Non-Supersymmetric Attractor Flow in Symmetric Spaces
aa r X i v : . [ h e p - t h ] D ec Non-Supersymmetric Attractor Flow in SymmetricSpaces
Davide Gaiotto, Wei Li and Megha Padi
Jefferson Physical Laboratory, Harvard University, Cambridge MA 02138, USA
Abstract
We derive extremal black hole solutions for a variety of four dimensional modelswhich, after Kaluza-Klein reduction, admit a description in terms of 3D gravity coupledto a sigma model with symmetric target space. The solutions are in correspondencewith certain nilpotent generators of the isometry group. In particular, we provide theexact solution for a non-BPS black hole with generic charges and asymptotic moduliin N = 2 supergravity coupled to one vector multiplet. Multi-centered solutions canalso be generated with this technique. It is shown that the non-supersymmetric solu-tions lack the intricate moduli space of bound configurations that are typical of thesupersymmetric case. ontents D -dimensional Pure Gravity 10 k . . . . . . . . . . . . . . . . . . . . 133.4.2 Full flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4.3 Example: 5D pure gravity compactified on a circle. . . . . . . . . . . 163.5 Multi-centered Solutions in Pure Gravity . . . . . . . . . . . . . . . . . . . . 17 G / ( SL (2 , R ) × SL (2 , R )) M D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Extracting the Coordinates from the Coset Elements . . . . . . . . . . . . . 204.3 Nilpotency of the Attractor Flow Generator k . . . . . . . . . . . . . . . . . . 234.4 Properties of Attractor Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Flow Generators in the G / ( SL (2 , R ) × SL (2 , R )) Model 26 k BP S using supersymmetry . . . . . . . . . . . . . . . . 265.1.2 Constructing k NonBP S . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2 Properties of Flow Generators . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2.1 Properties of k BP S . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2.2 Properties of k NonBP S . . . . . . . . . . . . . . . . . . . . . . . . . . 30 G / ( SL (2 , R ) × SL (2 , R )) Model 31 D . . . . . . . . . . . . . . . . . . . . . . . . . 326.1.2 4 D solution for given set of charges . . . . . . . . . . . . . . . . . . . 346.2 non-BPS Attractor Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.2.1 Lifting a geodesic to 4 D . . . . . . . . . . . . . . . . . . . . . . . . . 376.2.2 4 D solution for given set of charges . . . . . . . . . . . . . . . . . . . 39 G / ( SL (2 , R ) × SL (2 , R )) model 41 M D Introduction
Soon after the attractor mechanism was first discovered in supersymmetric (BPS) black holes[1], it was reformulated in terms of motion on an effective potential for the moduli [2]. Fer-rara et al demonstrated that the critical points of this potential correspond to the attractorvalues of the moduli. More recently, several groups used the effective potential to show thatnon-supersymmetric (non-BPS) extremal black holes can also exhibit the attractor mecha-nism, thereby creating a new and exciting field of research [3, 4]. Many connections betweennon-BPS attractors and other active areas of string theory soon revealed themselves. An-drianopoli et al found that both BPS and non-BPS black holes embedded in a supergravitywith a symmetric moduli space can be studied using the same formalism, and they uncov-ered many intricate relations between the two [5, 6]. Dabholkar, Sen and Trivedi proposeda microstate counting for non-BPS black holes (albeit subject to certain constraints [7]).Saraikin and Vafa suggested that a new extension of topological string theory generalizesthe Ooguri-Strominger-Vafa (OSV) formula such that it is also valid for non-supersymmetricblack holes [8]. Studying non-BPS attractors could also give insight into non-supersymmetricflux vacua. Given all these possible applications, it is important to characterize non-BPSblack holes as fully as possible.There has been a great deal of progress in understanding the near-horizon region of thesenon-BPS attractors. The second derivative of the effective potential at the critical pointdetermines whether the black hole is an attractor, and the location of the critical pointyields the values of the moduli at the horizon; in this way, one can compute the stabilityand attractor moduli for all models with cubic prepotential [9, 10]. However, the effectivepotential has only been formulated for the leading-order terms in the supergravity lagrangian.If one wants to include higher-derivative corrections, one can instead use Sen’s entropyformalism, which incorporates Wald’s formula, to characterize the near-horizon geometry ingreater generality [11]. Sen’s method has led to many new results [12, 13, 14, 15, 16, 17].The tradeoff is that this method cannot be used to determine any properties of the solutionaway from the horizon.The BPS attractor flow is constructed from the attractor value z ∗ BP S = z ∗ BP S ( p I , q I ) bysimply replacing the D-brane charges with the corresponding harmonic functions: z BP S ( ~x ) = z ∗ BP S ( p I → H I ( ~x ) , q I → H I ( ~x )) (1.1)where the harmonic functions are (cid:18) H I ( ~x ) H I ( ~x ) (cid:19) = (cid:18) h I h I (cid:19) + 1 | ~x | (cid:18) p I q I (cid:19) Moreover, this procedure can be applied to construct the multi-centered BPS attractor flowthat describes the supersymmetric black hole bound state [18], where the harmonic functions3re generalized to have multiple centers: (cid:18) H I ( ~x ) H I ( ~x ) (cid:19) = (cid:18) h I h I (cid:19) + X i | ~x − ~x i | (cid:18) ( p I ) i ( q I ) i (cid:19) It is conjectured in [19] that the non-BPS flow can be generated in the same fashion, namely,by replacing the charges in the attractor value with the corresponding harmonic functions.However, as will be proven in this paper, this procedure does not work for systems withgeneric charge and asymptotic moduli. In principle, one could construct the full non-BPS flow (the black hole metric, togetherwith the attractor flow of the moduli) by solving the equation of motion derived from thelagrangian. However, this is a second-order differential equation and only reduces to a first-order equation upon demanding the preservation of supersymmetry. Ceresole et al havewritten down an equivalent first-order equation in terms of a “fake superpotential”, butso far, the fake superpotential can only be explicitly constructed for special charges andasymptotic moduli [21, 20]. The most generic non-BPS equation of motion is complicatedenough that it has not yet been solved. Similarly, multi-centered non-BPS black holes havenot been studied.Our goal is to construct the full flow for non-BPS stationary black holes in four di-mensions. Instead of directly solving the equation of motion, we reduce the action on thetimelike isometry and dualize all 4D vectors to scalars. The new moduli space M D containsisometries corresponding to all the charges of the black hole, and the black hole solutionsare simply geodesics on M D . This method was introduced in [22] and has been used toconstruct static and rotating black holes in heterotic string theory [23, 24] and to study theclassical BPS single-centered flow and its radial quantization [25, 26].In this paper, we work in two specific theories of gravity, but we expect that this methodcan be used for any model whose M D is symmetric. The basic technique is reviewed inmore detail in Section 2. In Section 3, we show how this method works in a simple case: thetoroidal compactification of D -dimensional pure gravity. Section 4 serves as an introductionto single-centered attractor flow in N = 2 supergravity coupled to one vector multiplet, andSections 5 and 6 are dedicated to constructing the full flows for both BPS and non-BPSsingle-centered black holes with generic charges. We find that they are generated by theaction of different classes of nilpotent elements in the coset algebra. Both types of flows areshown to reach the correct attractor values at the horizon. In Section 7, the procedure isgeneralized straightforwardly to construct both BPS and non-BPS multi-centered solutions.We use a metric ansatz with a flat spatial slice and we are able to recover the BPS boundstates described by Bates and Denef. Using the same ansatz, we are able to build non-BPSmulti-centered solutions. Unfortunately, solutions generated this way turn out to always have This has also been shown in [20].
Here we outline the method we will use to construct black hole solutions. This method wasfirst described in [22]. We first reduce a general gravity action from four dimensions down tothree, and derive the equation of motion. We then specialize to certain theories which havea 3d description in terms of a symmetric coset space. In such situations, we can easily findsolutions to the equation of motion. The solutions are geodesics (or generalizations thereof)on the 3d moduli space and they are generated by elements of the coset algebra.
