A Note on Exact Solutions and Attractor Mechanism for Non-BPS Black Holes
aa r X i v : . [ h e p - t h ] D ec CAS-KITPC/ITP-022
A Note on Exact Solutions andAttractor Mechanismfor Non-BPS Black Holes
Rong-Gen Cai §† , Da-Wei Pang †‡ § Kavli Institute for Theoretical Physics China(KITPC)at the Chinese Academy of Sciences † Institute of Theoretical Physics,Chinese Academy of SciencesP.O.Box 2735, Beijing 100080, China ‡ Graduate University of the Chinese Academy of SciencesYuQuan Road 19A, Beijing 100049, China [email protected], [email protected]
Abstract
We obtain two extremal, spherically symmetric, non-BPS black hole solutions to 4D su-pergravity, one of which carries D2-D6 charges and the other carries D0-D2-D4 charges.For the D2-D6 case, rather than solving the equations of motion directly, we assume theform of the solution and then find that the assumption satisfies the equations of motionand the constraint. Our D2-D6 solution is manifestly dual to the solution presented in0710.4967. The D0-D2-D4 solution is obtained by performing certain [ SL (2 , Z )] dualitytransformations on the D0-D4 solution in 0710.4967. ontents Black holes have provided a variety of interesting research subjects in recent years, oneof which is the so-called “attractor mechanism”. It means that, in certain black holebackground, the moduli fields vary radially and “get attracted” to fixed values at the blackhole horizon, which depend only on the quantized charges carried by the black hole. As aresult, the entropy of the black hole is given only in terms of the charges and is independentof the asymptotic values of the moduli.The attractor mechanism was firstly discovered in the mid 1990s in the context ofN=2 extremal black holes [1] and was generalized to higher derivative theories in [2].The supersymmetric attractors were the main focus at first, but later it was realized thatthe attractor mechanism does not rely on supersymmetry in [3]. Non-supersymmetricattractors have been investigated extensively in recent years, see [4–8]. For reviews, see [9].It has been found that the non-BPS attractors share many interesting properties withtheir BPS cousins, on condition that the non-BPS black holes are extremal. Although wecan study the non-BPS attractors via the attractor equations, it is more useful to obtainthe solution for the moduli fields in the whole space. However, it is rather complicated toobtain the exact solutions for non-BPS black holes because the equations of motion aresecond-order differential equations, rather than the first-order equations appearing in BPScases. Usually people deal with this difficulty by making use of perturbative methods andnumerical analysis, such as in [4] and related references.1or STU black holes the situations have been better improved. It has been shownin [10–13] that for the BPS case, we can find solutions to the equations of motion for themoduli, allowing us to obtain their values everywhere, by replacing the charges appeared inthe attractive values of the moduli with the corresponding harmonic functions. A similarprocedure has been carried out in [6] for non-BPS black holes carrying D2-D6 charges.However, the solutions were still limited in the sense that the moduli fields were taken tobe purely imaginary.Recently some interesting papers appeared [14] [15], which directly solved the equa-tions of motion for the moduli in the STU model carrying D0-D4 charges. The moduliand charges were set to be equal in [14] while more general cases were discussed in [15].Compared to the exact solutions in [6], their solutions for the moduli were complex.We generalize the exact solutions in [14] and [15] to STU model carrying D2-D6 charges.It is difficult to solve the equations of motion directly because unlike the D0-D4 case, thesuperpotential contains a cubic term of the moduli fields. Instead of dealing with the firstorder flow