CERN-PH-TH/2009-150 DFPD-09/TH/16 =8 5d/4d U-dualities and black holes N 0 1 Anna Ceresolea, Sergio Ferrara b,c and Alessandra Gnecchid 0 2 n a J a INFN, Sezione di Torino & Dipartimento di Fisica Teorica 6 Universita` di Torino, Via Pietro Giuria 1, 10125 Torino, Italy 2 ] h b Theory Division - CERN, t CH 1211, Geneva 23, Switzerland - p e h c INFN - LNF, [ Via Enrico Fermi 40, I-00044 Frascati, Italy 3 v 9 d Dipartimento di Fisica “Galileo Galilei” & INFN, Sezione di Padova 6 Universita` di Padova, Via Marzolo 8, 35131 Padova, Italy 0 1 . 8 0 9 0 ABSTRACT : v i X r We use the connection between the U-duality groups in d = 5 and d = 4 to derive a properties of the = 8 black hole potential and its critical points (attractors). This N approach allows to study and compare the supersymmetry features of different solutions. Contents 1 Introduction 1 2 4d/5d relations for the = 8 extremal black hole potential 4 N 2.1 Symplectic sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Attractors in the 5 dimensional theory 9 4 Attractors in the 4 dimensional theory 11 4.1 Electric solution Q = (p0,q ) . . . . . . . . . . . . . . . . . . . . . . . . . 11 i 4.2 Magnetic solution Q = (p ,q0) . . . . . . . . . . . . . . . . . . . . . . . . 12 i 4.3 KK dyonic solution Q = (p0,q ) . . . . . . . . . . . . . . . . . . . . . . . 14 0 4.4 = 8 and = 2 attractive orbits at d = 5 and d = 4 . . . . . . . . . . 15 N N 5 Maurer-Cartan equations of the four dimensional theory 16 5.1 Attractor equations from Maurer-Cartan equations . . . . . . . . . . . . 17 1 Introduction The = 8 supergravity theory in d = 4 [1] and d = 5 [2] dimensions is a remarkable N theory which unifies the gravitational fields with other lower spin particles in a rather unique way, due to the high constraints of local = 8 supersymmetry, the maximal one N realized in a 4d Lagrangian field theory. These theories, particularly in four dimensions, are supposed to enjoy exceptional ultraviolet properties. For this reason, 4d supergravity has been advocated not only as the simplest quantum field theory [3] but also as a potential candidate for a finite theory of quantum gravity, even without its completion into a larger theory [4]. Maximal supergravity in highest dimensions has a large number of classical solutions [5] which may survive at the quantum level. Among them, there are black p-branes of several types[6] and interestingly, 4d black holes of different nature. On the other hand, theories with lower supersymmetries (such as = 2) emerging N from Calabi-Yau compactifications of M-theory or superstring theory, admit extremal black hole solutions that have been the subject of intense study, because of their wide rangeofclassicalandquantumaspects. Forasymptoticallyflat,stationaryandspherically symmetric extremal black holes, the attractor behaviour [7, 8] has played an important role not only in determining universal features of fields flows toward the horizon, but also to explore dynamical properties such as wall crossing[9] and split attractor flows[10], the connections with string topological partition functions[11] and relations with microstates counting[12] . Therefore, it has become natural to study the properties of extremal black holes not only in the context of = 2, but also in theories with higher supersymmetries, N up to = 8[13]-[22]. N 1 In = 8 supergravity, in