We will study stationary solutions in a theory with gravity coupled to scalar and vectormatter. Let the scalars be z i and the vectors be A I . Then the most general ansatz for astationary solution in four dimensions is: ds = − e U ( dt + ω ) + e − U g ab dx a dx b (2.1) F I = dA I = d (cid:0) A I ( dt + ω ) + A I (cid:1) (2.2)where a, b = 1 , , M D . This procedure is called the c ∗ -map.In three dimensions, a vector is Hodge-dual to a scalar. The equations of motion for ω andthe gauge fields allow us to define the dual scalars φ ω and φ A I .We then obtain the 3d lagrangian in terms of only scalars L = 12 √ g ( − κ R + ∂ a φ m ∂ a φ n g mn ) (2.3)where φ n are the moduli fields φ n = { U, z i , ¯ z ¯ i , φ ω , A I , φ A I } (2.4)and g ab is the space time metric and g mn is the metric of a manifold M D . The system is3 D gravity minimally coupled to a nonlinear sigma model with moduli space M D . Next,we will find the equation of motion in this theory.5 .2 Attractor Flow Equation The equation of motion of 3 D gravity is Einstein’s equation: R ab − g ab R = κT ab = κ ( ∂ a φ m ∂ b φ n g mn − g ab ∂ c φ m ∂ c φ n g mn ) (2.5)and the equation of motion of the moduli is: ∇ a ∇ a φ n + Γ nmp ∂ a φ m ∂ a φ p = 0 (2.6)For simplicity, we consider only the case where the 3 D spatial slice is flat (it is guaranteedto be flat only for extremal single-centered black holes). Then the dynamics of the moduliare decoupled from that of the 3 D gravity: R ab = 0 = ⇒ ∂ a φ m ∂ b φ n g mn = 0 (2.7)In the multi-centered case, we need to solve the full equations for the moduli as functions ofthe 3d coordinates ~x . For single-centered solutions, the moduli only depend on r ; to satisfythe above equations, the motion of the moduli must follow null geodesics inside M D .A generic null geodesic flows to the boundary of the moduli space M D . A single-centered attractor flow is defined as a null geodesic that terminates at a point on the U →−∞ boundary and in the interior region with respect to all other coordinates. This isguaranteed for the BPS attractor by the constraints imposed by the supersymmetry. To findthe single-centered non-supersymmetric attractor flow, one needs to find a way to constructnull geodesics and a constraint that pick out the ones that stop at this specific component ofthe boundary. In the next section, we will show that we can do this for models with specialproperties, and the method can be easily generalized to find the multi-centered attractorsolution. The problem of finding such a constraint in a generic model is not easy. To simplify, we studyany model whose moduli space is a symmetric homogeneous space: M D = G / H . When M D is a homogeneous space, the isometry group G acts transitively on M D . H denotesthe isotropy group, which is the maximal compact subgroup of G when one compactifieson a spatial isometry down to (1 ,
2) space, or the analytical continuation of the maximalcompact subgroup of G when one compactifies on the time isometry down to (0 ,
3) space.The Lie algebra g has the Cartan decomposition: g = h ⊕ k where[ h , h ] = h [ h , k ] = k (2.8)6hen G is semi-simple, the homogeneous space is symmetric, and[ k , k ] = h (2.9)The models with symmetric moduli space includes: D -dimensional gravity toroidallycompactified to four dimensions, certain models of 4D N = 2 supergravity coupled to vector-multiplet, and all 4D N > D -dimensional gravity toroidally compactified to four dimensions, and the 4D N = 2supergravity coupled to n V vector-multiplet.The left-invariant current is J = M − dM = J k + J h (2.10)where M is the coset representative, and J k is the projection of J onto the coset algebra k .The lagrangian density of the sigma-model with target space G / H is given by J k as: L = Tr( J k ∧ ∗ J k ) (2.11)The geodesic of the homogeneous space written in terms of the coset representative issimply M = M e kτ/ with k ∈ k (2.12)where M parameterizes the initial point, and the is for later convenience. A null geodesichas zero length: | k | = 0 (2.13)Therefore, in a homogeneous space, we can find the null geodesics that end at an attractorpoint by imposing the appropriate constraint on the null elements of the coset algebra.Since M is defined up to the action of the isotropy group H , in order to read off themoduli fields from M in an H -independent way, we construct the symmetric matrix usingthe metric signature matrix S : S ≡ M S M T (2.14)In all systems considered in the present paper, H is the maximal orthogonal subgroup of G with the correct signature: HS H T = S for ∀ H ∈ H (2.15)That is, the isotropy group H preserves the symmetric metric matrix S . Therefore, S isinvariant under M → M H with H ∈ H . Moreover, as the isotropy group H acts transitivelyon the space of of matrices with a given signature, the space of possible S is the same as the7ymmetric space G / H . That is, the moduli of G / H can be combined into the symmetricmatrix S . And the current of S is J S = S − dS (2.16)It is easy to perform the projection onto the coset algebra k . The (generalized) orthogo-nality condition of the isotropy group H can be expressed in terms of the subalgebra element h which is in H = e h as hS + S h T = 0 ∀ h ∈ h (2.17)In other words, ( h S ) is anti-symmetric: ( h S ) T = − ( h S ). Thus the coset algebra, beingthe compliment of h , can be defined as the k with ( k S ) being symmetric: ( k S ) T = ( k S ),i.e. k T = S − kS ∀ k ∈ k (2.18)Therefore, the projection of an element g in g onto the coset algebra k is: g k = g + S g T S − J = M − dM , the projection onto k is: J k = J + S J T S − S is related to the pro-jected left-invariant current J k by: J S = S − dS = 2( S M T ) − J k ( S M T ) (2.21)The lagrangian in terms of S is thus L = Tr( J S ∧ ∗ J S ). That is, the lagrangian density is L = 14 Tr( S − ∇ S · S − ∇ S ) (2.22)which is invariant under the action of the isometry group G : S → G − SG where G ∈ G (2.23)and whose conserved current is: J = S − ∇ S (2.24)where we have dropped the subscript S in J S , since we will only be dealing with this currentfrom now on. The equation of motion is the conservation of the current: ∇ · J = ∇ · ( S − ∇ S ) = 0 (2.25)8e now specialize to the single-centered solutions: they correspond to geodesics in thecoset manifold. The spherical symmetry allows the 3 d metric to be parameterized as ds = C ( r ) d~x (2.26)Then the equations of motion involve the operator d r r C ( r ) d r , and reduce to geodesic equa-tions in terms of a parameter τ such that drdτ = r C ( r ) . (2.27)The function C ( r ) is then determined from the equations of motion of 3 d gravity. Theequations of motion can be written as ddτ ( S − dSdτ ) = 0 (2.28)In the extremal limit the geodesics become null, the 3 d metric is flat and τ = − r (2.29)In the search for multi-centered extremal solutions, where the spherical symmetry isabsent, it is very convenient to restrict to solutions with a flat 3d metric. This is consistentwith the equations of motion as long as the 3 d energy momentum tensor is zero everywhere: T ab = T r ( J a J b ) = 0 (2.30)The coupled problem with generic non-flat 3 d metric is much harder, and exact solutionsare hard to find unless a second Killing vector is present.Since different values of the scalars at infinity are easily obtained by a G transformation,to start with, we will consider the flow starting from M = 1, and generalize to genericasymptotic moduli later. For a single-centered solution, the flow of M is M = M e kτ/ .Since all the coset representatives can be brought into the form e g with some g ∈ k by an H -action, we can write M = e g/ , so M = e g/ e kτ/ . And the flow of S is S ( τ ) = e g/ e kτ e g/ S (2.31)The charges of the solution are read from the conserved currents J ( r ) = S − ∇ S = S e − g/ ke g/ S r ˆ ~r (2.32)9 Torus Reduction of D -dimensional Pure Gravity Now we use the method introduced in the previous section to analyze pure gravity toroidallycompactified down to four dimensions. We explain why the attractor flow generator, k ,needs to be nilpotent, and we find the Jordan forms of k and k . Using this information, weconstruct single-centered attractor flows. We then generalize to multicentered black holesin pure gravity and show that these solutions have mutually local charges and no intrinsicangular momentum. The simplest example of a system that admits a 3 d description in terms of a sigma modelon a symmetric space is pure gravity in D dimensions, compactified on a D − D parameterizes the metric as ds D = ρ pq ( dy p + A pµ dx µ )( dy q + A qν dx ν ) + 1 √ det ρ ds ≤ p, q ≤ D − y p are the torus coordinates, x µ the coordinates on R , , and ρ pq the metric of the torus.A 4 D metric with one timelike Killing spinor is then parameterized as ds = − u ( dt + ω i dx i ) + 1 u ds (3.2)where u = e U , to connect with the parametrization in the later part of the paper; and i = 1 , , d space coordinates.The two expressions combine as ds D = G ab ( dy a + ˜ ω ai dx i )( dy b + ˜ ω bj dx j ) + 1 − det G ds ≤ a, b ≤ D − y a are the torus coordinates plus time, x i coordinates on R and G = (cid:18) ρ pq ρ pr A r A r ρ rq A r ρ rs A s − u √ det ρ (cid:19) (3.4)and ˜ ω a = (˜ ω p , ˜ ω ) is: ˜ ω p = ( A pi − A p ω i ) dx i ˜ ω = ω (3.5)If the forms ˜ ω a are dualized to scalars α a as dα a = − det GG ab ∗ d ˜ ω b (3.6)10he various scalars can be combined into a symmetric unimodular ( D − × ( D −