equations, we try to find the solution to the second order equation of motion.By observing the exact solution in [6], which was manifestly dual to the existed D0-D4system, we can assume that the D2-D6 solution is manifestly dual to the solution in [15]and check if it satisfies the equations of motion and the constraint. Fortunately, after atedious calculation we find that the assumption is correct.It has been known that the symplectic invariance of special geometry ensures that theLagrangian has an Sp (8 , Z ) symmetry, which reduces to [ SL (2 , Z )] at the level of the theequations of motion. Due to the SL (2 , Z ) duality, we can obtain new solutions by makinguse of the “seed solution”. We also obtain D0-D2-D4 solution from the D0-D4 solutionobtained in [15] by some simple SL (2 , Z ) duality transformations.The rest of the note is organized as follows: In Section 2 we give a brief review of theattractor mechanism in STU model as well as the main procedures and results in [15].Then we assume the form of the solution for STU model carrying D2-D6 charges withgeneral complex moduli and find that the solution satisfies both the equations of motionand the constraint. Next we obtain D0-D2-D4 solution by dualizing the D0-D4 solutionin [15] under [ SL (2 , Z )] symmetry. We summarize the results and discuss some relatedtopics in the final section. Type IIA string theory compactified on a CY manifold gives N = 2 supersymmetry. Themoduli fields belong to the vector multiplets and hypermultiplets of the resulting low-energy effective theory. The moduli in the vector multiplets are fixed according to theattractor mechanism while the moduli in the hypermultiplets play no role in the attractormechanism. The low-energy dynamics for the vector multiplets is completely determinedby a prepotential. If the Calabi-Yau manifold has h (1 ,
1) = N , we have N vector multiplets2nd N + 1 gauge fields, where the additional gauge field is the graviphoton coming fromthe gravity multiplet. The leading order prepotential, ignoring any α ′ corrections, is givenas F = D ABC X A X B X C X , (2.1)where A, B, C = 1 , · · · , N . The intersection number D abc are defined as6 D ABC = Z CY α A ∧ α B ∧ α C , (2.2)where α A denote the integer basis for H ( CY , Z ).Type IIA string theory admit D0, D2, D4 and D6 branes. D0 and D6 branes are elec-trically and magnetically charged with respect to the graviphoton, while D2 and D4 branesare electrically and magnetically charged with respect to the other N gauge fields. Onecan express the charge configuration collectively as ( q , q A , p , p A ), with A = 1 , , · · · , N .To be more precise, let Σ A be a basis of 4-cycles dual to α A introduced above, and let ˆΣ A be a basis dual to Σ A . Then the magnetic charge carried by the D4 brane wrapping Σ A is p A , while the electric charge carried by the brane wrapping ˆΣ A is q A .The K¨ahler potential is given by K = − ln[ i N X Λ=0 ( ¯ X Λ F Λ − X Λ ¯ F Λ )] , (2.3)where F Λ ≡ ∂ Λ F . If we choose the inhomogeneous coordinates z λ = X Λ X and set the gauge X = 1, the K¨ahler potential becomes K = − ln[ − iD ABC ( z A − ¯ z A )( z B − ¯ z B )( z C − ¯ z C )] . (2.4)The superpotential is given by W = q Λ X Λ − p Λ F Λ . (2.5)Here in the X = 1 gauge the superpotential becomes W = q + q A z A − D ABC z A z B p C + p D ABC z A z B z C . (2.6) Now let us concentrate on the so-called STU model, which can be interpreted in termsof type IIA string theory on a T of the form T × T × T . This model contains threevectormultiplets, i.e. N = 3. The prepotential and the K¨ahler potential can be obtainedimmediately. F = X X X X . (2.7)3 = − ln[ − i ( z − ¯ z )( z − ¯ z )( z − ¯ z )] = − ln(8 y y y ) , (2.8)where we have rewritten z a = x a − iy a ( a = 1 , ,