the Einsteinian approximation, there is a nice relation N between theclassification oflargeblack holeswhich undergo theattractorflowandcharge orbits which classify, in a duality invariant manner, the properties of the dyonic vector of electric and magnetic charges Q = (pΛ, q ) (Λ = 0,...,27 in d = 4) [23, 24]. The attractor Λ points are given by extrema of the 4d black hole potential, which is given by [16, 17] 1 V = Z Z AB = Q,V Q,VAB , (1.1) BH AB ∗ AB 2 h i h i where the central charge is the antisymmetric matrix (A,B = 1,...,8) Z = Q,V = QT ΩV = fΛ q h pΛ , (1.2) AB h ABi AB AB Λ − ΛAB the symplectic sections are V = (fΛ ,h ) , (1.3) AB AB ΛAB and Ω is the symplectic invariant metric. An important role is played by the Cartan quartic invariant I [25, 1] in that it only 4 depends on Q and not on the asymptotic values of the 70 scalar fields ϕ. This means that if we construct I as a combination of quartic powers of the central charge matrix 4 Z (q,p,ϕ) [26], the ϕ dependence drops out from the final expression AB ∂ I (Z ) = 0 . (1.4) 4 AB ∂ϕ Analogue (cubic) invariants I exist for black holes and/or (black) strings in d = 5[8, 23]. 3 These are given by 1 I (pI) = d pIpJpK , (1.5) 3 IJK 3! 1 I (q ) = dIJKq q q , (1.6) 3 I I J K 3! where d , dIJK are the (27)3 E invariants. Consequently, the d = 4 E quartic IJK 6(6) 7(7) invariant takes the form ∂I (q)∂I (p) I (Q) = (p0q +pIq )2 +4 p0I (q)+q I (p)+ 3 3 . (1.7) 4 − 0 I − 3 0 3 ∂q ∂pI (cid:20) I (cid:21) On the other hand, in terms of the central charge matrices Z (φ,q) (in d = 5 this ab is the 27 representation of USp(8)) and Z (φ,p,q) (in d = 4 this is the 28 of SU(8)), AB their expression is I (q) = Z ΩbcZ ΩdqZ Ωpa , Z Ωab = 0 , (1.8) 3 ab cd qp ab 1 I (p,q) = 4Tr(ZZ ZZ ) (TrZZ )2 +32Re(Pf Z ) , (1.9) 4 † † † AB 4 − (cid:2) (cid:3) 2 where ZZ = Z Z¯CB, Ωab is the 5d symplectic invariant metric, and the Pfaffian of the † AB central charge is [1] 1 Pf (Z ) = ǫABCDEFGHZ Z Z Z . (1.10) AB 244! AB CD EF GH In fact, these are simply the (totally symmetric) invariants which characterize the 27 dimensional representation of E and the 56 dimensional representation of E , which 6(6) 7(7) are the U-duality [27] symmetries of = 8 supergravity in d = 5 and d = 4, respectively. N When charges are chosen such that I and I are not vanishing, one has large black 4 3 holesandintheextremalcasetheattractorbehaviourmayoccur. However, whileatd = 5 there is a unique (1-BPS) attractor orbit with I = 0, associated to the space[24, 28] 8 3 6 E 6(6) = , (1.11) d=5 O F 4(4) at d = 4 two orbits emerge, the BPS one E 7(7) = , (1.12) d=4,BPS O E 6(2) and the non BPS one with different stabilizer E 7(7) = . (1.13) d=4,non BPS O − E6(6) Such orbits have further ramifications in theories with lower supersymmetry , but it is the aim of this paper to confine our attention to the = 8 theory. N In this paper, extending a previous result for = 2 theories [29], we elucidate the N connection between these configurations and we relate the critical points of the = 8 N black hole potential of the 5d and 4d theories. To achieve this goal we use a formula- tion of 4d supergravity in a E duality covariant basis [30], which is appropriate to 6(6) discuss a 4d/5d correspondence. This is not the same as the Cremmer-Julia[1] or de Wit- Nicolai[31] manifest SO(8) (and SL(8,R)) covariant formulation, but it is rather