Non-BPS Attractors in 5d and 6d Extended Supergravity
aa r X i v : . [ h e p - t h ] S e p CERN-PH-TH/164UCLA/07/TEP/19
Non-BPS Attractorsin 5d and 6d Extended Supergravity
L.Andrianopoli ♥♦♣ , S.Ferrara ♦♠ ♭ , A.Marrani ♥♠ and M.Trigiante ♣ ♥ Museo Storico della Fisica eCentro Studi e Ricerche “Enrico Fermi”Via Panisperna 89A, 00184 Roma, Italy ♦ Physics Department,Theory Unit, CERN,CH 1211, Geneva 23, Switzerland
[email protected], [email protected] ♣ Dipartimento di Fisica, Politecnico di Torino,Corso Duca degli Abruzzi 24, I-10129 Torino, Italyand INFN - sezione di Torino, Italy [email protected] ♠ INFN - Laboratori Nazionali di Frascati,Via Enrico Fermi 40,00044 Frascati, Italy [email protected] ♭ Department of Physics and Astronomy,University of California, Los Angeles, CA USA [email protected]
Abstract
We connect the attractor equations of a certain class of N = 2, d = 5 supergrav-ities with their (1 , d = 6 counterparts, by relating the moduli space of non-BPS d = 5 black hole/black string attractors to the moduli space of extremal dyonicblack string d = 6 non-BPS attractors. For d = 5 real special symmetric spacesand for N = 4 , , N = 4, we study the relation to the (2 , d = 6 theory. We finally describethe embedding of the N = 2, d = 5 magic models in N = 8, d = 5 supergravity aswell as the interconnection among the corresponding charge orbits. Introduction
Recently the study of the attractor equations for extremal black holes (BHs) [1]–[5] in fourdimensions received special attention, especially in relation with new results on non-BPS,non-supersymmetric solutions [6]–[48].Not much is known about non-BPS attractors in five dimensions, although generalresults for symmetric special geometries in BHs (and black strings) backgrounds werederived in [49]. More recently, it has been shown [36] that real special symmetric spaceshave, in the non-BPS case, a moduli space of vacua, as it was the case for their d = 4special K¨ahler descendants [21]. In four dimensions, massless Hessian modes for genericcubic geometries were shown to occur for the non-BPS case with non-vanishing centralcharge in [10, 34]. Some additional insight on the correspondence among (the supersym-metry preserving features of) extremal BH attractors in four and five dimensions havebeen gained in [40], by relating the d = 4 and 5 BH potentials and the correspondingattractor equations. In particular, it was shown that the moduli space of non-BPS at-tractors in d = 5 real special symmetric geometries must be in the intersection of themoduli spaces of non-BPS Z = 0 and non-BPS Z = 0 attractors in the corresponding d = 4 special K¨ahler homogeneous geometries.Aim of the present investigation is to perform concrete computations of the masslessmodes of the non-BPS d = 5 Hessian matrix, and further relate the d = 5 BH (or blackstring) potential to the d = 6 dyonic extremal black string potential and its BPS andnon-BPS critical points, following the approach of [49] and [50]. This analysis reveals anoteworthy feature of the relation between d = 5 and d = 6. Namely, the moduli space of d = 6 non-BPS (with vanishing central charge ) dyonic string attractors is a submanifoldof the moduli space of d = 5 non-BPS attractors of symmetric real special geometries.The only exception is provided by the cubic reducible sequence of real special geometries,for which the non-BPS d = 6 and d = 5 moduli spaces actually coincide. It is worthpointing out that moduli spaces also exist, for particular non-BPS-supporting chargeconfigurations, for all real special geometries with a d = 6 uplift [51]. This is the casefor the homogeneous non-symmetric real special geometries studied in [52]. For N = 2, d = 5 magic supergravities, with the exception of the octonionic case, the non-BPSmoduli spaces can also be obtained as suitable truncations of the moduli space of BPSsttractors of N = 8, d = 5 supergravity. In all cases, the Hessian matrix is semi-positivedefinite.It is worth pointing out that in this work we consider only extremal black p -extendedobjects which are asymptotically flat, spherically symmetric and with an horizon geom-etry AdS p +2 × S d − p − [53]. Thus, we do not deal with, for instance, black rings androtating BHs in d = 5, which however also exhibit an attractor behaviour (see e.g. [54]).The paper is organized as follows.In Sect. 2 we recall some relevant facts about N = 2, d = 6 self-dual black stringattractors and the properties of the black string effective potential in terms of the modulispace spanned by the tensor multiplets’ scalars. In Sect. 3 we discuss the d = 5 effectivepotential in a six-dimensional language for the d = 5 models admitting a d = 6 uplift(including all homogeneous real special geometries classified in [52]), in the absence (Sub- This means that non-BPS dyonic strings are neutral with respect to the central extension of the(1 , d = 6 supersymmetry algebra. d = 6 vector multiplets. In Subsubsect. 3.2.1 weperform an analysis of the attractors in d = 5, N = 2 magic supergravities, and commenton the moduli spaces of attractor solutions for such theories. Thence, in Subsects. 