2) matrix S = (cid:18) G ab + G α a α b G α a G α b G (cid:19) (3.7)In terms of the 4 D fields that is S = ρ pq − u √ det ρ α p α q ρ pr A r − u √ det ρ α p α − u √ det ρ α p A r ρ rq − u √ det ρ α α q A r ρ rs A s − u √ det ρ − u √ det ρ α α − u √ det ρ α − u √ det ρ α q − u √ det ρ α − u √ det ρ (3.8)The equations of motion derive from the lagrangian density L = T r ∇ SS − ∇ SS − , in-variant under S → U T SU for any U in SL ( D − SL ( D −
2) action is transitiveon the space of matrices with a given signature, the space of possible S is the same asthe symmetric space SL ( D − /SO ( D − , SO ( D − ,
2) is appropriate for the reduction from ( D − ,
1) to (3 ,
0) signature. The usualreduction from ( D − ,
1) to (2 ,
1) would give a SL ( D − /SO ( D − D,
0) to (3 ,
0) gives SL ( D − /SO ( D − ,
1) [27].The coset representative under the left SO ( D − ,
2) action can be described in terms ofa set of vielbeins e A = E Aa ( dy a + ω ai dx i ) e I = 1det M e I (3) (3.9)as M = (cid:18) E Aa E α b E (cid:19) (3.10)Then the symmetric SO ( D − ,
2) invariant matrix S = M S M T (3.11)can be used to read off the solution more easily. Without loss of generality, we can take S to be the signature matrix: S = Diag ( η, −
1) =
Diag (1 , · · · , , − , −
1) (3.12)The equations of motion are equivalent to the conservation of the SL ( D −
2) currents J = S − dS . Some of those currents correspond to the usual gauge currents in 4D: the first D − J i,D − are the KK monopole currents, the first D − J D − ,i are the KK momentum currents and the element J D − ,D − is the current for the 3 d gauge field ω . Regular 4 D solutions must have zero sourcesfor this current, otherwise ω will not be single valued.11 .2 Nilpotency We now show that all the attractor flows are generated by the nilpotent generators in thecoset algebra. To get extremal black hole solutions with a near horizon
AdS × S , thefunction u must scale as r as r goes to zero while the scalars go to a constant. This makes S diverge as r . The most natural way for S to diverge as τ for large τ is that k is nilpotent,with k = 0 . (3.13)This will be the crucial condition through the whole paper. Not every null geodesic corresponds to extremal black hole solutions. Let’s consider a simpleexample: hyperk¨ahler euclidean metrics in 4 D .Although this example is not about a black hole, it is still quite instructive. The 3 d sigmamodel is SL (2) /SO (1 , AdS . The coset representative is written as M = (cid:18) u / au / u / (cid:19) (3.14)and the symmetric invariant S = (cid:18) u − a u − au − au − u (cid:19) (3.15)A geodesic is the exponential of a Lie algebra element in the orthogonal to the stabilizer.The stabilizer SO (1 ,
1) is generated by σ . A null geodesic is hence the exponential of k = σ ± iσ . This is a nilpotent matrix, k = 0, hence M = 1 + τ k/
2. Take: k = σ + iσ = (cid:18) − − (cid:19) (3.16)then M = (cid:18) τ / τ / − τ / − τ / (cid:19) (3.17)and we can read off the geodesic solution from the invariant S = M T (cid:18) − (cid:19) M = (cid:18) τ ττ − τ (cid:19) (3.18)12ence u = 11 − τ , a = − τ − τ . (3.19)Dualizing a , we get ∗ dω = − dau = 1 r dr u − = 1 + 1 r (3.20)The 4D Euclidean metric is just the Taub-NUT metric. Notice that the multi-centeredTaub-NUT generalization of the metric is obtained by replacing τ above with some harmonicfunction P i q i | x − x i | . The sigma model equations of motion are equivalent to the conservationof the current J = S − ∇ S , and if S is given as above with τ = τ ( ~x ) then the equation ofmotion are ∇ τ ( ~x ) = 0 (3.21) k Now we look in detail at the single-centered black holes in pure gravity. Notice that as u goes to zero S tends asymptotically to a rank one matrix S = − u √ det ρ α p α q α p α α p α α q α α α α q α (3.22)hence the matrix k should also have rank 1. By inspection of S it is clear that a k of rankhigher than one gives a geodesic for which the matrix elements of ρ also diverge as τ so thatthe scalar fields do not converge to fixed attractor values.Notice that if k is nilpotent, then S is a polynomial in r , and the various scalars in thesolution will all be simple functions of r for such extremal solutions!The explicit form for k in terms of the charges is then straightforward to write. Considerthe Jordan form of k : as it is nilpotent, the eigenvalues are all zero. As it is rank one, ithas one single indecomposable block of size two: (cid:18) (cid:19) . It is written as k = − ηvv T f ( P, Q ) (3.23)with v null in the metric η , and f ( P, Q ) any degree-two homogeneous function of the charge(
P, Q ). This form is chosen so that v does not scale with the charge ( P, Q ).13hen k must have a Jordan form with all eigenvalues zero, one block of size 3: and possibly some other extra blocks of size two. Alternatively, there is a subspace V anni-hilated by k , a subspace V ′ whose image under k sits in V and has the same dimension as V ′ , and a single vector w such that kw ∈ V ′ and is non-zero. kw ⊂ V ′ , kV ′ ⊂ V, kV = 0 . (3.24)From the symmetry of ηK , it follows that the space kV ′ is made of null vectors only, andthat kw is orthogonal to it. Because ηk = − vv T , ( kw ) T ηkw is negative. Because of thesignature of η it is straightforward to see that V ′ can be of dimension at most one; hencethere are no blocks of size 2 in the Jordan form of k .Taking this into account, the final form of k is simply k = ηwv T + ηvw T (3.25)where v and w are two orthogonal ( D − v being null and w having norm − f ( P, Q ): wηv = 0 , vηv = 0 , wηw = − f ( P, Q ) . (3.26)Using the fact that in k , the first D − K i,D − are the magneticcharges, and the first D − K D − ,i are the electriccharges, and the element K D − ,D − is the Taub-NUT charge, which has to vanish, we have2( D −
4) + 1 = 2 D − D − v, w ).The full solution of ( v, w ) requires one to solve some degree-four equations, hence we’llleave it in a slightly implicit form. Let ( p, q ) be two ( D − P, Q ) of the 4 D gauge fields are P = p p + p · q p Q = p q + p · q q (3.27)And we choose f ( P, Q ) to be f ( P, Q ) = p · q (3.28)The solution of v and w written in terms of ( p, q ) is v = 1 √ p · q q + p − p q + p · q − p p + p · q , w = √ p · q q − p − p q + p · q p p + p · q . (3.29)14nd k can be written as k = qq T − pp T − p q + p · q q p p + p · q p p q + p · q q T − ( q + p · q ) 0 − p p + p · q p T p + p · q (3.30) First, for the full flow starting from M = 1, the scalars for the attractor solution generatedby this k can be read off from S ( τ ) = e kτ , by comparing with the form of S in terms of the4D fields: u − = [1 + ( p + p · q )( τ + p · q τ )][1 + ( q + p · q )( τ + p · q τ )] (3.31)and ρ = 1 + ( qq T − pp T ) τ + [( p + p · q ) qq T − p · q ( p + q )( p + q ) T ] τ p + p · q )( τ + p · q τ ) (3.32)Notice that as τ → ∞ , τ − e − U has the correct limit P · Q , which is the entropy where P and Q are the physical electric and magnetic charges.To generalize to arbitrary asymptotic moduli, M ( τ ) = e g/ e kτ/ , and the flow of S is(2.31), which can be written as S ( τ ) = e K ( τ ) S , where K ( τ ) is a matrix function. From nowon, we use lower case k to denote the coset algebra that generates the attractor flow, andcapital K to denote the function which we exponentiate directly to produce the solution.We will choose K ( τ ) to have the same properties as the generator k : K ( τ ) = 0 and K ( τ ) rank one (3.33)The equations of motion ∇ · ( S − ∇ S ) = 0 then simplify considerably with this ansatz. Ifone further requires that the subspace image of K ( τ ) remains constant everywhere, suchthat K ( τ ) ∇ K ( τ ) = ∇ K ( τ ) K ( τ ) = 0 (3.34)then the current reduces to J = S − ∇ S = S (cid:18) ∇ K ( τ ) + 12 [ ∇ K ( τ ) , K ( τ )] (cid:19) S (3.35)and the equations of motion are ∇ K ( τ ) + 12 [ ∇ K ( τ ) , K ( τ )] = 0 (3.36)which is solved by a harmonic K ( τ ). 15t might appear hard to find a K ( τ ) that is harmonic and satisfies all the requiredconstraints. However, by remembering that the constraints dictate K ( τ ) to have the form: K ( τ ) = ηV W T + ηW V T (3.37)with V being null and doesn’t scale with the charge ( P, Q ), and W orthogonal to V ev-erywhere, one can simply pick a constant null vector V = v ′ and a harmonic vector W ( τ )everywhere orthogonal to v ′ : W ( τ ) = w ′ τ + m with v ′ · W ( τ ) = 0 (3.38)Here m is a ( D − K ( τ ) is built: K ( τ ) = k ′ τ + g (3.39)where k ′ = ηv ′ w ′ T + ηw ′ v ′ T g = ηv ′ m T + ηmv ′ T (3.40)Now we need to solve for ( v ′ , w ′ ) for the same charge ( P, Q ) but in the presence of m .The form of g guaranteed that [ k ′ , g ] = 0 (3.41)where we have used the fact that v ′ is null and w ′ is orthogonal to v ′ . Therefore, shifting thestarting point of moduli does not change the current as a function of ( v, w ): J ( v ′ , w ′ ) = S (cid:18) k ′ r (cid:19) S = S (cid:18) ηv ′ ( w ′ ) T + ηw ′ ( v ′ ) T r ˆ ~r (cid:19) S (3.42)Thus, the solution of ( v ′ , w ′ ) in terms of charges solved from the current does not change aswe vary the starting point of the flow, i.e. they do not depend on the asymptotic moduli: v ′ ( Q ) = v ( Q ) w ′ ( Q ) = w ( Q ) (3.43)In summary, the flow with arbitrary starting point is simply generated by K ( τ ) = ηvW ( τ ) T + ηW ( τ ) v T with W = wτ + m (3.44)where ( v, w ) only depend on the charges ( P, Q ) and m gives the asymptotic moduli. Consider for example the case of extremal black holes in D = 5 pure gravity compactifiedon a circle. The 3 d sigma model is SL (3) /SO (1 , S gr = ρ − u √ ρ α α ρA − u √ ρ α α − u √ ρ α ρA − u √ ρ α α ρ ( A ) − u √ ρ − u √ ρ α α − u √ ρ α − u √ ρ α − u √ ρ α − u √ ρ (3.45)16hen we can calculate S = S e kτ using (3.30) and