related to the Sezgin-Van Nieuwenhuizen 5d/4d dimensional reduction[32]. These two formula- tions are related to one another by dualizing several of the vector fields and therefore they interchange electric and magnetic charges of some of the 28 vector fields of the final theory. The precise relation between these theories was recently discussed in [33]. The paper is organized as follows. In sec. 2 we rewrite the 4d black hole potential in terms of central charges. This is essential in order to discuss the supersymmetry properties of the solutions. In fact, in the specific solutions we consider in sec. 3 and 4, BPS and non-BPS critical points are simply obtained by some charges sign flip. This will manifest in completely different symmetry properties of the central charge matrix, in the normal frame, at the fixed point. These properties reflect the different character of the BPS and non BPS charge orbits. 3 The solutions of the critical point equations are particularly simple in the “axion free” case, discussed in sec. 3 and 4, which only occur for some chosen charge configurations. In sec. 3 we derive critical point equations that are completely general and that may be used to study any solution. The formula for the = 8 potential given in sec. 2 was obtained in an earlier work N [33], and it is identical to the = 2 case [29]. The only difference relies in the kinetic N matrix a which, in = 2 is given by real special geometry while in = 8 is given IJ N N in terms of the E coset representatives [32, 16]. However, in the normal frame, when 6(6) we suitably restrict to two moduli, this matrix does indeed become an = 2 matrix, N although the interpretation in terms of central charges is completely different. The supersymmetry properties of the solutions in the = 8 and = 2 theories are N N compared in subsection 4.4. We will see that in the = 2 interpretation, depending on N the sign of the charges, both a BPS and a non-BPS branch exist in d = 5 while two non BPS branches exist in the d = 4 theory. In = 8, the occurrence of one less branch N in both dimensions is due to the fact that the central and matter charges of the = 2 N theory are all embedded in the central charge matrix of the = 8 theory. The higher N number of attractive orbits can also be explained by the different form of the relevant non compact groups and their stabilizers for the moduli space of solutions. = 8 2 4d/5d relations for the extremal black hole N potential In this section we remind the reader how the = 8 potential was derived in a basis that N illustrates the relation between 4 and 5 dimensions [33]. Using known identities [17, 34], the black hole potential can be written as a quadratic form in terms of the charge vector Q and the symplectic 56 56 matrix ( ), related × M N to the 4d vector kinetic matrix ΛΣ N 1 V = QT ( )Q, (2.1) BH −2 M N where is M Im +Re (ImN) 1Re Re (Im ) 1 − − N N N − N N ( ) = . (2.2) M N   (Im ) 1Re (Im ) 1 − − − N N N   The indices Λ,Σ of are now split as (0,I), according to the decomposition of 4d ΛΣ N charges with respect to 5d ones, thus assumes the block form ΛΣ N 00 0J = N N , (2.3) ΛΣ N NI0 NIJ ! 