3) for later convenience. The metric andconnection on the moduli space are given by G a ¯ b = − δ ab ( z a − ¯ z a ) = δ ab (2 y a ) , G a ¯ b = − δ ab ( z a − ¯ z a ) = δ ab (2 y a ) , Γ aaa = − z a − ¯ z a . (2.9)The superpotential can be written as W = q + q a z a − p z z − p z z − p z z + p z z z . (2.10)Consider a static, spherically symmetric four-dimensional spacetime with the metric ds = − e U ( τ ) dt + e − U ( τ ) d~x , (2.11)where τ = 1 / | ~x | . Note that the horizon locates at τ = 0 and the asymptotic infinity tendsto τ → ∞ . The effective Lagrangian describing the system is L eff = ˙ U + G a ¯ b ˙ z a ˙¯ z ¯ b + e U V BH , (2.12)where the dot denotes differentiation with respect to τ . The term V BH in the above effectiveLagrangian is the so-called “effective potential”, which is given by V BH = | DZ | + | Z | = G a ¯ b ( D a Z )( ¯ D ¯ b ¯ Z ) + Z ¯ Z. (2.13) Z is the central charge of the SUSY algebra and is expressed as Z = e K/ W in our case.The covariant derivative of the central charge is D a Z = e K/ [ ∂ a + ( ∂ a K )] W. (2.14)One can obtain the equations of motion by varying the above effective Lagrangian.¨ U = e U V BH , (2.15)¨ z a + Γ abc ˙ z b ˙ z c = e U G a ¯ b ∂ ¯ b V BH . (2.16)In addition, there is a Hamiltonian constraint on the solutions˙ U + G a ¯ b ˙ z a ˙¯ z ¯ b − e U V BH = c , (2.17)where c = 0 for extremal black holes.When the black hole in the solution is extremal, the values of the moduli z a will befixed at the horizon, irrespective of their values at asymptotic infinity. The attractor valuescan be obtained by minimizing the effective potential, either directly or via the attractorequations. The entropy of the extremal black hole, whether BPS or not, is given by theeffective potential evaluated at the extremum: S = A πV BH | ext . (2.18)4 .3 The D0-D4 solution with complex moduli The simplest solution of the D0-D4 system is the case of a D0-D4-D4-D4 black hole withoutB-fields, where q > p a > p = q a = 0, which results in a BPS configuration.The solution to the effective Lagrangian is e − U = 4 H H H H , (2.19) z a = − i r H H a s abc H b H c , (2.20)where s abc = | ǫ abc | and the harmonic functions are given as follows H a = 1 √ p a τ, H = 1 √ q τ. (2.21)The attractor values of the moduli become z a = − i r q p a s abc p b p c . (2.22)One can obtain a non-BPS solution by simply analytic continuation from the BPS case,that is, we assume q < , p a >
0. Thus the harmonic functions turn out to be H a = 1 √ p a τ, H = − √ q τ, (2.23)and we can write down the solution e − U = | H H H H | , (2.24) z a = − i r − H H a s abc H b H c . (2.25)The attractor values of the moduli z a = − i r − q p a s abc p b p c . (2.26)The authors of [15] generalized the simple non-BPS solution to situations where theasymptotic moduli are more general and/or there are more charges present. In particular,they normalized the asymptotic volume moduli so that y a | ∞ = 1 but kept the asymptoticB-fields x a ∞ = B a as free variables. They obtained the following solution by solving theequations of motion directly, e − U = − H H H H − B , (2.27) z a = B − ie − U s abc H b H c , (2.28)where the harmonic functions are given by H a = 1 √ p a τ, H = − √ B ) + q τ. (2.29)5 Exact Solution of the D2-D6 System
The exact solutions to the D0-D4 system with general complex moduli were obtained in [14]and [15] by solving the equations of motion directly. However, it would be more difficultto carry out similar procedures for the D2-D6 system due to the cubic term involving theD6 charge p in the superpotential. However, the exact solutions to the D2-D6 systemwith purely imaginary moduli were obtained in [6]. The basic idea was that one could takethe horizon values of the moduli and replace the charges with corresponding harmonicfunctions. Then one could check if the ansatz satisfies the equations of motion as well asthe constraint. Fortunately after a somewhat more involved calculation one found that theansatz satisfied all the requirements. The result was manifestly dual to the known D0-D4system.Inspired by such a method, we make a similar ansatz for the moduli of the D2-D6system and find that the ansatz also satisfies the equations of motion and the constraint.Furthermore, the solution to the D2-D6 system is also manifestly dual to the solutionobtained in [15]. In this subsection we will list the main results of [6], which leads to our ansatz. Thesuperpotential for the D2-D6 system is W = q a z a + p z z z . (3.1)If p q q q <