4.1,4.2 and 4.3 we recall a similar analysis of the attractors respectively in d = 5, N = 8, 6and 4 supegravities [49, 55, 56]. The analysis holds for all N = 2 symmetric spaces, aswell as for homogeneous spaces by considering particular charge configurations. In Sect.5 we comment on the conditions to be satisfied in order to obtain an anomaly-free (1 , d = 6 supergravity by uplifting N = 2, d = 5 theories. Sect. 6 is devoted to final remarksand conclusions.The Appendix discusses some group embeddings, relevant in order to elucidate therelation between the N = 8, d = 5 BPS unique orbit and the non-BPS orbits of the N = 2, d = 5 theories obtained as consistent truncations of N = 8 supergravity. Such N = 2 theories include the magic supergravities based on the Jordan algebras J H , J C , J R with n H = 0 , , (1 , , d = 6 attractors for extremal dyonic strings In d = 6, (1 ,
0) and (2 ,
0) chiral supergravities there are no BPS BH states, because thecentral extension of the corresponding d = 6 superalgebras does not contain scalar centralcharges [58]. However, there are BPS (dyonic) string configurations, as allowed from thesuperalgebra, and extremal black string BPS attractors exist [50, 49]. Such attractorspreserve 4 supersymmetries, so they are the d = 6 analogue of d = 5 and d = 4 -BPSextremal BH attractors. Interestingly enough, extremal black string non-BPS attractorsalso exist in such d = 6 theories [49], as it is the case for (extremal BH attractors) in d = 5 and d = 4. The next sections are partially devoted to such an issue.Let us start by recalling the general structure of the minimal supergravity in d = 6,the chiral (1 ,
0) theory. The field content of the minimal theory is: • Gravitational multiplet:( V aµ , ψ Aµ , B + µν ) ; ; ( µ = 0 , , · · · , A = 1 ,
2) ; (2.1) • Tensor multiplets: ( B − µν , χ A , φ ) i ; ( i = 1 , · · · , q + 1) ; (2.2)The scalar fields in the tensor multiplets sit in the coset space [59] GH = O (1 , q + 1) O ( q + 1) . (2.3)They may be parametrized in terms of q + 2 fields X Λ , (Λ = 0 , , · · · , q + 1),contstrained by the relation X Λ X Σ η ΛΣ ≡ X Λ X Λ = 1 , (2.4) In the literature they are sometimes referred to as (2 ,
0) and (4 ,
0) respectively [57]. η ΛΣ = diag [1 , − , · · · , − G ΛΣ = 2 X Λ X Σ − η ΛΣ (2.5)whose inverse matrix is: G ΛΣ = 2 X Λ X Σ − η ΛΣ . (2.6)As for any d = 6 theory, the field strengths of the antisymmetric tensors H Λ = dB Λ have definite self-duality properties: G ΛΣ ⋆ H Σ = η ΛΣ H Σ ; (2.7)As a consequence, there is no distinction between the associated electric and mag-netic charges e Λ = η ΛΣ e Σ = Z S H Λ . (2.8) • Vector multiplets: ( A µ , λ A ) α ; ( α = 1 , · · · , m ) ; (2.9)The kinetic matrix for the vector field strengths is given in terms of a given constantmatrix C Λ αβ by [60]: N αβ = X Λ C Λ αβ . (2.10) • Hypermultiplets: ( ζ A , q ) ℓ ; ( ℓ = 1 , · · · , p ) . (2.11)The hypermultiplets do not play any role in the attractor mechanism, and will notbe discussed further here.Since the vector multiplets do not contain scalar fields, the only contribution to theblack string effective potential comes from the tensor multiplets, and reads [50]: V (6) = G ΛΣ e Λ e Σ = 2( X Λ e Λ ) − e Λ e Λ (2.12)or equivalently, in terms of the dressed central and matter charges Z = ( X Λ e Λ ) and Z i = P i Λ e Λ (where P ΛΣ , P ΛΣ X Σ = 0 is the projector orthogonal to the central charge): V (6) = Z + Z i Z i . (2.13)The criticality conditions for the effective black string potential (2.13) reads ∂ i V (6) = 0 ⇔ ZZ i = 0 , ∀ i, (2.14)and therefore two different extrema are allowed, the BPS one for Z i = 0 ∀ i , and a non-BPSone for Z = 0, both yielding the following critical value of V (6) : V (6) | extr = | e Λ e Λ | . (2.15)3 N = 2 , d = 5 attractors with a six dimensional in-terpretation In the absence of gauging, the minimal five dimensional theory generally admits thefollowing field content (omitting hypermultiplets): • Gravitational multiplet:( V aµ , ψ Aµ , A µ ) ; ( µ = 0 , , · · · , A = 1 ,
2) ; (3.1) • Vector multiplets: ( A µ , χ A , φ ) a ; ( a = 1 , · · · , n ) . (3.2)The scalar fields do not necessarily belong to a coset manifold, but their σ -model isdescribed by real-special geometry. In particular, the scalar manifold is described by thelocus V ( L ) = 1 (3.3)where L I ( φ ), I = 0 , , · · · , n are function of the scalars and V is the cubic polynomial: V ( L ) = 13! d IJK L I L J L K , (3.4)written in terms of an appropriate totally symmetric, constant matrix d IJK . Note thatin order to have a d = 6 uplift the real special geometry must have a certain structure,as discussed in [62]. Namely V = zX Λ η ΛΣ X Σ + X Λ C Λ αβ X α X β . (3.5)This is always the case for the homogeneous spaces discussed in [52], where C Λ αβ iswritten in terms of the γ -matrices of SO ( q + 1) Clifford algebras.The kinetic matrix for the vector field-strengths has the general form: a IJ = − ∂ I ∂ J log V| V =1 . (3.6)The BH effective potential in five dimensions is given by V (5) = a IJ q I q J (3.7)where q I = R S ∂ L ∂F I are the electric charges and a IJ the inverse of (3.6). d = 6 vector multiplets We are interested in finding the relation of the six dimensional