compare the result to S gr above to solvefor all the scalars. We find that e − U = r [1 + ( q + pq )( τ + pq τ )][1 + ( p + pq )( τ + pq τ )] (3.46) ρ = 1 + ( q + pq )( τ + pq τ )1 + ( p + pq )( τ + pq τ ) (3.47)when starting from the identity. If we allow arbitrary g the flow is too complicated to writeexplicitly here, but the attractor value of ρ is the same: q/p . In the context of pure gravity compactified on a torus, we can also give some examples ofmulti-centered solutions in the same spirit as the ones for BPS solutions in N = 2 super-gravity, though some important features of the latter are not present here.We are interested in solutions given in terms of harmonic functions which can generalizethe single-centered extremal solutions presented above. Similar to the single-centered case,we exponentiate a matrix function K ( ~x ): S ( ~x ) = e K ( ~x ) S (3.48)We will choose K ( ~x ) to have the same properties of the generator k : K ( ~x ) = 0 and K ( ~x ) rank one (3.49)Using a similar argument to the single-centered flow, we require that the subspace imageof K ( ~x ) remains constant everywhere, such that K ( ~x ) ∇ K ( ~x ) = ∇ K ( ~x ) K ( ~x ) = 0, thenthe equations of motion are ∇ K ( ~x ) + 12 [ ∇ K ( ~x ) , K ( ~x )] = 0 (3.50)which is solved by a harmonic K ( ~x ).A multi-centered K ( ~x ) that is harmonic and satisfies all the required constraints can thenbe built in the same way as the single-centered one: K ( ~x ) = ηvW ( ~x ) T + ηW ( ~x ) v T (3.51)where v is the same constant null vector as in k , and W ( ~x ) is the multi-centered harmonicfunction: W ( ~x ) = X i w i | ~x − ~x i | + m (3.52)17here ~x i is the position of the i th center, and w i is determined by the charges at the i thcenter, and m is related to the moduli at infinity. Requiring W ( ~x ) to be orthogonal to v everywhere gives the following constraints on the { w i , m } : First, taking ~x to infinity, it gives v · m = 0 (3.53)Second, w i are orthogonal to v : v · w i = 0 (3.54)In addition to the constraint from the zero Taub-NUT charge condition − bz − cy = 0, thismakes the space of each possible w i only ( D − W is a( D − D −
4) independent harmonic functions to work with, becauseof the orthogonality to v and the requirement of no timelike NUT charges. This makes thesolution relatively boring.The multi-centered solution in pure gravity does not have the characteristic features ofthe typical BPS multi-centered solution in N = 2 supergravity, where many centers withrelatively non-local charges form bound states which carry intrinsic angular momentum.The basic reason is that when the ansatz K ( ~x ) = ηvW T ( ~x ) + ηW ( ~x ) v T is used, thesecond term of the conserved currents J = S ( ∇ K + [ ∇ K, K ]) S drops out. The firstresult is that the charges of the various centers in the solution can be read off directly from( v, w i ), and they do not depend on the positions, charges of the other centers. Thus, thereis no constraint on the position of each center as in the N = 2 BPS multi-centered solution;centers can be moved around freely.Moreover, the condition of no timelike Taub-NUT charge is a linear constraint on thecharges at each center which results in a static 4 D solution, as ∗ dω = 0 leads to ω = 0.Therefore, no angular momentum is present. G / ( SL (2 , R ) × SL (2 , R )) We now tackle a more complicated subject: N = 2 supergravity coupled to one vectormultiplet. First, we reduce the theory down to three dimensions and derive the metricfor the resulting moduli space, which is the coset G / ( SL (2 , R ) × SL (2 , R )). We thendiscuss the Cartan and Iwasawa decompositions of the group G , which we use to constructthe coset algebra and translate the flow of coset representative into the flow of the modulifields, respectively. We then specify the representation of G we will be working with, anddescribe the form of attractor flow generators in this representation. Other work on this coset space has appeared recently, including [28, 29, 30]. .1 The moduli space M D The 3 d moduli space for N = 2 , d = 4 supergravity coupled to n V vector multiplets is well-studied, for example in [31, 32, 33]. Some of the main results are compiled in the Appendix.Here we briefly review the essential points.The bosonic part of the action is: S = − π Z d x p g (4) h R − g i ¯ j dz i ∧ ∗ d ¯ z ¯ j − F I ∧ G I i (4.1)where I = 0 , ...n V , and G I = ( Re N ) IJ F J + ( Im N ) IJ ∗ F J . For a model endowed with aprepotential F ( X ), N IJ = F IJ + 2 i (Im F · X ) I (Im F · X ) J X · Im F · X (4.2)where F IJ = ∂ I ∂ J F ( X ). We reduce to three dimensions, dualizing the vectors ( ω , A I ) to thescalars ( σ , B I ), and renaming A I as A I . The metric of M ∗ D is then ds = d U · d U + 14 e − U ( d σ + A I d B I − B I d A I ) · ( d σ + A I d B I − B I d A I ) + g i ¯ j ( z, ¯ z ) d z i · d ¯ z ¯ j + 12 e − U [( Im N − ) IJ ( d B I + N IK d A K ) · ( d B J + N JL d A L )] (4.3)It is a para-quaternionic-K¨ahler manifold. Since the holonomy is reduced from SO (4 n V +4)) to Sp (2 , R ) × Sp (2 n V + 2 , R ), the vielbein has two indices ( α, A ) transforming under Sp (2 , R ) and Sp (2 n V + 2 , R ), respectively. The para-quaternionic vielbein is the analyticalcontinuation of the quaternionic vielbein computed in [34]. The explicit form is given in theappendix.For n V = 1, X I = ( X , X ). For our purpose, we choose the prepotential F ( X ) = − ( X ) X (4.4)The metric of M ∗ D with one-modulus is (4.3) with g z ¯ z = y and N and ( Im N ) − being N = (cid:18) − (2 x − iy )( x + iy ) x ( x + iy )3 x ( x + iy ) − x + iy ) (cid:19) Im N − = − y (cid:18) xx x + y (cid:19) The isometries of the M ∗ D descend from the symmetries of the 4D system. The gaugesymmetries in 4D give shifting isometries of M ∗ D , whose associated conserved charges are: q I dτ = J A I = P A I − B I P σ , p I dτ = J B I = P B I + A I P σ , kdτ = J σ = P σ (4.5)19here the momenta { P σ , P A I , P B I } are P σ = 12 e − U ( d σ + A I d B I − B I d A I ) (4.6) P A I = e − U [( Im N ) IJ d A J + ( Re N ) IJ ( Im N − ) JK ( d B K + ( Re N ) KL d A L )] − B I P σ (4.7) P B I = e − U [( Im N − ) IJ ( d B J + ( Re N ) JK d A K )] + A I P σ (4.8)Here τ is the affine parameter defined as dτ ≡ − ∗ sin θdθdφ . ( p , p , q , q ) are the D6-D4-D2-D0 charges, and k the Taub-NUT charge. A non-zero k gives rise to closed time-likecurves, so we will set k = 0 from now on.Note that the time translational invariance in 4D gives rise to the conserved current J U = P U + 2 σJ σ + A I J A I + B I J B I (4.9)where P U = 2 dU . The corresponding conserved charge is the ADM mass: 2 M ADM dτ = J U . The metric (4.3) for the case n V = 1 describes an eight-dimensional manifold with coordi-nates φ n = { u, x, y, σ, A , A , B , B } . This manifold is the coset space G / ( SL (2 , R ) × SL (2 , R )). The root diagram for the Cartan decomposition of G is shown in Figure 1. Thesix roots that lie on the horizontal and vertical axes { L ± h , L h , L ± v , L v } are the six non-compactgenerators of the subgroup H = SL (2 , R ) h × SL (2 , R ) v :[ L h/v , L ± h/v ] = ∓ L ± h/v , [ L + h/v , L − h/v ] = 2 L h/v (4.10)and the two vertical columns of eight roots { a αA } are the basis of the subspace K . { a A , a A } for each A is a spin-1 / SL (2 , R ):[ L h , (cid:18) a A a A (cid:19) ] = (cid:18) − a A a A (cid:19) [ L + h , (cid:18) a A a A (cid:19) ] = (cid:18) a A (cid:19) [ L − h , (cid:18) a A a A (cid:19) ] = (cid:18) − a A (cid:19) And { a α , a α , a α , a α } for each α span a spin-3 / SL (2 , R ):[ L v , a α a α a α a α ] = − a α − a α a α a α [ L + v , a α a α a α a α ] = a α a α a α [ L − v , a α a α a α a α ] = − a α − a α − a α All the commutators can be easily read off from the Root diagram (1), we will only writedown the following ones which will be useful later.[ a , a ] = −
13 [ a , a ] = − L + h [ a , a ] = −
13 [ a , a ] = − L − h (4.11)20 h - a a a a L v + L v3 L h3 L v - a a a a L h + Figure 1: Root Diagram of Cartan Decomposition of G Being semisimple, the algebra of G has the Iwasawa decomposition g = h ⊕ a ⊕ n , where a is the maximal abelian subspace of k , and n is the nilpotent subspace of the positive rootspace Σ + of a . In Figure 2, we show the Iwasawa decomposition of G . The two Cartangenerators in a are { u , y } , and { x , σ , A , A , B , B } span a nilpotent subspace n : n = 0for n ∈ n . a and n together generate the solvable subgroup Solv of G , which act transitivelyon M D = G /SL (2 , R ) × SL (2 , R ). In particular, y generates the rescaling of y , and { u , x , σ , A , A , B , B } generates the translation of { U, x, σ, A , A , B , B } . The modulispace M D can be parameterized by the solvable elements:Σ( φ ) = e U u +(ln y ) y e x x + A I A I + B I B I + σ σ (4.12)The origin of the moduli space a = A = A = B = B = 0 x = 0 y = u = 1 (4.13)correspond to Σ( φ ) = 1.In Fig 2, the isometries are plotted according to their eigenvalues under the two Cartan21 ` B ` B ` A ` A ` x y ux ` A A B B a Figure 2: Root Diagram of Isometry of M D = G / ( SL (2 , R ) × SL (2 , R )). { u , y , x , σ , A , A , B , B } generates the solvable subgroup.generators u and y [31]. { u , y } are related to a αA by : u = −
18 [( a + a ) − ( a + a )] y = 18 [3( a + a ) + ( a + a )] (4.15)The three generators { σ , u , ˆ σ } on the horizontal axis and { x , y , ˆx } on the vertical axis formthe horizontal and vertical SL (2 , R ), respectively. The vertical SL (2 , R ) generate the dualityinvariance. Denote the two vertical columns of eight isometries as ξ ξ ξ ξ ξ ξ ξ ξ ≡ − ˆA B − ˆA B ˆB − A − ˆB A { ξ A , ξ A } for each A span a spin-1 / SL (2 , R ), and { ξ α , ξ α , ξ α , ξ α } for each α span a spin-3 / SL (2 , R ). The matrix representation of u and y are u = Diag [0 , , − , , − , , y = Diag [1 , − , − , − , , ,