0, the resulting configuration is BPS, while p q q q > z = − i r q q p q , z = − i r q q p q , z = − i r q q p q . (3.2)As pointed out in [6], for both BPS and non-BPS STU black holes, we can find solutionsto the equations of motion for the moduli, allowing us to obtain their values everywhere.Such solutions can be obtained by replacing the charges in the attractor values of themoduli with the corresponding harmonic functions. Furthermore, we have to check that ifsuch solutions satisfy the equations of motion and the constraint.For the D2-D6 case, after replacing the charges with harmonic functions, the solutionsturn out to be e − U = 2 p H H H H , (3.3) z = − i r H H H H , z = − i r H H H H , z = − i r H H H H . (3.4)Note that the above equations can also be expressed as z = − i e − U H H , z = − i e − U H H , z = − i e − U H H . (3.5)6hey proved that such an ansatz did satisfy the equations of motion and the constraint byworking out the terms in these equations explicitly. In this subsection we will show that our ansatz satisfies the equations of motion and theconstraint of the D2-D6 system. Furthermore, the exact solution is also manifestly dual tothe solution obtained in [15]. We would like to set z a = x a − iy a for convenience.The effective Lagrangian becomes L eff = ( ˙ U ) + G a ¯ b ˙ z a ˙¯ z ¯ b + e U V BH = ( ˙ U ) + 14 X a =1 ( ˙ x a ) + ( ˙ y a ) ( y a ) + e U V BH . (3.6)The expression for the effective potential is given by V BH = | DZ | + | Z | = G a ¯ b D a Z ¯ D ¯ b ¯ Z + Z ¯ Z = G a ¯ b ( e K/ ( ∂ a + ∂ a K ) W )( e K/ ( ∂ b + ∂ b K ) W ) + e K W W . (3.7)We can obtain the following explicit expression for the effective potential by making useof (2.8), (2.9) and (3.1), V BH = 12 y y y { q [( x ) + ( y ) ] + q [( x ) + ( y ) ] + q [( x ) + ( y ) ]+( p ) [( x ) + ( y ) ][( x ) + ( y ) ][( x ) + ( y ) ] + 2 p q x x [( x ) + ( y ) ]+2 p q x x [( x ) + ( y ) ] + 2 p q x x [( x ) + ( y ) ]+2 q q x x + 2 q q x x + 2 q q x x } . (3.8)Then the equations of motion and the constraint turn out to be¨ U = e U V BH , (3.9)12 ddτ [ ˙ x ( y ) ] = e U ∂V BH ∂x , ddτ [ ˙ x ( y ) ] = e U ∂V BH ∂x , ddτ [ ˙ x ( y ) ] = e U ∂V BH ∂x , (3.10)72 ddτ [ ˙ y ( y ) ] + 12( y ) [( ˙ x ) + ( ˙ y ) ] = e U ∂V BH ∂y , ddτ [ ˙ y ( y ) ] + 12( y ) [( ˙ x ) + ( ˙ y ) ] = e U ∂V BH ∂y , ddτ [ ˙ y ( y ) ] + 12( y ) [( ˙ x ) + ( ˙ y ) ] = e U ∂V BH ∂y . (3.11)Assume the solutions take the following form e − U = 4 H H H H − c , (3.12) z = x − i e U H H , z = x − i e U H H , z = x − i e U H H , (3.13) H = a + p τ, H = a + q τ, H = a + q τ, H = a + q τ, (3.14)where c , a and a i (i=1,2,3) are numerical constants. Let us solve (3.9) first. From (3.12)we have the following expression¨ U = 4 e U ( p H H H + q H H H + q H H H + q H H H ) − e U (2 p q H H + 2 p q H H + 2 p q H H + 2 q q H H + 2 q q H H + 2 q q H H ) . (3.15)Thus we can expand both ¨ U and e U V BH using (3.8) and (3.15) then compare the corre-sponding terms to find the solutions to x , x , x . Consider the q term for example, weobtain 4 e U ( H H H ) = e U y y y [( x ) + ( y ) ] (3.16)Solving this equation gives x = c H H . (3.17)The solutions to x and x can be obtained in a similar way, which gives x = c H H , x = c H H . (3.18)One can check that the above solutions solve (3.9) completely.The next task is to check if the above solutions satisfy the remaining equations ofmotion and the constraint. Due to the cyclic symmetries of x i and