attractor behavior tothe five dimensional one. Let us first consider the simplest case of a six dimensionalsupergravity theory only coupled to q + 1 tensor multiplets (no vector multiplets). In thiscase, n = q + 1 and the scalar content is given by the six dimensional scalars X Λ plus4he Kaluza–Klein (KK) dilaton z . The five dimensional scalar fields are related by theconstraint (3.3), where the surface expression (3.4) takes here the simple form : V ( L ) = V ( z, X ) = 12 zX Λ X Σ η ΛΣ . (3.1.1)The constraint (3.3) then becomes: 12 X Λ X Λ = z − . (3.1.2)The components of the kinetic matrix are in this case: a IJ = a zz = z − a z Λ = 0 a ΛΣ = z e G ΛΣ (3.1.3)where the matrix e G e G ΛΣ ( X ) = 2 X Λ X Σ X Γ X Γ − η ΛΣ (3.1.4)is related to G in (2.5) by e G ΛΣ | X Λ X Λ =1 = G ΛΣ . (3.1.5)More precisely, setting: ˆ X Λ ≡ X Λ √ X Λ X Λ , ( ˆ X Λ ˆ X Λ = 1) (3.1.6)we have: e G ΛΣ ( X ) = G ΛΣ ( ˆ X ) . (3.1.7)The matrix (3.1.3) is easily inverted giving: a IJ = a zz = z a z Λ = 0 a ΛΣ = z − e G ΛΣ (3.1.8)Then, in this case the BH effective potential takes the form: V (5) = z e z + z − e G ΛΣ ( X ) e Λ e Σ = z e z + z − V (6) ( ˆ X ) . (3.1.9)where ( e z , e Λ ) ≡ q I denote the electric charges and, to obtain the last expression, wemade use of (3.1.7). The physical interpretation of the charges e z and e Λ is the following: e z is the Kaluza-Klein charge and e Λ are the charges of dyonic strings wrapped around S . The extrema of V (5) are found for: ∂V (5) ∂z = 0 ⇒ ze z − z − V (6) ( ˆ X ) = 0 (3.1.10) This corresponds to the d = 5 symmetric real spaces of the “ generic sequence” SO (1 , × SO (1 ,q +1) SO ( q +1) [61]. z = (cid:18) V (6) | extr e z (cid:19) (3.1.11)and for: ∂V (5) ∂ ˆ X Λ = 0 ⇒ ∂V (6) ∂ ˆ X Λ = 0 (3.1.12)which shows that in this case the attractor solutions of the five dimensional theory areprecisely the same of the parent six dimensional theory.The BH entropy is now given by [49]: (cid:16) S (5) BH (cid:17) / = V (5) | extr = 3 (cid:18) e z V (6) | extr (cid:19) = 3 (cid:18) e z e Λ e Λ (cid:19) . (3.1.13)The solution of Eqs. (3.1.12) depends on whether the d = 6 attractor is BPS or not.As previously mentioned, the d = 6 BPS attractors correspond to Z i = 0 ∀ i , whereas thenon-BPS ones are given by Z = 0 (and Z i = 0 for at least some i ) [50, 49]. Thus, all q + 1 d = 6 BPS moduli are fixed, while there are q non-BPS flat directions, spanningthe d = 6 non-BPS moduli space SO (1 ,q ) SO ( q ) [49].The supersymmetry-preserving features (BPS or non-BPS) of the d = 6 attractorssolutions depend on the sign of e Λ e Λ : it is BPS for e Λ e Λ > e Λ e Λ < d = 5 solution is non-BPS, because in a given frame [51] e z e Λ e Λ = e z e + e − (with e ± ≡ e ± e ), and if e + e − < e Λ e Λ > d = 5 solutions [40].Thus, we can conclude that for the “generic sequence” of d = 5 symmetric real specialspaces the non-BPS moduli space, predicted in [36], does indeed coincide with the abovementioned d = 6 (tensor multiplets’) non-BPS moduli space, found in [49]. d = 6 vector multiplets Let us now generalize the discussion to the case where s extra vector multiplets:( A µ , λ A , Y ) α , α = 1 , · · · , s , (3.2.1)corresponding to the dimensional reduction of six dimensional ones, are present [62]. Thereduction may be done preserving the SO (1 , q + 1) symmetry when the number s of d = 6vector multiplets coincides with the dimension of the spinor representation of SO (1 , q +1): s = dim [spin SO (1 , q + 1)] . (3.2.2)This implies that the kinetic matrix of the d = 6 vector fields is positive definite and nophase transitions, as discussed in [63, 62], occur in this class of models.The extra scalars contribute to the general relations (3.6) and (3.7) via a modificationof the cubic form V into [52]: V = 12 zX Λ X Σ η ΛΣ + 12 X Λ Y α Y β Γ Λ αβ . (3.2.3)6he total number of five dimensional scalars is then q + 2 + s . Of particular interestare the four magic models which are associated with the simple Jordan algebras havingan irreducible norm form (displayed in Table 4 of [36]). In these cases q = 1 , , , s = 2 q . Also the “ generic sequence” L (0 , P ) can be viewed as a particular case of Eq.(3.2.3) with q = 0 and s = P .The d = 6 origin of the second term in Eq. (3.2.3) is the kinetic term of the d = 6vector fields, which reads [60, 62] (Λ = 0 , , ..., q + 1, α = 1 , ..., s , C Λ αβ = C Λ βα ) X Λ C Λ αβ F α ∧ ∗ F β . (3.2.4)Thus, in the presence of d = 6 BH charges Q α , it originates an effective d = 6 BHpotential of the form V (6) BH = X Λ C Λ αβ Q α Q β . (3.2.5)Such a potential has run-away extrema at d = 6 [57]. This can be seen for instance inthe case n T = 1 ⇔ q = 0, where Eq. (3.2.5) reduces to ( α = 1 , ..., P , X = coshφ , X = sinhφ ) V (6) BH ( φ ) = coshφ C αβ Q α Q β + sinhφ C αβ Q α Q β = e φ Q α Q α , (3.2.6)(in the last step we used the fact that in the n T = 1 case we may set C αβ = C αβ = δ αβ without loss of generality). Consequently ∂V (6) BH ( φ ) ∂φ = 0 ⇔ V (6) BH ( φ ) = 0 ⇔ φ = −∞ . (3.2.7)We then conclude that, besides BPS BH attractors, also non-BPS extremal BH attrac-tors are excluded in (1 ,
