18) c t + g ( ψ ) t + ∂ ψ f ( ψ ) t (cid:19) + I ( r, ψ ) . (A.26)By (A.26), Eqs.(A.24) and (A.25) simplify to ∂ ψ g ( ψ ) + 2 Lr p ∂ r I ( r, ψ ) − Lpr c = 0 , (A.27) − (cid:0) ∂ ψ g ( ψ ) + ∂ ψ f ( ψ ) (cid:1) t + ∂ ψ I ( r, ψ ) + 2 Lpr ∂ ψ f ( ψ ) = 0 . (A.28)From (A.27) one obtains I ( r, ψ ) = − r c + p Lr ∂ ψ g ( ψ ) + I ( ψ ) , (A.29)and so (A.28) becomes − t (cid:0) ∂ ψ g ( ψ ) + ∂ ψ f ( ψ ) (cid:1) + p Lr ∂ ψ g ( ψ ) + ∂ ψ I ( ψ ) + 2 Lpr ∂ ψ f ( ψ ) = 0 , (A.30)which implies that, ∂ ψ g ( ψ ) + ∂ ψ f ( ψ ) = 0 , (A.31) p L ∂ ψ g ( ψ ) + 2 Lp ∂ ψ f ( ψ ) = 0 , (A.32) ∂ ψ I ( ψ ) = 0 . (A.33)Thus, g ( ψ ) + ∂ ψ f ( ψ ) = c ′ , (A.34) ∂ ψ g ( ψ ) + 4 L p f ( ψ ) = c , (A.35) I ( ψ ) = c . (A.36)Now we can simplify our results. From (A.34) and (A.35) one obtains, − ∂ ψ f ( ψ ) + 4 L p f ( ψ ) = c ⇒ f ( ψ ) = c e Lp ψ + c e − Lp ψ + p L c , (A.37)Therefore, g ( ψ ) = c + 2 Lp ( c e − Lp ψ − c e Lp ψ ) . (A.38)– 20 –lternatively we can solve g ( ψ ) as, − ∂ ψ g ( ψ ) + 4 L p g ( ψ ) = 4 L p c ⇒ g ( ψ ) = c e Lp ψ + c e − Lp ψ + c , (A.39)which is consistent with the first solution. c .In summary, I ( r, ψ ) = − r c − Lpr (cid:16) c e Lp ψ + c e − Lp ψ (cid:17) + c , (A.40) g ( t, ψ ) = c t + c + 2 Lp ( c e − Lp ψ − c e Lp ψ ) , (A.41) f ( r, ψ ) = p Lr c + c e Lp ψ + c e − Lp ψ + p L c . (A.42)(A.43)So, X t = − (cid:18) c t + c t (cid:19) − r c − Lpr (cid:16) c e Lp ψ + c e − Lp ψ (cid:17) + c , (A.44) X r = r (cid:18) c t + c + 2 Lp ( c e − Lp ψ − c e Lp ψ ) (cid:19) , (A.45) X ψ = p Lr c + c e Lp ψ + c e − Lp ψ + p L c . (A.46)Thus the killing vector expand as follows, X = X t ∂ t + X r ∂ r + X ψ ∂ ψ = − c (cid:18) 12 ( t + r − ) ∂ t − rt∂ r − p Lr ∂ ψ (cid:19) − c ( t∂ t − r∂ r ) + c p L ∂ ψ + c ∂ t − c Lp e Lp ψ (cid:18) r ∂ t + r∂ r − p L ∂ ψ (cid:19) − c Lp e − Lp ψ (cid:18) r ∂ t − r∂ r − p L ∂ ψ (cid:19) , (A.47)and consequently, there are six isometries generated by, K = 12 ( t + r − ) ∂ t − rt∂ r − p Lr ∂ ψ , K = t∂ t − r∂ r , (A.48) K = ∂ ψ , K = ∂ t K = e Lp ψ (cid:18) r ∂ t + r∂ r − p L ∂ ψ (cid:19) , K = e − Lp ψ (cid:18) r ∂ t − r∂ r − p L ∂ ψ (cid:19) . – 21 – eferences [1] J. P. Gauntlett, J. B. Gutowski, C. M. Hull, S. Pakis and H. S. Reall, Allsupersymmetric solutions of minimal supergravity in five dimensions , Class. Quant.Grav. (2003) 4587 [hep-th/0209114v3].[2] A. Castro, J. L. Davis P. Kraus and F. Larsen, String Theory Effects onFive-Dimensional Black Hole Physics , Int. J. Mod. Phys. A , 2008[arXiv:0801.1863].[3] R. Emparana and H. S. 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