0] (4.14) M as the solvable elements, the symmetric matrix S can be written in terms of the eight coordinates φ n , from the solvable elements Σ S ( φ ) = Σ( φ ) S Σ( φ ) T (4.16)The coordinates are read off from the symmetric matrix S . k . The near-horizon geometry of the 4 D attractor is AdS × S , i.e. e − U → p V BH | ∗ τ as τ → ∞ (4.17)In terms of the variable u ≡ e U u → V BH τ − as τ → ∞ (4.18)The solvable element is M = e U u + ... ∼ u u (4.19)As the flow goes to the near-horizon, u → M ( τ ) ∼ u − ℓ/ ∼ τ ℓ (4.20)where − ℓ is the lowest eigenvalue of u . That is, M ( τ ) is a polynomial function of τ .On the other hand, since the geodesic flow is generated by k via M ( τ ) = M (0) e kτ/ (4.21)i.e., M ( τ ) is an exponential function of τ . To reconcile the two statements, k must benilpotent: k ℓ +1 = 0 (4.22)That is, the element in k that generates the attractor flow is nilpotent. Moreover, by lookingat the weights of the fundamental representation of G , we see that ℓ = 2 k ∗ = 0 (4.23)Moreover, the nilpotency of the attractor flow generators guarantees that it is null: k = 0 = ⇒ ( k ) = 0 = ⇒ T r ( k ) = 0 (4.24)which means that k is null. 23 .4 Properties of Attractor Flow The scalar moduli space is parameterized by a symmetric 7 × S which sits in G ,i.e. preserves a non-degenerate three form w ijk such that η is = w ijk w stu w mno ǫ jktumno is ametric with signature (4 ,
3) normalized so that η = 1. To facilitate the comparison with thepure 5 D gravity case we decompose 7 as 3 ⊕ ¯3 + 1 of SL (3) and pick as non-zero componentsof w the 3 ∧ ∧
3, ¯3 ∧ ¯3 ∧ ¯3 and 3 ⊗ ¯3 ⊗ w = dx ∧ dx ∧ dx + dy ∧ dy ∧ dy − √ dx a ∧ dy a ∧ dz (4.25)The resulting expression for η is η = dx a dy a − dz (4.26)We know that k must be an element of G , hence also of SO (4 , S kS = k T S k S = ( k ) T (4.27)In this base a G Lie algebra element is given as k = A j i ǫ i j k v k √ w i ǫ i j k w k − A i j −√ v i −√ v j √ w j (4.28)Here A is a traceless 3 × S is a symmetric element in G with signature { , − , − , , − , − , } , i.e. S = M S M T with S = η η
00 0 1 = Diag (1 , − , − , , − , − ,
1) (4.29)where η is the one for pure gravity.If the gauge field is turned off, then S is block diagonal S | F =0 = S gr S − gr
00 0 1 (4.30)where S gr is the same as the one for pure 5 D gravity. Turning on a non-zero 5 D vector fieldcorresponds to a more general S : S = e k T ( S | F =0 ) e k (4.31)24ith k a G Lie algebra matrix with w equal to the fifth component of the gauge field, v equal to the time component of the gauge field and w equal to the scalar dual to thethree-dimensional part of the gauge field.In this representation, ( x, y ) can be extracted from symmetric matrix S via: x ( τ ) = − S ( τ ) S ( τ ) y = S ( τ ) S ( τ ) − S ( τ ) S ( τ ) (4.32)And u via: u = 1 p S ( τ ) S ( τ ) − S ( τ ) (4.33)The 4 D gauge currents sit in J = S − ∇ S , where J ( J ) is again the electric(magnetic)current for the KK photon, J the timelike NUT current, and J ( J ) the electric(magnetic)current for the reduction of the 5 D gauge field. J = − J σ J = √ J A J = 23 J A J = −√ J B J = √ J B (4.34)Moreover, J − J = 2 J U (4.35)We use Q to denote the charge matrix, where it relates to the D-brane charge { p , p , q , q } and the vanishing NUT charge k by( Q , Q , Q , Q ) = ( √ p , −√ p , q , √ q ) Q = − k = 0 (4.36)Since k is nilpotent: k = 0, S = e kτ S = (1 + kτ + 12 k τ ) S (4.37)The AdS × S near-horizon geometry of the 4 D attractor dictates u = V BH | ∗ τ − as τ → ∞ .Therefore, the flow generator k can be obtained by k = 2 V BH | ∗ ( uS | u → ) S (4.38)Computing k using S constructed from the solvable elements Σ( φ ) shows that k is of ranktwo, its Jordan form has two blocks of size 3. It can be written as k = X a,b =1 , v a v Tb c ab S (4.39)25ith v a null and orthogonal to each other: v a · v b ≡ v Ta S v b = 0, and c ab depends on theparticular choice of k . Thus k can be expressed as: k = X a =1 , ( v a w Ta + w a v Ta ) S (4.40)where each w a is orthogonal to both v a : w a · v b = 0, and w a satisfy w a · w b = c ab (4.41)Parallel to the pure gravity case, the single-centered attractor flow is constructed as S ( τ ) = e K ( τ ) S , where we choose K ( τ ) to have the same properties as the generator k : K ( τ ) = 0 and K ( τ ) rank two (4.42)This determines K ( τ ) = kτ + g where k = X a =1 , [ v a w Ta + w a v Ta ] S and g = X a =1 , [ v a m Ta + m a v Ta ] S (4.43)where the two 7-vectors m a ’s are orthogonal to v a and contain the information of asymptoticmoduli. Using [[ k, g ] , g ] = 0, the current is reduced to J = S ( k + [ k, g ]) S r ˆ ~r (4.44)from which we obtain v a and w a in terms of the charges and the asymptotic moduli. G / ( SL (2 , R ) × SL (2 , R )) Model
We now explicitly construct the generators of single-centered attractor flows. We start withthe BPS flow which is associated with a specific combination of the coset algebra generators a αA . It can be derived from the condition of preservation of supersymmetry. We thenconstruct the non-BPS attractor flow generator in analogy with the BPS one. In Section5.2, we write k BP S and k nonBP S in terms of the v a and w a vectors. This form will be especiallyhelpful in generalizing to the multi-centered case. k BP S using supersymmetry
To describe BPS trajectories it is useful to remember that the stabilizer of S in G is SO (1 , × SO (1 , G which are antisymmetric after26ultiplication by S . Geodesics are exponentials of elements that are symmetric after mul-tiplication by S . Such elements sit in a ( , ) representation of SO (1 , × SO (1 , SO (1 , a αA under the two SO (1 ,
2) groups, a BPS trajectory is generated by k BP S = a αA C A z α . (5.1)The twistor z and the coefficients C A are fixed in terms of the charges of the extremal BPSblack hole and the condition of zero time-like NUT charge.To see why this is true, expand the coset element k BP S that generates the BPS attractorflow using a αA : k BP S = a αA C αA (5.2)where C αA are conserved along the flow. On the other hand, the conserved currents in thehomogeneous space are constructed by projecting the one-form valued Lie algebra g − · dg onto k , which gives the vielbein in the symmetric space: g − dg | k = a αA V αA (5.3)where V αA is conserved: ddτ (cid:16) V αAa ˙ φ a (cid:17) = 0 (5.4)Therefore, the expansion coefficients of k BP S are C αA = V αAa ˙ φ a (5.5)In terms of the vielbein, the supersymmetry condition that gives the BPS geodesics arewritten as [33]: V αA z α = 0 (5.6)That is: V αAa ˙ φ a z α = 0 = ⇒ C αA z α = 0 (5.7)Define z α = ǫ αβ z β , C αA = C A z α (5.8)Therefore, the coset element k BP S is expanded by the coset algebra basis a αA as k BP S = a αA C A z α .Note that k BP S has five parameters ( C A , z ) where A = 1 , . . . ,
4. As will be shown later, z can actually be determined by ( C A ) and moduli at infinity. So the geodesic generatedby k BP S is indeed a four-parameter family. It is easy to show that k BP S is null, but moreimportantly, it is nilpotent: k BP S = 0 (5.9)As will be shown later, k BP S indeed gives the correct BPS attractor flow.27 .1.2 Constructing k NonBP S
To construct the non-BPS attractor flow, one needs to find an element in the coset algebradistinct from k BP S that satisfies: k NonBP S = 0 (5.10)The hint again comes from the BPS generator. Note that k BP S = a αA P A z α can be writtenas: k BP S = e − zL − h k BP S e zL − h (5.11)where k BP S spans only the right four coset generators a A : k BP S = a A C A (5.12)That is, k BP S is generated by starting with the element spanning the four generators anni-hilated by the horizontal SL (2) raising operator L + h , then conjugating with the horizontal SL (2) lowering operator L − h . And it is very easy to show that ( k BP S ) = 0 which proves( k BP S ) = 0.In G /SL (2 , R ) , there are two third-degree nilpotent generators in total. And sincethere are only two SL (2 , R )’s inside H , a natural guess for a non-BPS solution is to look atvectors with fixed properties under the second SL (2 , R ) group. An interesting condition is tohave positive charge under some rotation of L v , i.e. an SL (2 , R ) rotation of P A =1 , a αA C αA .Therefore, this suggests to us to start with the element spanning the four generators annihi-lated by the square of the vertical SL (2 , R ) raising operator ( L + v ) and then conjugate withthe vertical SL (2 , R ) lowering operator L − v : k NonBP S ( z ) = e − zL − v k NonBP S e zL − v (5.13)where k NonBP S = a αa C αa where α, a = 1 , . (5.14)And one can show that: ( k NonBP S ) = 0 which proves ( k NonBP S ) = 0. Moreover, ( k NonBP S ) is rank two.As long as one can pick the coefficients C αA and the twistor z that describes the SO (1 , .2 Properties of Flow Generators k BP S
We now turn to solving for v a and w a in (4.40) in terms of C A and z . First, from (4.39)we know that the null space of k is five-dimensional and the v a span the two-dimensionalcomplement of this null space. For k BP S = a αA C A z α the null space of ( k BP S ) does notdepend on C A . Therefore, the v a depend only on the twistor z = z /z .Recall that we are using the basis where k has the form (4.28). From inspection of k BP S ,we find that ( v , v ) can always be chosen to have the form: v = ( V , − η V , v = ( − V , η V , √