0) supergravity in six dimensions. However, we can have a 0-dimensional black object by an intersection of a d = 6 BH with a d = 6 black string. Itsreduction to d = 5 gives a BH which carries both the string charge and the BH charge,with cubic invariant of the form [64] I = e z e Λ e Λ + e Λ C Λ αβ Q α Q β , (3.2.8)and d = 5 resulting BH entropy S (5) BH ∼ p | I | . Thus, even if the KK charge e z vanishes,one gets a contribution from the second term of Eq. (3.2.8). This is in contrast withthe case of the d = 6 dyonic extremal black string treated in Subsect. 3.1, where thenon-vanishing of the KK charge e z was needed in order to get a non-vanishing entropyfor the corresponding d = 5 BH, obtained by wrapping the d = 6 string on S .The inclusion of extra multiplets corresponding to d = 6 vector multiplets induces asignificative complication in the model. In particular, the moduli space of the non-BPSattractors drastically changes with respect to the case described in section 3.1. As weshall prove below, in the magic models the number of moduli becomes equal to s = 2 q instead of q as it was in the absence of these extra multiplets.Before entering into the detail of the magic models, let us argue the existence, atleast for the homogeneous spaces L ( q, P ) (and, for q = 4 m , L ( q, P, P ′ )) [52], of particularnon-BPS critical points where the same results of section 3.1 may still be directly applied.Indeed, it turns out that for the four magic models the non-BPS attractor moduli spaces7f dimension 2 q always contain as a subspace precisely the coset SO (1 ,q ) SO ( q ) (that is the modulispace of d = 6 non-BPS attractors for q +1 strings, as discussed above). Such submanifoldof the moduli space may be obtained by considering the particular critical point where Y α = 0. This critical point may always be reached because, as (3.2.3) and (3.6) show,the Y coordinates always appear quadratically in the effective potential (3.7). Then,for Y α = 0 the effective potential reduces to the one previously considered (Eq. (3.1.9)),whose non-BPS attractor solution is known to have q flat directions belonging to the coset SO (1 ,q ) SO ( q ) . This is in fact only half the total number of flat directions for these solutions. Itmay be understood because the non compact stabilizer of the non-BPS orbit (that is forexample F − ⊃ SO (1 ,
8) for q = 8 [65, 66]), mixes the X with Y variables, so that therestriction { Y α } = 0 implies the reduction of the orbit to its subgroup SO (1 , q ). The sameconsiderations may be directly extended, for charge configurations where the spinorialcharges are set to zero, to the series of homogeneous non-symmetric spaces L ( q, P ) (and,for q = 4 m , L ( q, P, P ′ )) [52], which always admit a non-BPS attractor point where all thespinorial moduli are zero. As before, this condition selects the submanifold SO (1 ,q ) SO ( q ) of thenon-BPS attractor moduli space, with the only difference that in this case the number q is not directly related to the number of spinorial moduli. N = 2 magic models For N = 2 supergravity, one can apply the general relations of real special geometry[61, 49], so that the efffective potential V ( φ, q ) = a IJ q I q J (3.2.1.1)takes a simpler form. Indeed, for N = 2 supergravity the vector kinetic matrix a IJ isrelated to the metric g xy of the scalar manifold via a IJ = h I h J + 32 h I,x h J,y g xy (3.2.1.2) a IJ = h I h J + h I,x h J,y g xy or conversely g xy = 32 h I,x h J,y a IJ . (3.2.1.3)In terms of these quantities the central charge is Z = q I h I (3.2.1.4)and we can write the potential as V ( q, φ ) = Z + 32 g xy ∂ x Z∂ y Z (3.2.1.5)where ∂ x Z = q I h I,x = P ax Z a are the matter charges. The index x = 1 , · · · , n V is a worldindex labelling the scalar fields while a is the corresponding rigid index. P ax denotes thescalar vielbein. The matter charges obey the differential relations: ∇ Z = P a Z a ; ∇ Z a = 23 g ab P b Z − r T abc P b g cd Z d . (3.2.1.6)8o make explicit computations of the attractor points of the potential and of the corre-sponding Hessian matrix, let us use the property that both T abc and g ab , written in rigidindices, are invariants of the group SO( q + 1), where q = 1 , , , L ( q, H –representation R of the scalarfields branch with respect to SO( q + 1) in the following way R → + ( q + ) + R s , (3.2.1.7)where R s is the real Clifford module of SO( q + 1) of dimensions dim( R s ) = 2 , , , q . The index a split into the indices 1 , m, α , where m = 1 , . . . , q + 1 and α = 1 , . . . , dim( R s ). Let us write the general form for T abc and g ab : g = α ; g mn = β δ mn ; g αβ = γ δ αβ ; T = r α g ; T mn = − r α g mn ; T αβ = 12 r α g αβ ; T nαβ = − γ r β Γ nαβ , (3.2.1.8)where Γ n are the (symmetric, real) SO( q + 1) gamma matrices in the R s representation.The coefficients of T abc are determined in terms of the coefficients of g ab by the followingrelation: T a ( bc T aef ) = 12 g ( bc g ef ) . (3.2.1.9)The potential V can be written in the following useful form: V = Z + 32 g ab Z a Z b = Z + 32 (cid:0) Z Z + Z n Z n + Z α Z α (cid:1) , (3.2.1.10)where the following short-hand notation is used: Z a ≡ g ab Z b . Let us now compute theextrema of V . Using eqs. (3.2.1.6) we find ∇ V = P (cid:20) Z Z − √ α (cid:18) Z Z − Z n Z n + 12 Z α Z α (cid:19)(cid:21) ++ P n Z Z n + 2 r α Z Z n + 32 γ p β Γ nαβ Z α Z β ! ++ P α Z Z α − r α Z Z α + 3 √ β Γ nαβ Z n Z β ! . (3.2.1.11)It is straightforward to see that the above expression has two zeroes corresponding to thetwo attractors: • BPS attractor: Z n = Z α = Z = 0 and the potential at the extremum reads V = Z ; • non-BPS attractor: Z n = Z α = 0, Z = q α Z and the potential at theextremum reads V = 9 Z . 