y i appear in V BH , it isnecessary to check the first two equations in (3.10) and (3.11). The left hand side of thefirst equation in (3.10) gives12 ddτ [ ˙ x ( y ) ] = − cp q e U − c ( p H + q H ) e U ˙ U . (3.19)Note that ˙ U = − e U ( p H H H + q H H H + q H H H + q H H H ) (3.20)8nd ∂V BH ∂x = 12 y y y { q + 2 q q x + 2 q q x + 2( p ) x [( x ) + ( y ) ][( x ) + ( y ) ]+4 p q x x x + 2 p q [( x ) + ( y ) ] x + 2 p q [( x ) + ( y ) ] x } . (3.21)Substituting the expressions of x and y to the above equations, one can find that bothsides match after a lengthy calculation.Subsequently we rewrite the first equation of (3.11) as follows:12 ¨ y ( y ) + 12( y ) ( ˙ x ) − y ) ( ˙ y ) = e U ∂V BH ∂y . (3.22)Note that ˙ x = − x ( p H + q H ) , (3.23)˙ y = − y (2 ˙ U + p H + q H ) , (3.24)¨ y = y [4 ˙ U + 4 ˙ U ( p H + q H ) + ( p H + q H ) − U + ( p ) ( H ) + q H ] . (3.25)Thus the left hand side of (3.22) can be given as12 ¨ y ( y ) + 12( y ) ( ˙ x ) − y ) ( ˙ y ) = − ¨ Uy + 12 y [ ( p ) ( H ) + q H ] + ( x ) y ) ( p H + q H ) (3.26)while ∂V BH ∂y = − V BH y + 12 y y y { q y + 2( p ) y [( x ) + ( y ) ][( x ) + ( y ) ] + 4 p q x x y } . (3.27)One can find that our solutions also satisfy this equation by making use of the equation ofmotion (3.9).Finally, we have to check the constraint (2.17), which can be rewritten as follows( ˙ U ) + 14 X a =1 ( ˙ x a ) + ( ˙ y a ) ( y a ) = e U V BH . (3.28)By making use of (3.20), (3.23), (3.24) as well as the cyclic permutations of the last twoequations, one can find that the left hand side of (3.28) can be simplified dramatically,( ˙ U ) + 14 X a =1 ( ˙ x a ) + ( ˙ y a ) ( y a ) = ¨ U . (3.29)Thus the constraint is also satisfied according to the equation of motion (3.9).9ow we would like to summarize our main result. The solution to the non-BPS D2-D6system can be expressed as e − U = 4 H H H H − c , (3.30) z = c H H − i e U H H , z = c H H − i e U H H , z = c H H − i e U H H , (3.31) H = a + p τ, H = a + q τ, H = a + q τ, H = a + q τ. (3.32)It can be easily seen that the above solutions have a manifestly dual form with respect tothe solution obtained in [15]. The numerical constants are left undetermined due to subtlepoints which will be discussed in the last section. One can see that the moduli exhibit thesame attractor values at the horizon as those of the simple D2-D6 non-BPS black hole.The entropy is given by S = πV | ext = 2 π p p q q q , (3.33)which is also the same as that of the simple D2-D6 case. In this section we obtain new solutions carrying D0-D2-D4 charges by transforming theoriginal D0-D4 solution in [15] under SL (2 , Z ) duality. The symplectic structure of N = 2 supergravity admits a symplectic invariant I , which isgiven by I = | Z | + | DZ | (4.1) I becomes a function of charges when restricted to STU model, which is given by I = p |W (Γ) | , where W (Γ) = 4(( p q )( p q ) + ( p q )( p q ) + ( p q )( p q )) − ( p Λ q Λ ) − p q q q + 4 q p p p (4.2)and Γ = ( p Λ q Λ ) , Λ = 0 , , ,
3. The symplectic invariance of special geometry ensures thatthe Lagrangian has an Sp (8 , Z ) symmetry, which reduces to [ SL (2 , Z )] at the level of thethe equations of motion. Given an SL (2 , Z ) matrix (cid:18) a bc d (cid:19) , with ad − bc = 1, the moduli change as˜ z a = az a + bcz a + d . (4.3)10n [6], the authors changed the charges appeared in the attractor values of the moduli tothe corresponding harmonic functions. Consider the spherically symmetric, static metricansatz: ds = − e U dt + e − U d~x , (4.4)where the metric is given by e − U = p |W ( H ) | . (4.5)In order to obtain general (D0, D2, D4, D6) system from their D2-D6 solution, they tooka specific element of [ SL (2 , Z )] and obtained the general solutions via duality transforma-tions. We will find the D0-D2-D4 solution in a similar way in the next section. First consider the general superpotential (2.6). For a configuration carrying