4 The kinetic matrix depends on the 70 scalars of the = 8 theory, which are given, N in the 5d/4d KK reduction, by the 42 scalars of the 5d theory (encoded in the 5d vector kinetic matrix a = a ), by the 27 axions aI and the dilaton field eφ. In a normalization IJ JI that is suitable for comparison to = 2 , it has the form N 1d i e2φa aIaJ +e6φ 1d +ie2φa aK  3 − IJ −2 J KJ  = (cid:0) (cid:1) , (2.4) ΛΣ N      1d +ie2φa aK d ie2φa   −2 I IK IJ − IJ        where d d aIaJaK , d d aJaK , d d aK . (2.5) IJK I IJK IJ IJK ≡ ≡ ≡ The black hole potential of [33], computed from (2.1) using the above formulas, can be rearranged as 2 1 1 1 d V = p0eφaI a p0eφaJ + p0e3φ 2 + p0e 3φ + BH IJ − 2 2 2 6 (cid:18) (cid:19) (cid:0)1 1 (cid:1) (cid:0) (cid:1)1 (cid:0) (cid:1) 1 + e φp0d aIJ e φp0d + 2 p0eφa a pJeφ + − I − J I IJ 2 2 2 2 × − (cid:18) (cid:19) (cid:18) (cid:19) 1 d 1 1 (cid:0) 1 (cid:1) (cid:0) (cid:1) + 2 p0e 3φ pId e 3φ 2 p0e φd aIJ pKd e φ + − I − − I KJ − 2 × 6 −2 − 2 × 2 (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) 2 (cid:0) (cid:1) 1 1 1 + eφpI a eφpJ + e 3φpKd + IJ − K 2 2 2 (cid:18) (cid:19) 1 (cid:0) (cid:1) (cid:0) (cid:1) 1 d + e φpKd aIJ e φpLd + 2 q e 3φ p0e 3φ + − KI − JL 0 − − 2 2 × 6 (cid:18) (cid:19) 1 (cid:0) (cid:1) (cid:0)d (cid:1) 1 (cid:0) (cid:1) 1 + 2 q aIe 3φ p0e 3φ + 2 q e φ aIJ p0d e φ + I − − I − J − 2 × 6 2 × 2 (cid:18) (cid:19) (cid:18) (cid:19) 1 (cid:0) (cid:1)1 1 (cid:0) (cid:1) 1 2 q e 3φ pId e 3φ 2 q aIe 3φ pJd e 3φ + 0 − I − I − J − −2 × 2 − 2 × 2 (cid:18) (cid:19) (cid:18) (cid:19) 1 (cid:0) (cid:1) 1 (cid:0) (cid:1)1 2 q e φ aIJ pKd e φ + q e 3φ 2 + 2 q e 3φ q aIe 3φ + I − KJ − 0 − 0 − I − −2 × 2 2 × +1 q a(cid:0)Ie 3φ 2(cid:1)+ 1(cid:0) q e φ aIJ(cid:1) q e(cid:0)φ , (cid:1) (cid:0) (cid:1)(cid:0) (cid:1) I − I − J − 2 2 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (2.6) with aIJ = a 1. This form shows that it can be written in terms of squares of electric −IJ and magnetic components as 1 1 1 1 V = (Ze)2 + Z0 2 + ZeaIJZe + ZI a ZJ , (2.7) BH 2 0 2 m 2 I J 2 m IJ m (cid:0) (cid:1) 5 provided one defines, d 1 Ze = e 3φq +e 3φq aI +e 3φ p0 e 3φpId , 0 − 0 − I − 6 − 2 − I Z0 = e3φp0 , m 1 Ze = e φp0d pJd e φ +q e φ , I 2 − I − IJ − I − ZI = eφpI eφp0aI . (2.8) m − In order to get the symplectic embedding of the four dimensional theory, we still need to complexify the central charges. To this end, we define the two complex vectors 1 Z (Ze +iZ0) , 0 ≡ √2 0 m 1 Z (Ze +iZa) , (2.9) a ≡ √2 a m where Ze = Ze(a 1/2)I , Za = ZI (a1/2)a (2.10) a I − a m m I such that V = Z 2 +Z Z¯ , (2.11) BH 0 a a | | where now a = 1,...,27 is a flat index, which can be regarded as a USp(8) antisymmetric traceless matrix. The potential at the critical point gives the black hole entropy corresponding to the given solution, which in d = 4 reads S BH = I = Vcrit. , (2.12) π | 4| BH p while in d = 5 it is [38] S BH = 33/2 I 1/2 = 3Vcrit 3/4 , (2.13) π | 3| 5 (cid:0) (cid:1) where I and I are the invariants of the = 8 theory in d = 4 and d = 5 respectively. 4 3 N 2.1 Symplectic sections In virtue of the previous discussion, we can trade the central charge (1.2)for the 28- component vector Z = fΛ q h pΛ , (2.14) A A Λ − ΛA where f and h are symplectic sections satisfying the following properties [40, 41] 6 a) = h (f 1)A , NΛΣ ΛA − Σ b) i(f h h f) = Id , † † − c) fTh hTf = 0 . − Notice that one still has the freedom of a further transformation h hM , → f fM , (2.15) → as it leaves invariant the vector kinetic matrix , as well as relations a) c), when M is N − a unitary matrix MM = 1 . (2.16) † Indeed, when the central