9et us now compute the Hessian matrix: ∇ V = ( P ) α Z − r α Z ! + 8 r α Z ! + 2 αβ Z n + α γ Z α ++ P P n (cid:18) Z Z n + 16 r α Z Z n − √ αβγ Γ nαβ Z α Z β (cid:19) ++ P P α (cid:18) Z Z α − r α Z Z α + r αβ Γ nαβ Z n Z β (cid:19) ++ P n P m Z n Z m + β Z + 2 14 r α Z ! + 3 β γ Z α δ mn ++ P n P α Z n Z α + 8 p β Γ nαβ ZZ α + r βα Γ nαβ Z β Z + 3 (Γ m Γ n ) αβ Z m Z β ! ++ P α P β γ Z − r α Z ! δ αβ + 3 γ β Z n δ αβ + 4 γ √ β Γ nαβ ZZ n −− γ r αβ Γ nαβ Z Z n + 32 Γ nαδ Γ nβγ Z δ Z γ + 92 Z α Z β (cid:21) (3.2.1.12)At the BPS critical point it is straightforward to check that: ∇ V = 83 Z g ab P a P b . (3.2.1.13)As expected, the BPS critical point is a stable attractor. At the non-BPS attractor theHessian reads: ∇ V = 24 Z (cid:2) g ( P ) + g mn P n P m (cid:3) . (3.2.1.14)The moduli space is therefore spanned by the scalar fields in the R s representation. Thesecan be regarded as particular coordinates of the moduli spaces of the N = 2, d = 5 non-BPS solutions of the magic models J O , J H , J C and J R , which respectively are F − SO (9) , USp (4 , USp (4) × USp (2) , SU (2 , SU (2) × U (1) and SL (2 , R ) SO (2) (see Table 4 of [36]). It is worth pointing out that,with the exception of J O , all such spaces can be obtained as consistent truncations ofthe N = 8, d = 5 BPS attractor moduli space F USp (6) × USp (2) (quaternionic K¨ahler), byperforming an analysis which is the d = 5 counterpart of the d = 4 analysis exploited in[33]. Since for J C and J R the N = 8 −→ N = 2 reduction preserves n H = 1 and n H = 2hypermultiplets respectively, the following inclusions must hold: J C : F ⊃ ( SU (2 , = ⇒ F U Sp (6) × U Sp (2) ⊃ SU (2 , SU (2) × U (1) × SU (2 , SU (2) × U (1) ;(3.2.1.15) J R : F ⊃ SL (2 , R ) × G = ⇒ F U Sp (6) × U Sp (2) ⊃ SL (2 , R ) SO (2) × G SO (4) . (3.2.1.16)The two group embeddings given by Eqs. (3.2.1.15) and (3.2.1.16) are discussed inAppendix. 10n the other hand, the truncation generating J H implies J H : F ⊃ U Sp (4 ,
2) = ⇒ F U Sp (6) × U Sp (2) ⊃ U Sp (4 , U Sp (4) × U Sp (2) . (3.2.1.17)In this case, the of U Sp (8) decomposes along
U Sp (6) × U Sp (2) as −→ ( , ) ⊕ ( ′ , ). The and ′ of U Sp (6) further decompose with respect to
U Sp (4) × U Sp (2)(maximal compact subgroup of the stabilizer
U Sp (4 ,
2) of the non-BPS orbit) as follows: −→ ( , ) ⊕ ( , ) ⊕ ( , ) ; ′ −→ ( , ) ⊕ ( , ) . (3.2.1.18)Thus, the decomposition of the ( , ) and ( ′ , ) of U Sp (6) × U Sp (2) with respect to
U Sp (4) × U Sp (2) × U Sp (2) read: massive : ( , ) −→ ( , , ) ⊕ ( , , ) ⊕ ( , , ) ; massless : ( ′ , ) −→ ( , , ) ⊕ ( , , ) . (3.2.1.19)Since in the non-BPS case the N = 2 R -symmetry is the U Sp (2) ∼ SU (2) inside U Sp (6) ( i.e. the first
U Sp (2) in the decomposition (3.2.1.19)) one obtains 8 massive and20 massless hypermultiplets’ degrees of freedom, and 6 massive and 8 massless vectors’degrees of freedom. Notice that, since in the BPS case the N = 2 R -symmetry is the U Sp (2) ∼ SU (2) commuting with U Sp (6) ( i.e. the second
U Sp (2) in the decomposition(3.2.1.19)), the non-BPS case differs from the BPS case only by an exchange of the ( , , )representation with the ( , , ) one. N -extended theories For any extended supergravity in five dimensions the BH potential enjoys the generalexpression in terms of the dressed charges [55, 56]: V ( φ, q ) = 12 Z AB Z AB + X + Z I Z I (4.1)where Z AB ( A, B = 1 , · · · N ) are the antisymmetric, Sp ( N )-traceless graviphoton centralcharges, X the trace part while Z I ( I = 1 , · · · , n ) denote the matter charges (which onlyappear for N ≤ ∂V∂φ i = 0 . (4.2)We are going to study in the following the BPS and non-BPS attractors for the variouscases. 11 .1 N = 8 , d = 5 and (2 , , d = 6 The scalar manifold is the coset
G/H = E Sp (8) . (4.1.1)and the BH potential takes the form: V = 12 Z AB Z AB . (4.1.2)The differential relations among the 27 central charges Z AB (satisfying Z AB Ω AB = 0),are: ∇ Z AB = 12 Z CD P ABCD , (4.1.3)where the vielbein P ABCD = P ABCD,i dφ i satisfies the conditions P ABCD = P [ ABCD ] , P ABCD Ω AB = 0 . (4.1.4)The extremum condition is then ∇ V = Z AB ∇ Z AB = 12 P ABCD Z AB Z CD = 0 . (4.1.5)To explicitly find the solution, it is convenient to put the central-charge matrix in normalform: Z AB = e e e