D0-D4 charges W = q − D AB z A z B , (4.6)where D AB ≡ D ABC p C . If we add D2 charges to the system, the superpotential turns outto be W = q + q A z A − D AB z A z B , (4.7)However, if we do the following transformationsˆ q = q + 112 D AB q A q B , ˆ z A = z A − D AB q B , (4.8)where D AB ≡ ( D AB ) − . Then the D0-D2-D4 superpotential becomes W = ˆ q − D AB ˆ z A ˆ z B , (4.9)which has the same form as the D0-D4 case.Now let us specialize to the STU model. The matrices D ab and D ab are given explicitlyas follows: D ab = 16 p p p p p p (4.10)The inverse matrix D ab = − p p p p p p − p p p p p p − p p p (4.11)Then from (4.8) we have the following transformations z ′ a = z a + k a , (4.12)11here the quantities with primes belong to the D0-D2-D4 system and the quantities withoutprimes belong to the original D0-D4 system from now on. k a can be written as k ≡ D b q b = q p + q p − p q p p ,k ≡ D b q b = q p + q p − p q p p ,k ≡ D b q b = q p + q p − p q p p . (4.13)Note that the charge configuration of the D0-D4 system is expressed as (ˆ q , , , p a ) whilefor the D0-D2-D4 system we have ( q , q a , , p ′ a ).According to (4.3) and (4.12), we can write down the [ SL (2 , Z )] matrices M = (cid:18) k (cid:19) , M = (cid:18) k (cid:19) , M = (cid:18) k (cid:19) . (4.14)Our task is to generalize new solutions by making use of these [ SL (2 , Z )] matrices.Take the same notations as those in [15] p = a , q = − a , p = a , q = a ,p = a , q = a , p = a , q = a , (4.15)which transform as a ′ i ′ j ′ k ′ = ( M ) i ′ i ( M ) j ′ j ( M ) k ′ k a ijk i, j, k = 0 , . (4.16)Using (4.14), one can check that( ˆ q , , , p a ) ⇒ ( q , q a , , p a ) . (4.17)Similarly, the harmonic functions transform as H ′ = H , H ′ = H , H ′ = H . (4.18) H ′ = k H + k H = 1 √ k + k ) + q τ,H ′ = k H + k H = 1 √ k + k ) + q τ,H ′ = k H + k H = 1 √ k + k ) + q τ, (4.19) H ′ = H − k k H + k k H + k k H = 1 √ B ) − ( k k + k k + k k )] + q τ. (4.20)12e can see that the duality invariant does not change indeed I (Γ) = 4ˆ q p p p = 4 q p p p − ( p ) q − ( p ) q − ( p ) q +2 q q p p + 2 q q p p + 2 q q p p . (4.21) I ′ (Γ) = 4 q p p p − ( p q + p q + p q ) + 4( p q p q + p q p q + p q p q )= 4 q p p p − ( p ) q − ( p ) q − ( p ) q +2 q q p p + 2 q q p p + 2 q q p p = I (4.22)Furthermore, we can see that W ( H ) is also invariant. W ( H ) = 4 H H H H . (4.23) W ′ ( H ′ ) = 4(( H ′ H ′ )( H ′ H ′ ) + ( H ′ H ′ )( H ′ H ′ ) + ( H ′ H ′ )( H ′ H ′ )) − ( H ′ H ′ + H ′ H ′ + H ′ H ′ ) + 4 H ′ H ′ H ′ H ′ . (4.24)After substituting the expressions for H ′ harmonic functions (4.18)- (4.20), one can findthat W ′ ( H ′ ) = W ( H ) . (4.25)Here the metric is given by e − U = |W ( H ) | − c . (4.26)Thus e − U ′ = e − U . (4.27)Now we have to check whether the new solution satisfies the equations of motion. From (4.12)and (4.27) we can see that the left hand side of the equations of motion and the constraintremain unchanged. Thus we just need to check if the effective potential V BH on the righthand side remains invariant. It can be easily seen that K ′ = K, W = W ′ , D a Z = D a ′ Z ′ , (4.28)after taking all the relevant formulae into account. Thus by the definition of V BH , we have V ′ BH = V BH . (4.29)Then our new solution also satisfies the equations of motion and the constraint.One can easily obtain the attractor values of the moduli z ′ a = k a − i r − q p a s abc p b p c , (4.30)which is the same as those discussed in previous examples. The entropy is given by S BH = πV BH | ext = 2 π p ˆ q p p p , (4.31)which also agrees with the previously known D0-D2-D4 entropy.13 Summary and Discussion