2) (5.17)where η is a 3d metric of signature (1 , − , − V , V are two three-vectors with V · V = 0 V · V = 0 V · V = − v , v ) forms a new set of ( v , v ), this means in particularthat any v + cv gives a new v . Looking at the forms of ( v , v ), we see that V is definedup to a shifting of V as V = V − cV .An explicit computation shows that V and V are given by the twistor z and u as V = ( z ) + ( z ) ( z ) − ( z ) z z V = 1 z u − z u z u + z u z u − z u z u + z u . (5.19)where the twistor u = u u is related to c by u = − cz − cz z (5.20)The twistor representation of V and V are V αβ = 2 z α z β V αβ = z α u β + z β u α (5.24) When solving for ( v, w ), there are some freedom on the choice of ( v , v ) and ( w , w ): firstly, a rotationalfreedom ( v , v ) → ( v , v ) (cid:18) R R R R (cid:19) and ( w , w ) → ( w , w ) (cid:18) R R R R (cid:19) (5.15)where R is orthogonal: RR T = 1 Secondly, a rescaling freedom: v a → rv a and w a → r w a (5.16) With the inner product of three-vectors defined as v a · v b = v Ta η v b (5.21) z u − z u to be 1. Note that for the BPScase, the twistor u is totally arbitrary.Now we solve for w a . The condition that w a are orthogonal to v a dictates that they havethe form: w = ( W , η W , w = ( W , η W ,
0) (5.25)where W and W are linearly independent, and are related to the charges by w a · w b = c ab : W · W = 12 c W · W = 12 c W · W = 12 c (5.26)Recall that V is defined up to a shift by V : V = V − cV . The consequence is that W isdefined up to a shift by W : W = W + cW . Note that the numerical factors in front of V and W are opposite. Write down ( W , W ) in terms of ( C A , z ): W = 14 z ( C + C ) + ( C + C ) z ( C − C ) + ( C − C ) z C + 2 C z W = 12 − ( C + C ) + ( C + C ) z − ( C − C ) + ( C − C ) z − C + 2 C z (5.27)The twistor representations of W and W are W = (cid:18) C u − C u C u − C u C u − C u C u − C u (cid:19) W = (cid:18) C z − C z C z − C z C z − C z C z − C z (cid:19) (5.28)Define the totally symmetric P αβγ : P = C P = C P = C P = C (5.29)Then the three-vectors ( W , W ) span the four dimensional space( W α , W α ) BP S = ( P αβγ u γ , P αβγ z γ ) (5.30) k NonBP S
The form of v a for the non-BPS case is only slightly different from the BPS case: the twovectors v a can be chosen to have the form: v = ( V , η V , v = ( V , − η V , √
2) (5.31)
The twistor representation of a three-vector v = ( x, y, z ) is σ v = xσ + yσ + zσ = (cid:18) x + y zz x − y (cid:19) (5.22)It’s length is v T η v = det( σ v ) = x − y − z (5.23) V , V are two three-vectors satisfying the same condition as the BPS ones (5.18).Again, the vectors V and V can be written as (5.19), and the twistor representations aregiven in (5.24) with one major difference: u is no longer arbitrary, but is determined by C αA as: u = u u = C C (5.32)The form of ( w , w ) are also slightly different from the BPS one (5.25) w = ( W , − η W , w = ( W , η W ,
0) (5.33)The ( W , W ) can be written in terms of ( C αa , z ) thus: W = 12( C z − C z ) [( C C − C C ) z + ( C ) ] + [ C C − C C + ( C ) ] − [( C C − C C ) z + ( C ) ] + [ C C − C C + ( C ) ]2[( C C − C C ) z + C C ] (5.34) W = − z [ C z + (3 C − C ) z − C ] + [ C z + C − C ] − z [ C z + (3 C − C ) z − C ] + [ C z + C − C ]2[ C z + (2 C − C ) z − C ] (5.35)In terms of u = u u = C C , the twistor representation of W and W are: W αβ = u α u β + ( C u − C u ) z α z β (5.36) W αβ = ( z α u β + u α z β ) + ( C − C z − u ) z α z β (5.37)As a consequence, the precise value of u is an extra constraint on the vectors w a , and thereis only a three-dimensional space of them, with ( W , W ) a linear combination of (0 , V ),( V ,
0) and ( u α u β , V ).( W , W ) NonBP S = m (0 , V ) + n ( V ,
0) + ℓ ( u α u β , V ) (5.38) G / ( SL (2 , R ) × SL (2 , R )) Model
Now that we have completely characterized the generators of single-centered attractor flow,we can lift the geodesics to four-dimensional black hole solutions. After some calculationaleffort, we find that the BPS solution is given in terms of harmonic functions. Next, we showthat the non-BPS case is qualitatively different, and the final solutions cannot be formulatedso simply. 31 .1 BPS Attractor Flow D We have already noted that the flow starting from generic asymptotic moduli ( x , y ) isgenerated by M ( τ ) = e ( kτ + g ) / , with g defined before in (4.43). The matrix g has the sameform as k . Therefore, in the BPS case, it has the expansion g BP S = a αA z α G A (6.1)where the twistor z is the same as the one in k BP S . The flow of ( x, y ) and u can be extractedfrom the symmetric matrix S ( τ ) = M ( τ ) S M ( τ ) T via (4.32) and (4.33). Using k BP S = 0and g = 0, S ( τ ) = S e kτ + g is a quadratic function of τ : S ( τ ) BP S = α B ( τ ) + β B ( τ ) − S ( τ ) BP S = γ B ( τ ) + δ B ( τ ) (6.2) S ( τ ) BP S = ǫ B ( τ ) + ζ B ( τ ) − { α B ( τ ) , γ B ( τ ) , ǫ B ( τ ) } are quadratic functions of τ , and { β B , δ B , ζ B } are linear func-tions of τ : α B ( τ ) = z ( H H − ( H ) ) + z ( H H − H H ) + ( H H − ( H ) ) β B ( τ ) = ( H − H ) z + ( H − H ) γ B ( τ ) = − (cid:0) ( H H − H H )( z −
1) + 2( H ( H + H ) − H ( H + H )) z (cid:1) δ B ( τ ) = −
12 (( H + H ) z − ( H + H )) ǫ B ( τ ) = ( H H − ( H ) ) z + ( H H − H H ) z + ( H H − ( H ) ) ζ B ( τ ) = 2( H z + H ) (6.3)where H A is a linear function of τ defined as H A ( τ ) ≡ C A τ + G A .The attractor values are reached when τ → ∞ along the geodesic: x ∗ BP S = − ( k S ) ( k S ) y ∗ BP S = p ( k S ) ( k S ) − ( k S ) ( k S ) (6.4)and u ∗ BP S = 1 p ( k S ) ( k S ) − ( k S ) (6.5)The asymptotic moduli ( x , y ) can be expressed in terms of ( G A , z ) by extraction from S = e g S . 32 x * , y * L H x , y L H x , y LH x , y LH x , y L - - Figure 3: Sample BPS flow. The attractor point is labeled ( x ∗ , y ∗ ). The initial points ofeach flow are given by ( x = 1 . , y = 0 . , ( x = 2 , y = 4) , ( x = − . , y = 0 . , ( x = − , y = 3) 33sing this technique, one can construct all BPS single-centered black holes. The chargesof each black hole can be read off from the current J using (4.34). One example is givenin Figure 3, where we parametrically plot x ( τ ) and y ( τ ) for a BPS black hole with charges( p , p , q , q ) = (5 , , , −
3) and attractor point ( x ∗ , y ∗ ) = (0 . , . D solution for given set of charges To get the solution for a specific set of charges requires more effort. In this section, wepresent the analytical result for any set of charges ( p I , q I ).The ten parameters in k BP S and g are { z, u, C A , G A } , among which the twistor u isarbitrary, corresponding to the freedom of the shift by ( v , w ) in the definition of ( v , w ):( v , w ) → ( v + cv , w − cw ). The remaining true parameters are enough to parameterizethe four D-brane charges ( p I , q I ) and the arbitrary asymptotic moduli ( x , y ) under thecondition of vanishing Taub-NUT charge and fixing u = 1. We now solve for k BP S and g for the given D-brane charges and ( x , y ), using the eight constraints, namely, 4 chargesand zero Taub-NUT charge plus 3 asymptotic moduli, to fix C A and G A , leaving the othertwistor z unfixed. For the sake of simplicity, we will denote k BP S by k for the rest of this section. Then thecurrent J ( Q ) = k T r gives the five coupled equations: Q = S ( k + 12 [ k, g ]) S (6.6)In order to show that the BPS flow can be expressed in terms of harmonic functions: H ( τ ) = Qτ + h with Q = ( p I , q I ) and h = ( h I , h I ) (6.7)we will solve g in terms of h instead of ( x , y ). The four h ’s relate to the asymptotic moduliby x = x ∗ ( Q → h ) y = y ∗ ( Q → h ) u = u ∗ ( Q → h ) (6.8)and there is one extra degree of freedom to be fixed later.To evaluate [ k, g ], we first use the commutation relation (4.11) to obtain[ a A C A , a B G B ] = h C, G i ( − L + h ) (6.9) The discriminant of the charge (5 , , , −
3) is positive, so this is indeed a BPS solution. The twistor z can be left unfixed because we will not specify the asymptotic values of the scalars withtranslational invariance, namely, ( { a, A I , B I } ). Fixing them can fix the twistor z . C A and G A is defined as h C, G i ≡ C G − C G +3 C G − C G .Then twisting Eq (6.9) with the twistor z as in (5.11) gives the commutator of k and g withthe same twistor z : [ k, g ] = [ a αA z α C A , a βB z β G B ] = h C, G i Θ (6.10)where Θ is defined as Θ ≡ − z e − zL − h L + h e zL − h . On the other hand, using (4.43),[ k, g ] = ( v v T − v v T ) S ( w · m − w · m ) (6.11)Θ can also be written as Θ = ( v v T − v v T ) S , and we can check that ( w · m − w · m ) = h C, G i .First, separate from G A the piece which has the same dependence on ( h, z ) as C A on( Q, z ): G A = G Ah + E A with G Ah ≡ C A ( Q → h, z ) (6.12)That is, g contains two pieces: g = g h + Λ with g h = a αA z α G Ah and Λ = a αA z α E A (6.13)We need to solve for E A .There are three constraints from (6.8). The ( x , y ) and u are extracted from the sym-metric matrix S = e g S via (4.32) and (4.33). On the other hand, requiring (6.8) gives( x , y , u ) in terms of h : x = − ( g h S ) ( g h S ) y = p ( g h S ) ( g h S ) − ( g h S ) ( g h S ) u = 1 p ( g h S ) ( g h S ) − ( g h S ) (6.14)Therefore, defining Π ≡ ( e g − g h ) S , Π has to satisfy three constaints:Π = Π = Π = 0 (6.15)in order for (6.14) to hold for arbitrary h . Using the unfixed degree of freedom in h ’s to set h C, G h i = 0, (6.6) becomes Q = S ( k + 12 [ k, Λ]) S (6.16)The zero Taub-NUT charge condition in (6.16) imposes the fourth constraint on Λ: the(3,2)-element of S ( k + [ k, Λ]) S for arbitrary k has to vanish. Combining with (6.15), wehave 4 constraints to fix E A to be: E = − E = −