00 0 0 − e − e − e ⊗ (cid:18) − (cid:19) , (4.1.6)and to truncate the theory to the “charged” submanifold spanned by the vielbein com-ponents that couple to the dressed charge in normal form, that is: P ≡ P = P , P ≡ P = P while P = P = − P − P . (4.1.7)In this way, the covariant derivatives of the charges (4.1.3) become: ∇ e = ( e + 2 e + e ) P + ( e + e + 2 e ) P ∇ e = ( e − e ) P + ( − e − e − e ) P ∇ e = ( − e − e − e ) P + ( e − e ) P . (4.1.8)Using these relations, the extremum condition of V becomes ∇ V = 4 { P ( e − e )( e + 2 e + e ) + P ( e − e )( e + e + 2 e ) } = 0 . (4.1.9)It admits only one solution with finite area, which breaks the symmetry Sp (8) → Sp (2) × Sp (6). Up to Sp (6) rotations it is: e = e = − e ; V extr = 43 e = 43 M extr . (4.1.10)12his is a BPS attractor, supported by the unique BPS orbit [49] E F , and the maximumamount of supersymmetry preserved by the solution at the horizon is 1/4 ( N = 8 → N =2). As mentioned above, the two vielbein-components P and P span the submanifoldof the moduli space which couples to the proper values of the central charge. Thisautomatically projects out, in the N = 2 reduced theory, the 28 scalar degrees of freedomcorresponding to the hypermultiplets.The Hessian matrix reads H ij ≡ ∇ i ∇ j V = 14 P ABLM P CDLM Z AB Z CD . (4.1.11)To have the complete spectrum of massive plus flat directions, we have to consider in(4.1.11) the complete vielbein P ABCD . On the solution, where Sp (8) → Sp (2) × Sp (6)( A → ( α, a ), α = 1 , a = 1 , · · · , → ( , ) + ( ′ , ) P ABCD → P αβab + P αabc (4.1.12)where P αβab = ǫ αβ P ab (satisfying P ab Ω ab = 0) is the vielbein of the SU ∗ (6) Sp (6) N = 2 vectormultiplet sigma model, while P αabc (satisfying P αabc Ω ab = 0) spans the N = 2 hyperscalarsector. Note that, at the horizon, from (4.1.6) and (4.1.10) we find, for the central chargein normal form: Z AB → ( Z ab = e Ω ab ; Z αβ = − e ǫ αβ ) (4.1.13)The Hessian matrix (4.1.11) is then: H ij = 14 (cid:0) P abLM Z ab + P αβLM Z αβ (cid:1) (cid:0) P cdLM Z cd + P γδLM Z γδ (cid:1) = 9 e P LM,i P LM,j . (4.1.14)The hyperscalar vielbein P αabc do not appear in (4.1.14) so that the corresponding direc-tions do not acquire a mass. The moduli space of the solution is then [36] F USp (6) ⊗ USp (2) .The N = 8, d = 5 theory has an uplift to (2 , d = 6 supergravity, whose scalarmanifold is SO (5 , SO (5) × SO (5) [50]. In such a theory, the unique orbit with non-vanishing area isthe -BPS orbit SO (5 , SO (5 , [67], specified by an SO (5 ,
5) charge vector e Λ with non-vanishingnorm e Λ e Λ = 0. The corresponding moduli space of -BPS attractors is SO (5 , SO (5) × SO (4) , andit is indeed contained [66] in the N = 8, d = 5 -BPS moduli space F USp (6) × USp (2) ,as implied by our analysis. Note that the two non-compact forms of F which occurin N = 2 and N = 8, d = 5 supergravities precisely contain the two non-compactforms of SO (9) present in the corresponding moduli spaces [66]: F − ⊃ SO (1 ,
8) and F ⊃ SO (5 , N = 6 ( N = 2 , J H ) The scalar manifold is the coset
G/H = SU ∗ (6) U Sp (6) , (4.2.1)13he BH potential takes the form: V = 12 Z AB Z AB + 13 X , (4.2.2)and the differential relations among the 14+1 central charges Z AB (satisfying Z AB Ω AB =0) and X , are: ∇ Z AB = Ω CD Z C [ A P B ] D + 16 Ω AB Z CD P CD + 13 XP AB ∇ X = 12 Z AB P AB , (4.2.3)where P AB = P AB,i dφ i is the Ω-traceless vielbein of G/H satisfying the conditions P AB = P [ AB ] , P AB Ω AB = 0 . (4.2.4)To study the attractors, it is convenient to put the central-charge matrix in normal form: Z AB = e e
00 0 − e − e ⊗ (cid:18) − (cid:19) , (4.2.5)so that the BH potential takes the form V = e + e + ( e + e ) + 13 X . (4.2.6)The vielbein components that couple to the dressed charges in normal form are: P ≡ P , P ≡ P while P = − P − P . (4.2.7)In this way, the covariant derivatives of the charges (4.2.3) become: ∇ e = 13 ( − e + e + X ) P + 13 ( e + 2 e ) P ∇ e = 13 (2 e + e ) P + 13 ( e − e + X ) P ∇ X = (2 e + e ) P + ( e + 2 e ) P . (4.2.8)Using these relations, the extremum condition of V becomes ∇ V = 23 XZ AB P AB + Ω CD Z CA Z AB P BD = 2 n P (2 e + e )( e + 23 X ) + P ( e + 2 e )( e + 23 X ) o = 0 (4.2.9)Two inequivalent solutions with finite area are there:1. e = e = − X , giving for the Bekenstein–Hawking entropy V extr = 3 X .This is the N = 6 1/6-BPS solution and breaks the symmetry of the theory to Sp (4) × Sp (2). 14. e = e = 0, with Bekenstein–Hawking entropy V extr = X .It is a non-BPS attractor of the N = 6 theory, and leaves all the Sp (6) symmetryof the theory unbroken.Since the bosonic sector of this theory coincides with the one of an N = 2 theory basedon the same coset space [49], these are also the attractor solutions of the corresponding N = 2 model. In the N = 2 version, however, the interpretation of the attractor solutionsas BPS and non-BPS are interchanged.To study the stability of the solutions, let us consider the Hessian matrix H ij ≡ ∇ i ∇ j V = P AB P CD h Z AC Z BD + 29 X Ω AC Ω BD − XZ AC Ω BD ++ Z AL Z LM Ω BC Ω MD i = P AB P CD (cid:18) Z AC − X Ω AC (cid:19) (cid:18) Z BD − X Ω BD (cid:19) + − P AB P DB (cid:18) Z AC − X Ω AC (cid:19) (cid:18) Z CD − X Ω CD (cid:19) (4.2.10)and evaluate it on the two extrema. In the first case (BPS N = 6, non-BPS N = 2) thesolution breaks the symmetry to Sp (4) × Sp (2), ( A → ( α, a ), α = 1 , a = 1 , · · · , Z AB → ( Z ab = − X Ω ab ; Z αβ = 43 X ǫ αβ ) , (4.2.11)so that Z AB − X Ω AB → (cid:26) Z ab − X Ω ab = − X Ω ab Z αβ − Xǫ αβ = X ǫ αβ . (4.2.12)Corresponding to the group decomposition of the degrees of freedom: → ( , ) + ( , ) + ( , ) P AB → ( P