charge transforms as Z ZM , → ZZ ZMM Z = ZZ , (2.17) † † † † → the black hole potential V ZZ (2.18) BH † ≡ isleftinvariant. Inourcase, werearrangethe28indicesintoasinglecomplexvectorindex, tobeidentified, forasuitablechoiceofM, withthetwo-foldantisymmetric representation ofSU(8), accordingtothedecomposition28 27+1ofSU(8) USp(8); wethushave → → Z = fΛ q h pΛ = 0 0 Λ − Λ0 = f0 q +fJ q h p0 h pJ , 0 0 0 J − 00 − J0 Z = fΛ q h pΛ = a a Λ − Λa = f0 q +fJ q h p0 h pJ ; a 0 a J − 0a − Ja (2.19) which, from the definition in (2.9) yields 1 d 1 Z = e 3φq +e 3φaIq + e 3φ +ie3φ p0 e 3φd pI , 0 − 0 − I − − I √2 6 − 2 (cid:20) (cid:18) (cid:19) (cid:21) 1 1 (cid:0) (cid:1) Z = e φq (a 1/2)I + e φd (a 1/2)I ieφaJ(a1/2) a p0+ a √2 − I − a 2 − I − a − J (cid:20) (cid:18) (cid:19) e φd (a 1/2)I ieφ(a1/2) a pJ . − − IJ − a − J (cid:21) (cid:0) (cid:1) (2.20) 7 Thus we consider e 3φ 0 −   1 fΛ = , (2.21) A √2     e 3φaI e φ(a 1/2)I   − − − a       e 3φd ie3φ 1e φd (a 1/2)K +ieφaK(a1/2) a − − 6 − −2 − K − a K  1 h = . (2.22) ΛA √2      1e 3φd e φd (a 1/2)J ieφ(a1/2) a   2 − I − IJ − a − I        From f 1 − e3φ 0   (f 1) A = √2 , (2.23) − Λ      eφaI(a1/2) a eφ(a1/2) a − I I        by matrix multiplication, we find that relations a) b) and c) are fulfilled by f and h, that we now recognize to be the symplectic sections. We finally perform the transformation f = fM (where M = f 1f = h 1h), with M ′ − ′ − ′ unitary matrix, in virtue of identities a), b) and c), valid for both (f,h) and (f ,h). A ′ ′ model independent formula for M valid for any = 2 d-geometry (in particular, for any N truncation of = 8 to an = 2 geometry, such as the models treated in this paper) is N N given by the matrix [42] M = A1/2MˆG 1/2 , (2.24) − with 1 0...0 1 0...0 0 0     1 A = . , G = . , g = e 4φa , (2.25) a g IJ 4 − IJ  . IJ   . IJ       0   0          where Mˆ is given by 1 1 ∂ K Mˆ = J¯ , (2.26) 2 (cid:18)−iλIe−2φ e−2φδJI¯+ie−2φλI∂J¯K(cid:19) 8 where “ λI” are the imaginary parts of the complex moduli zI = aI iλI, and K is − − the Ka¨hler potential K = ln(8 ), with = 1d λIλJλK; the matrix Mˆ satisfies the − V V 3! IJK properties AMˆG 1Mˆ = Id , − † G 1Mˆ AMˆ = Id . (2.27) − † For the models considered below, this matrix M does indeed reproduce, for the given special configurations, the formula in eq. (4.7). Note that Mˆ performs the change of basis between the central charges defined as 1 Z = (Ze +iZ0) , 0 √2 0 m 1 Z = (Ze +ia ZJ) , (2.28) I √2 I IJ m and the special geometry charges (Z, Z), that is the charges in “curved” rather than DI¯ the “flat” indices. 3 Attractors in the 5 dimensional theory It was shown in [23] that the cubic invariant of the five dimensions can be written as 5 5 5 I = Z Z Z , (3.1) 3 1 2 3 5 where Z ’s are related to the skew eigenvalues of the USp(8) central charge matrix in a the normal frame 5 5 5 Z +Z Z 0 0 0 1 2 − 3 5 5 5 0 Z +Z Z 0 0 0 1 eab =  0 1 03 − 2 Z5 +Z5 Z5 0 ⊗ 1 0 . 2 3 − 1 5 5 5 (cid:18)− (cid:19)  0 0 0 (Z +Z +Z )  − 1 2 3    (3.2) We consider a configuration of only three non-vanishing electric charges (q ,q ,q ), that 1 2 3 we can take all non-negative. We further confine to two moduli λ ,λ , describing a 1 2 geodesic submanifold SO(1,1)2 E /USp(8) whose special geometry is determined by 6(6) ∈ the constraint 1 d λˆIλˆJλˆK = λˆ1λˆ2λˆ3 = 1 , (3.3) IJK 3! where λˆI = 1/3λI, defining the stu model [29]. − V − 9