In this note we obtain the non-BPS, extremal, spherically symmetric black hole solutions offour-dimensional supergravity, carrying D2-D6 and D0-D2-D4 charges. The D2-D6 solutioncontains general complex moduli and is manifestly dual to the D0-D4 cousin appearedin [15]. The D0-D2-D4 solution is obtained by [ SL (2 , Z )] duality transformations fromthe D0-D4 solution. Both of our solutions give the same attractor values of the moduliand the same entropies as those of previously known examples carrying the same charges.One may obtain new solutions carrying general (D0, D2,D4,D6) charges from the knownsolutions by duality transformations.One subtle point is the determination of the numerical constants in the D2-D6 solution.Of course one can take the same values as those in [15], that is, c = B, , a = 1 √ B ) , a = a = a = 1 √ . (5.1)Thus one can obtain the mass by expanding the warp factor,2 G N M non − BPS = 1 √ | p | + X a q a (1 + B )) , (5.2)which has a similar form as that given in [15]. This can be interpreted as the sum of themasses of the D2 and D6-branes, which also exhibits a marginal bound state behavior.However, in such cases the asymptotic values of the moduli are different, z a = B − i B , (5.3)which means that the normalization of the asymptotic moduli should be different fromthat in [15].Another interesting non-BPS configuration is the D0-D6 system, which has been ex-tensively studied in recent years, see e.g. [17] [18] [19] [20]. The attractor mechanism forD0-D6 Kaluza-Klein black holes has been discussed in [21] using the entropy function for-malism. Since our solution is manifest dual to the D0-D4 solution, it will be interesting tostudy the relations between our solution and the D0-D6 system discussed in [15].The STU model has been discussed in [22] in another interesting way, that is, sucha model can be tackled in the context of quantum information theory. The use of thisformalism expresses the black hole potential in an especially elegant form as the normsquared of a suitable tripartite entangled state. Then the classification of solutions proceedswith analysing the charge codes using some elements of quantum error correction. However,only doubly extremal solutions were discussed in that paper for illustration. So it wouldbe interesting to extend similar analysis using the more general solutions.A further direction is to generalize the famous “OSV” conjecture [23], which relates thepartition function of BPS black holes to the partition function of topological strings, tonon-BPS case. In a recent paper [24], Sarakin and Vafa pointed out that there was some14ubtle points when generalizing the original “OSV” conjecture to non-BPS cases. Thusan extension of OSV that can be applied simultaneously to both BPS and non-BPS blackholes is needed, which is more difficult to realize. However, the various exact solutions ofnon-BPS black holes provide concrete examples for testing their conjecture. We would liketo study this problem in the near future. Note Added : After the first version appearing on arXiv, we were informed with [16].In that interesting paper the authors constructed interpolating solutions describing single-center static extremal non-supersymetric black holes in four dimensional N = 2 supergrav-ity with cubic prepotentials. They derived and solved the first-order flow equations for 5Drotating electrically charged extremal black holes in a Taub-NUT geometry. Then usingthe 4D/5D connections they obtained the corresponding 4D solutions. One key point forthese results was that the 5D geometry was assumed to be a time fibration over a Hyper-K¨ahler base. When the 4D prepotential contains a cubic term, the corresponding solutionsto the first-order flow equations are e − U = 49 N ( H A f − / X A ) − c , (5.4) z A = 32 ( c + ie − U N H B f − / X B ) f − / X A . (5.5)One can find that our D2-D6 solution agrees with their solution when restricted to STUmodel . Acknowledgements
We thank G. Cardoso and P. Levay for correspondence. The work was supported in partby a grant from Chinese Academy of Sciences, by NSFC under grants No. 10325525 andNo. 90403029.
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