ab ; P ; P aα ) , (4.2.13)the scalar vielbein decomposes as P AB → ǫ αβ PP αa ≡ − P aα P ab − Ω ab P (4.2.14)where P ab is the SO (1 , SO (5) vielbein, satisfying P ab Ω ab = 0.On the solution, the Hessian matrix (4.2.10) is then: H ij = 2 X (cid:0) P ab P ab + 3 P (cid:1) . (4.2.15)As expected, the directions corresponding to the scalars in the ( , ) of Sp (4 , Sp (4) × Sp (2) areflat. When the theory is interpreted as an N = 6 one, this is the BPS solution whose15tates, regarded as N = 2 BPS multiplets, have flat directions corresponding to thehyperscalar sector. On the other hand, in the N = 2 interpretation this is instead thenon-BPS solution, and now the flat directions correspond to degrees of freedom in thevector multiplets’ moduli space.The second solution (non-BPS N = 6, BPS N = 2) leaves all the Sp (6) symmetryunbroken since the horizon value of the central charge matrix in normal form is now: Z AB → . (4.2.16)Now the vielbein degrees of freedom do not decompose at all → P AB → P AB (4.2.17)and correspondingly all the scalar degrees of freedom become massive.Let us end this section by writing the quantities used here in the N = 2 formalismadopted in subsubsection 3.2.1. In this case the rigid index a labelling the tangent spacedirections are replaced by the antisymmetric traceless couple [ AB ] (recall that we use theconvention that any summation over an antisymetrized couple always requires a factor1 /
2) : T A A ,B B ,C C = 2 r (cid:18) Ω A B Ω B C Ω C A −
16 Ω A A Ω B C Ω B C −−
16 Ω B B Ω A C Ω A C −
16 Ω C C Ω B A Ω B A + 118 Ω A A Ω B B Ω C C (cid:19) ; g A A ,B B = Ω B A Ω B A −
16 Ω A A Ω B B , (4.2.18)where antisymmetrization in the couples ( A , A ) , ( B , B ) , ( C , C ) is understood. Asfar as the central charges are concerned, we have the following correspondence: Z = 1 √ X ; P a Z a = 12 √ P AB Z AB . (4.2.19) N = 4 , d = 5 and (2 , , d = 6 The scalar manifold is the coset
G/H = O (1 , × SO (5 , n ) Sp (4) × SO ( n ) , (4.3.1)spanned by the vielbein dσ , P IAB ( A, B = 1 , · · · , I = 1 , · · · , n ), where dσ = ∂ i σdφ i is the vielbein of the O (1 ,
1) factor while P IAB = P IAB,i dφ i is the Ω-traceless vielbein of SO (5 ,n ) Sp (4) × SO ( n ) satisfying the conditions P IAB = P I [ AB ] , P IAB Ω AB = 0 . (4.3.2)16he bare electric charges are a SO (5 , n )-singlet e and a SO (5 , n )-vector e Λ (the weightwith respect to SO (1 ,
1) is +2 for e and − e Λ ).The BH potential reads: V = 12 Z AB Z AB + 4 X + Z I Z I , (4.3.3)and the differential relations among the 5 central charges Z AB (satisfying Z AB Ω AB = 0),the singlet X and the n matter charges Z I are [55]: ∇ Z AB = Z I P IAB − Z AB dσ ; (4.3.4) ∇ X = 2 Xdσ ; (4.3.5) ∇ Z I = 12 Z AB P IAB − Z I dσ , (4.3.6)yielding ∇ V = 2 P IAB (cid:0) Z AB Z I (cid:1) + 2 dσ (cid:18) X − Z AB Z AB − Z I Z I (cid:19) . (4.3.7)The central charge matrix may be put in normal form: Z AB = (cid:18) e − e (cid:19) ⊗ (cid:18) − (cid:19) , (4.3.8)so that the BH potential takes the form V = 2 e + 4 X + Z I Z I , (4.3.9)and the differential relations among the dressed charges become ( dσ and P I ≡ P I = − P I are the components of the scalar vielbein coupling to the charges in normal form): ∇ e = Z I P I − e dσ (4.3.10) ∇ X = 2 Xdσ (4.3.11) ∇ Z I = 2 e P I − Z I dσ . (4.3.12)Then the extremization of the BH potential takes the form ∇ V = 8 P I (cid:0) e Z I (cid:1) + 2 dσ (cid:0) X − e − Z I Z I (cid:1) = 0 . (4.3.13)Two inequivalent solutions with finite area are there:1. Z I = 0 ; e = 2 X .This is the N = 4 1/4-BPS solution and breaks the Sp (4) R -symmetry of thetheory to Sp (2) × Sp (2), leaving the SO ( n ) symmetry unbroken. It corresponds toan SO (5 ,n ) SO (4 ,n ) orbit of the charge vector.2. Z AB = 0 ; Z I Z I = 8 X .It is a non-BPS attractor of the N = 4 theory, corresponding to choose the vector Z I to point in a given direction, say 1, in the space of charges: Z I = 2 √ δ I . Thissolution breaks the symmetry of the theory to Sp (4) × SO ( n − SO (5 ,n ) SO (5 ,n − orbit of the charge vector.17n both cases the Bekenstein–Hawking entropy turns out to satisfy [55] S (5) BH = ( V | extr ) / = p | e e Λ e Λ | . (4.3.14)To study the stability of the solutions, let us consider the Hessian matrix H ij ≡ ∇ i ∇ j V (4.3.15)= P IAB,i P JCD,j (cid:18) Z AB Z CD δ IJ + 12 Z I Z J δ CDAB (cid:19) + − P IAB, ( i ∂ j ) σZ AB Z I + ∂ i σ∂ j σ (cid:18) Z AB Z AB + Z I Z I + 16 X (cid:19) (4.3.16)and evaluate it on the two extrema.On the BPS attractor solution, the R -symmetry Sp (4) is broken to Sp (2) × Sp (2)( A → ( α ˜ α )) and the dressed charges in normal form become Z AB → X (cid:18) ǫ αβ − ǫ ˜ α ˜ β (cid:19) , Z I → . (4.3.17)Correspondingly, the vielbein P IAB decomposes to ( P I ǫ αβ , − P I ǫ ˜ α ˜ β , P Iα ˜ α ) where P I and P Iα ˜ α are the vielbein of the submanifold SO (1 ,n ) SO ( n ) (spanning N = 2 vector multiplets) and SO (4 ,n ) Sp (2) × Sp (2) × SO ( n ) (spanning N = 2 hypermultiplets) respectively. Since on the solution Z AB P IAB → XP I , the Hessian matrix (4.3.15) then becomes: H ij = 8 X (cid:0) P I,i P I,j + 3 ∂ i σ∂ j σ (cid:1) (4.3.18)showing that the 4 n scalars parametrized by P Iα ˜ α , which correspond to N = 2 hyper-multiplets, have massless Hessian modes.On the other hand, the non-BPS solution breaks the symmetry SO ( n ) to SO ( n − I → , k ; k = 1 , · · · n −