11 + z E = − E = z z (6.17)35he remaining 4 conditions in the coupled equations (6.16) determine C A in the BPSgenerator k BP S = a αA z α C A to be C = √ − q − q z − p z + p z (1 + z ) C = √ − q − (2 p − v ) z + ( p + 2 q ) z + p z (1 + z ) C = √ − p + ( p + 2 q ) z + (2 p − v ) z − q z (1 + z ) C = √ p + 3 p z − q z + q z (1 + z ) (6.18)The G Ah are then determined by G Ah = a αA z A C A ( Q → h, z ). Using the solution of C A and G Ah , we see the product h C A , G Ah i is proportional to the symplectic product of ( p I , q I ) and( h I , h I ): h C A , G Ah i = 21 + z < Q, h > where < Q, h > ≡ p h + p h − q h − q h (6.19)The condition h C A , G Ah i = 0 is then the integrability condition on h : < Q, h > = p h + p h − q h − q h = 0 (6.20)Substituting the expressions of C A and G A in terms of ( p I , q I ) into (6.2), we obtain theBPS attractor flow in terms of the charges ( p I , q I ). In particular, the attractor values are x ∗ BP S = − p q + p q p ) + p q ] y ∗ BP S = p J ( p , p , q , q )2[( p ) + p q ] (6.21)where J ( p , p , q , q ) is the quartic E invariant: J ( p , p , q , q ) = 3( p q ) − p q )( p q ) − ( p q ) − p ) q + 4 p ( q ) (6.22)thus J ( p , p , q , q ) is the discriminant of charge. The attractor values match those fromthe compactification of Type II string theory on diagonal T , with q → q . The attractorvalue of u is u ∗ BP S = 1 p J ( p , p , q , q ) (6.23)The constraint on h from u = 1 is then J ( h , h , h , h ) = 1.Now we will show that the geodesic we constructed above indeed reproduces the attractorflow given by replacing charges by the corresponding harmonic functions in the attractor36oduli. Using the properties of Λ, we have proved that, in terms of k and g , the flow of( x, y ) can be generated from the attractor value by replacing k with the harmonic function kτ + g h : x ( τ ) = x ∗ ( k → kτ + g h ) y ( τ ) = y ∗ ( k → kτ + g h ) (6.24)Since k and g h have the same twistor z , this is equivalent to replacing the C A with theharmonic function C A τ + G Ah while leaving the twistor z fixed: x ( τ ) = x ∗ ( C A → C A τ + G Ah , z ) y ( τ ) = y ∗ ( C A → C A τ + G Ah , z ) (6.25)Since C A is linear in Q and G Ah linear in h , and since z drops out after plugging in the solutionof C A in terms of ( Q, z ) and G Ah in terms of ( h, z ), this proves that the flow of ( x , y ) is givenby replacing the charges in the attractor moduli by the corresponding harmonic functions: x BP S ( τ ) = x ∗ BP S ( Q → Qτ + h ) y BP S ( τ ) = y ∗ BP S ( Q → Qτ + h ) (6.26)The integrability condition < Q, h > = 0, in terms of H = Qτ + h , is < H, dH > = 0 (6.27) D Similar to the BPS attractor flow, the non-BPS flow is generated by M ( τ ) = e ( kτ + g ) / ,and ( x, y ) can be extracted from the symmetric matrix S ( τ ) by (4.32) and the relevantelements of S ( τ ) are given by (6.2). The only difference is that now { α B , β B , γ B , δ B , ǫ B , ζ B } are changed into the non-BPS counterparts { α NB , β NB , γ NB , δ NB , ǫ NB , ζ NB } , which can bewritten in terms of H αa ≡ C αa τ + G αa and z : α NB ( τ ) = − (( H H − H H )( z − + ( H ) z − zH H + ( H ) ) β NB ( τ ) = ( z − H − H z ) − z H + 2 zH + H γ NB ( τ ) = ( z − z ( H H − H H ) + H H ) + z (( H ) − ( H ) ) δ NB ( τ ) = 12 ( z (1 + z ) H + z (3 H − H ) − H z + ( H − H )) ǫ NB ( τ ) = (4 H H − H ( H + 4 H )) z − H H z − ( H ) ζ NB ( τ ) = 2( H z + (2 H + H ) z + H ) (6.28)Note that H H = u is fixed, independent of τ . The non-BPS flow written in terms of ( H αa , z )has the same simple form as the BPS flow, i.e. the scalars are rational functions with boththe numerator and denominator being only quadratic. This is due to the nilpotency of the37enerator: k = 0. Again, the attractor values are reached when τ → ∞ , and the asymptoticmoduli can be expressed in terms of ( G αa , z ) by extraction from S = e g S .Unlike the BPS case, there are only eight parameters in k NonBP S and g NonBP S : the twotwistors ( z, u ) and ( C αa , G αa ) under the constraints that u = C C = G G (6.29)Therefore, while k BP S and g BP S can parameterize arbitrary ( p I , q I ) and ( x , y ) while leaving( z, u ) free, all the parameters in k NonBP S and g NonBP S , including ( z, u ), will be fixed. H x * , y * L H x , y L H x , y LH x , y L H x , y LH x , y L - Figure 4: Sample non-BPS flow. The attractor point is labeled ( x ∗ , y ∗ ). The initial points ofeach flow are given by: ( x = 0 . , y = 5 . , ( x = 1 . , y = 0 . , ( x = − . , y = 0 . , ( x = 1 . , y = 1 . , ( x = − . , y = 0 . C αa is x ∗ NonBP S = γ NB ( H αa → C αa ) ǫ NB ( H αa → C αa ) y ∗ NonBP S = p α NB ( H αa → C αa ) ǫ NB ( H αa → C αa ) − γ NB ( H αa → C αa ) ǫ NB ( H αa → C αa ) (6.30)38ith z given by12 (cid:0) − C − C + z (cid:0) ( z − C + 3 z ( C + C ) + 6 C (cid:1)(cid:1) = 0 (6.31)As in the BPS case, the charges of the black hole are read off from the current J using (4.34).We have checked that the attractor point is a non-supersymmetric critical point of the blackhole potential V BH = | Z | + | DZ | : ∂V BH = 0 and DZ = 0 (6.32)It reproduces the results reported in the literature [9]. An example of the non-BPS attractorflow is shown in Figure 4, with ( p , p , q , q ) = (5 , , ,
3) and attractor point ( x ∗ , y ∗ ) =( − . , . J (5 , , / , <
0, so this is indeed a non-BPS black hole.Unlike the BPS attractor flow, all the non-BPS flows starting from different asymptoticmoduli have the same tangent direction at the attractor point. The mass matrix of the black-hole potential at a BPS critical point has two identical eigenvalues, whereas the eigenvaluesat a non-BPS critical point are different. The common tangent direction for the non-BPSflows corresponds to the eigenvector associated with the smaller mass. D solution for given set of charges We now discuss how to construct the non-BPS black hole solution for a specific set of charges( p I , q I ).One major difference between the non-BPS case and the BPS case is that[ k NonBP S , g
NonBP S ] = 0 (6.33)automatically, since the forms of ( w , w ) and ( m , m ) guarantee that w · m = w · m = 0.Thus the charge equation (6.6) becomes simply Q NonBP S = S ( k NonBP S ) S (6.34)These five coupled equations determine the two twistors ( z, u ) and C αa in terms of ( p I , q I ).Similar to the BPS case, the four equations which determine the D-brane charge allow us towrite C αa in terms of the charges ( p , p , q , q ) and the twistor z via C = ( − q + 6( p − q ) z + 4( p + q ) z − p z ) √ z ) C = ( p + q ) − p − q ) z − ( p + 5 q ) z + 2 p z √ z ) C = ( p − q ) − q z + 4(3 p − q ) z + (3 p + q ) z √ z ) C = 2 p + ( p + 5 q ) z − p − q ) z − ( p + q ) z √ z ) (6.35)39nd u = C C . In contrast to the BPS case, the G αa do not enter the equations and thereforecannot be used to eliminate the twistor z . Requiring the Taub-NUT charge to vanish givesthe following degree-six equation for the z : p z + 6 p z − (3 p + 4 q ) z − p − q ) z + (3 p + 4 q ) z + 6 p z − p = 0 (6.36)The three parameters in g NonBP S , namely, G αa with the constraint G G = u are then fixedby the given asymptotic moduli ( x , y ) and u = 1.Similar to the BPS flow, the full non-BPS flow can be generated from the attractorvalue by replacing C αa with the harmonic function H αa ( τ ) = C αa τ + G αa , while leaving z unchanged as in (6.25). However, there are two major differences. First, the harmonicfunctions H αa have to satisfy the constraint H ( τ ) H ( τ ) = u = C C = G G (6.37)Note that this does not impose any constraint on the allowed asymptotic moduli since thereare still three degrees of freedom in G αa to account for ( x , y , u ). We will see later that itinstead imposes a stringent constraint on the allowed D-brane charges in the multi-centerednon-BPS solution.Secondly, unlike the BPS flow, replacing C αa in the attractor moduli by the harmonicfunction H αa ( τ ) is not equivalent to replacing the charges Q with H = Qτ + h as in (6.26).The twistor z here is no longer free, but is determined in terms of the charges as a root of thedegree-six equation (6.36), so replacing Q by Qτ + h , for generic Q and h , would not leave z invariant. Therefore, the generic non-BPS flow cannot be given by the naive harmonicfunction procedure, as proposed by Kallosh et al [19]. Next, we will define the subset of theNonBPS single-centered flow that can be constructed by the harmonic function procedure.When the attractor has only D − D Q = (0 , p , , q ), (6.36) has aroot z = 0, which is independent of the value of charges. If the asymptotic moduli h is alsoof the form of h = (0 , h , , h ), replacing Q by Q τ + h would leave the solution z = 0invariant. Now we will use the duality symmetry to extend the subset to a generic chargesystem with restricted asymptotic moduli.The one-modulus system can be considered as the STU attractor with the three moduli( S, T, U ) identified. Since the STU model has SL (2 , Z ) duality symmetry at the level ofthe equations of motion, the one-modulus system has an SL (2 , Z ) duality symmetry comingfrom identifying the three SL (2 , Z ) symmetries of the STU model. That is, the one-modulussystem is invariant under the following element of SL (2 , Z ) ˆΓ = (cid:18) a bc d (cid:19) ⊗ (cid:18) a bc d (cid:19) ⊗ (cid:18) a bc d (cid:19) with ad − bc = 1 (6.38)40he modulus z = x + iy transforms as z → ˆΓ z = az + bcz + d (6.39)and the transformation on the charges is given by [35]. A generic charge ( p , p , q , q ) canbe reached by applying the transformation ˆΓ on a D − D D − D Q Q = p q → ˆΓ Q = − c (3 d p + c q ) d (2 bc + ad ) p + ac q b ( bc + 2 ad ) p + a cq ) a (3 b p + a q ) The solution of the twistor z with the new charges ˆΓ Q is z = a ± √ a + c c (6.40)independent of the D − D Q ,there exists a transformation ˆΓ Q such that Q = ˆΓ Q Q for some Q . The twistor z remainsinvariant under Q → Qτ + ˆΓ Q h for arbitrary h . We conclude that the non-BPS single-centered black holes that can be constructed via the naive harmonic function procedure areonly those with ( Q, h ) being the image of a single transformation ˆΓ on the ( Q , h ) from a D − D x NB ( τ ) = x ∗ NB (ˆΓ Q → ˆΓ Q τ + ˆΓ h ) y NB ( τ ) = y ∗ NB (ˆΓ Q → ˆΓ Q τ + ˆΓ h ) (6.41)Since we are considering arbitrary charge system, the constraint is on the allowed values of h . G / ( SL (2 , R ) × SL (2 , R )) model As proven in the pure gravity system, the multi-centered attractor solutions are given byexponentiating the matrix harmonic function K ( ~x ): S ( ~x ) = e K ( ~x ) S (7.1)with K ( ~x ) having the same properties as the generator k : K ( ~x ) = 0 and K ( ~x ) rank two (7.2)41e now describe how to formulate K ( ~x ) for multi-centered solutions in G .The K ( ~x ) satisfying all the above constraints is constructed as K ( ~x ) = X a =1 , [ v a W a ( ~x ) T + W a ( ~x ) v Ta ] S (7.3)with v a being the same two constant null vectors in k , and the multi-centered harmonicfunction W a ( ~x ) = X i ( w a ) i | ~x − ~x i | + m a (7.4)is everywhere orthogonal to v a . The two 7-vectors ( m , m ) contain the information ofasymptotic moduli and has the same form as ( w , w ). Write K ( ~x ) as K ( ~x ) = P i k i | ~x − ~x i | + g where k i = X a =1 , [ v a ( w a ) Ti + ( w a ) i v Ta ] S and g = X a =1 , [ v a m Ta + m a v Ta ] S (7.5)Since v only depends on the twistor ( z, u ), and ( w , w ) are linear in C A or C αa , the abovegenerating procedure is equivalent to replace C A or C αa by the multi-centered harmonicfunctions while keeping the twistor ( z, u ) fixed.Next we discuss the properties of the BPS multi-centered attractor solution and non-BPSones separately, since they are very different in character. In constrast with the multi-centered solutions in pure gravity, now the second term of thecurrent J = ∇ K + [ ∇ K, K ] does not vanish automatically since[ k BP Si , k
BP Sj ] = 0 and [ k BP Si , g
BP S ] = 0 (7.6)Therefore, the centers are no longer free, and we cannot simply read off the charges from J .Instead, we need to solve for C Ai and G A in a set of 5 N coupled equations. The divergenceof the current is ∇ · J = 4 π X i δ ( ~x − ~x i ) S ( k i + 12 [ k i , g ] + 12 X j [ k i , k j ] | ~x i − ~x j | ) S (7.7)Using Q i to denote the charge matrix which relates to the D-brane charge { p , p , q , q } i asin (4.36), and with Q as the vanishing NUT charge, we have 5 N coupled equations from Q i = π R i ∇ · J : Q i = S ( k i + 12 [ k i , g ] + 12 X j [ k i , k j ] | ~x i − ~x j | ) S (7.8)42he generators of the multi-centered BPS attractor solution { k i } and g have 4( N + 1) + 2parameters in total: the two twistors ( z, u ) and { C Ai , G A } . On the left hand side of (7.8),there are also 3 N − ~x i . On the otherhand, a generic N -centered attractor solution has 4 N D-brane charges ( p I , q I ), and threeadditional constraints from the asymptotic moduli ( x , y ) and u = 1. As we will show, likethe single-centered BPS solution, the three asymptotic moduli, together with the vanishingof the total Taub-NUT charge, determine the 4 G A inside g . Moreover, as in the single-centered case, we can solve C Ai in terms of the 4 D D-brane charges Q i while leaving ( z, u )unfixed. The remaining N − N − N -centers ~x i .First, integrating over the circle at the infinity, P i Q i = π R ∇ · J gives the sum of theabove N matrix equations: Q tot = X i Q i = S ( X i k i + 12 [ X i k i , g ]) S (7.9)which is the same as the one for the single-center attractor with charge Q tot . This determines g to be g = g h + Λ, same as the one for single-centered attractor in (6.13). As in the singlecentered case, h is fixed by the asymptotic moduli ( x , y ) by x = x ∗ BP S ( Q → h ) y = y ∗ BP S ( Q → h ) (7.10)and the two constraints: < Q tot , h > = 0 J ( h , h , h , h ) = 1 (7.11)We have used the vanishing of the total Taub-NUT charge to determine Λ. Next, we willuse the remaining coupled 5 N − N { C Ai } and impose N − N centers { ~x i } where i = 1 , · · · , N .The tentative solutions of C Ai are given by (6.18) with ( p I , q I ) replaced by ( p Ii , q I,i ). Theflow generator of each center k i is then k i = a αA z α C Ai . Substituting the solution of k i and g = g h + Λ into (7.8), and using[ k i , g h ] = 2 < Q i , h > Θ [ k i , k j ] = 2 < Q i , Q j > Θ (7.12)where all the k i ’s and g h have the same value for the twistor z , we get Q i = S ( k i + < Q i , h > Θ + 12 [ k i , Λ] + X j < Q i , Q j > | ~x i − ~x j | Θ) S (7.13)Just as in the single-centered case, the solution of k i and the form of Λ guarantee that Q i = S ( k i + 12 [ k i , Λ]) S (7.14)43e see that as long as the following integrability condition is satisfied: < Q i , h > + X j < Q i , Q j > | ~x i − ~x j | = 0 (7.15)the k i and g given above indeed produce the correct multi-centered attractor solution. Justlike in the single-centered case, the multi-centered solution flows to the correct attractormoduli ( x ∗ i , y ∗ i ) near each center, independent of the value of z . It also follows that themulti-centered solution can be generated by replacing the charges inside the attractor valueby the multi-centered harmonic function: x BP S ( ~x ) = x ∗ BP S ( Q → X i Q i | ~x − ~x i | + h ) y BP S ( ~x ) = y ∗ BP S ( Q → X i Q i | ~x − ~x i | + h ) (7.16)The sum of the N equations in the integrability condition (7.15) reproduces the constrainton h : < Q tot , h > = 0. Thus the remaining N − N − N centers { ~x i } with i = 1 , · · · , N . From (A.5) and (4.6), wesee that ∗ d ω is given by J . Defining the angular momentum ~J by ω i = 2 ǫ ijk J j x k r as r → ∞ (7.17)we see that there exists a nonzero angular momentum given by ~J = 12 X i We thank A. Neitzke, J. Lapan and A. Strominger for helpful discussions. The work issupported by DOE grant DE-FG02-91ER40654. M.P. is also supported by an NSF GraduateFellowship. A Derivation of the moduli space M D Here we briefly review the derivation of the 3 d moduli space M D from the c ∗ -map of the4 D supergravity coupled to n V vector multiplets [31] [32] [33].The bosonic part of the action for the N = 2 supergravity coupled to n V vector-multipletsis: S = − π Z d x p g (4) h R − g i ¯ j dz i ∧ ∗ d ¯ z ¯ j − F I ∧ G I i (A.1)where the ranges of the indices are i, j = 1 , . . . , n V and I = 0 , , . . . , n V , and G I =47 Re N ) IJ F J + ( Im N ) IJ ∗ F J . The complex symmetric matrix N IJ is defined by F I = N IJ X J D i F I = N IJ D i X J (A.2)For model endowed with a prepotential F ( X ), N IJ = F IJ + 2 i (Im F · X ) I (Im F · X ) J X · Im F · X (A.3)where F IJ = ∂ I ∂ J F ( X ).After reduction on the time-like isometry, the action is S = − π R dt R d x L . The 3Dlagrangian L has three parts: L = L gravity + L moduli + L e.m where L gravity = − √ g R + d U ∧ ∗ d U − e U d ω ∧ ∗ d ω L moduli = g i ¯ j d z i ∧ ∗ d ¯ z ¯ j (A.4) L e.m. = 12 e − U ( Im N ) IJ d A I ∧ ∗ d A J + 12 e U ( Im N ) IJ ( dA I + A I d ω ) ∧ ∗ ( dA J + A J d ω )+( Re N ) IJ d A I ∧ ( dA J + A J d ω )The dual scalars for ω and A I are defined by: e U ( Im N ) IJ ∗ ( dA J + A J d ω ) + ( Re N ) IJ d A J = − d φ A I e U ∗ d ω + ( A I d φ A I − φ A I d A I ) = − d φ ω (A.5)After renaming the variables φ ω → σ, A I → A I , φ A I → B I , we obtain the 3d lagrangian interms of scalars only: L = − √ g R + d U ∧ ∗ d U + 14 e − U ( d σ + A I d B I − B I d A I ) ∧ ∗ ( d σ + A I d B I − B I d A I )+ g i ¯ j ( z, ¯ z ) d z i ∧ ∗ d ¯ z ¯ j + 12 e − U ( Im N ) IJ d A I ∧ ∗ d A J + 12 e − U ( Im N − ) IJ ( d B I + ( Re N ) IK d A K ) ∧ ∗ ( d B J + ( Re N ) JL d A L )= − √ g R + g mn ∂ a φ m ∂ a φ n (A.6)where, as before, φ n are the 4( n V + 1) moduli fields: φ n = { U, z i , ¯ z ¯ i , σ, A I , B I } , and g ab isthe space time metric, g mn is the moduli space metric. Therefore, the moduli space M D has metric: ds = d U · d U + 14 e − U ( d σ + A I d B I − B I d A I ) · ( d σ + A I d B I − B I d A I ) + g i ¯ j ( z, ¯ z ) d z i · d ¯ z ¯ j + 12 e − U [( Im N − ) IJ ( d B I + N IK d A K ) · ( d B J + N JL d A L )] (A.7)48t is a para-quaternionic-K¨ahler manifold. Since the holonomy is reduced from SO (4 n V +4)) to Sp (2 , R ) × Sp (2 n V + 2 , R ), the vielbein has two indices ( α, A ) transforming under Sp (2 , R ) and Sp (2 n V + 2 , R ), respectively. The para-quaternionic vielbein is the analyticalcontinuation of the quaternionic vielbein computed in [34]: V αA = iu ve a iE a − i ¯ E ¯ a ¯ e ¯ a − ¯ v i ¯ u The 1-forms are defined as u ≡ e K / − U X I ( d B I + N IJ dA J ) e a ≡ e ai d z i E a ≡ e − U e ai g i ¯ j e K / ¯ D ¯ j X I ( d B I + N IJ d A J ) v ≡ − d U + i e − U ( d a + A I d B I − B I d A I ) (A.8)where e ai is the veilbein of the 4D moduli space, and the bar denotes complex conjugate.The line element is related to the vielbein by ds = − u · ¯ u + g a ¯ b e a · ¯ e ¯ b − g a ¯ b E a · ¯ E ¯ b + v · ¯ v = ǫ αβ ǫ AB V αA ⊗ V βB (A.9)where ǫ αβ and ǫ AB are the anti-symmetric tensors invariant under Sp (2 , R ) ∼ = SL (2 , R ) and Sp (2 n v + 2 , R ).The isometries of the M ∗ D descends from the symmetry of the 4D system. 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