1) so that the vielbein P IAB decomposes into ( P AB , P kAB ). TheHessian matrix on the solution is: H ij = 8 X (cid:18) P AB,i P AB,j + 3 ∂ i σ∂ j σ (cid:19) (4.3.19)Note in particular that the 5( n −
1) scalars corresponding to the vielbein P kAB , spanningthe submanifold SO (5 ,n − SO (5) × SO ( n − , are flat directions.For the N = 4 theory it is easy to find a six dimensional uplift in terms of the IIB ,(2 ,
0) chiral d = 6 theory coupled to n tensor multiplets [50] (at least for the anomaly-free case n = 21) on similar lines as performed in section 3. Indeed, similarly to thedimensional reduction of the N = 2 theory coupled to tensor multiplets only, in thedimensional reduction of the IIB theory from six to five dimensions the scalar con-tent is incremented only by the KK-dilaton, which provides a O (1 ,
1) factor commutingwith the SO (5 ,n ) SO (5) × SO ( n ) coset. Moreover, the vector content in the gravitational multipletis also incremented by one graviphoton (whose integral corresponds to the singlet charge X ). Since the KK-dilaton is stabilized on the attractor solutions, then the five dimen-sional attractors are in one to one correspondence with the six dimensional ones: on theBPS attractor there are 4 n flat directions (corresponding to the quaternionic manifold SO (4 ,n ) SO (4) × SO ( n ) ), while on the non-BPS solution there are 5( n −
1) flat directions (spanningthe coset SO (5) × SO ( n − Anomaly free (1 , , d = 6 supergravity with neutralmatter In this section we comment on the constraints that an N = 2, d = 5 supergravity shouldsatisfy in order to be uplifted to an an anomaly-free N = (1 , d = 6 theory.It is well known that in a (1 ,
0) supergravity with neutral matter the absence of thegravitational anomaly demands a relation among the triple n T , n V , n H of possible mattermultiplets (tensor, vector and hyper multiplets, respectively), namely [68, 69] n H − n V + 29 n T = 273 . (5.1)Moreover, the consistency of the gauge invariance of tensor and (Abelian) vectormultiplets requires that the gauge vector current is conserved, i.e. [62, 51, 70, 71, 72] d ∗ J α = η ΛΣ C Λ αβ C Σ γδ F β ∧ F γ ∧ F δ = 0 , (5.2)implying that ( C Λ αβ = C Λ( αβ ) ) η ΛΣ C Λ( αβ C Σ γδ ) = 0 . (5.3)Such a condition holds true for all symmetric real special manifolds [52], with the ex-ception of the sequence L ( − , P ), P > e.g. the Table 2 of [52]) the symmetric spaces are L ( q,
0) = L (0 , P ), q, P > “ generic sequence ” , extended to consider also the d = 5 up-lift of the so-called d = 4 stu model), L ( q,
1) for q = 1 , , , J R , J C , J H and J O , respectively) and L ( − ,
0) (the d = 5 uplift of the so-called d = 4 st model).The condition (5.1) for the magic models respectively gives the following allowedtriples ( n T , n V , n H ) [51]: J R : (2 , , J C : (3 , , J H : (5 , , J O : (9 , , . (5.4)Notice that for the J O -based supergravity n H = 28, so its corresponding quaternionicmanifold could be identified with the exceptional quaternionic K¨ahler coset [74] E − E × SU (2) (which is the quaternionic reduction - or equivalently the hypermultiplets’ scalar manifold- of the d = 4 J O -based supergravity [74, 52]).On the other hand, for the “ generic sequence ” there are two possible uplifts to d = 6,depending whether one starts with L ( q,
0) or L (0 , P ). Indeed, starting from L ( q,
0) thecondition (5.1) implies n H = 244 − q, (5.5)which demands 0 q
8, whereas starting from L (0 , P ) the same anomaly-free condi-tion yields n H = 244 + P, (5.6)which always admits a solution. 19he (1 , d = 6 theory obtained by uplifting the real special symmetric sequence L ( q,
0) has n V = 0 and n T = q + 1, and thus 1 n T
9. On the other hand, theanomaly-free (1 , d = 6 uplift of the real special symmetric sequence L (0 , P ) has n T = 1 and n V arbitrary, thus it may be obtained from the standard compactification ofheterotic superstrings on K manifolds (see e.g. [75]).The model L ( − ,
0) admits an anomaly-free uplift to d = 6, having n V = n T = 0 and n H = 273.All other homogeneous non-symmetric real special spaces do not fulfill the condition(5.2)-(5.3) in presence of only neutral matter, so they seemingly have a d = 6 uplift to(1 ,
0) supergravity which is not anomaly-free, unless they are embedded in a model wherea non-trivial gauge group is present, with charged matter [76, 77].
There are three theories with eight supercharges which admit black hole/black stringattractors, namely N = 2 supergravity in d = 4, 5, 6 dimensions. For symmetric specialgeometries, the entropy is respectively given by the quartic, cubic and quadratic invariantof the corresponding U -duality group in the three diverse dimensions. In this paperwe extend previous work [40] on the investigation of the BPS and non-BPS attractorequations of such theories, by relating them as well as the corresponding moduli spacesof (non-BPS) critical points.Furthermore, we related the moduli space of the N = 8, d = 5 BPS unique orbit tothe moduli space of N = 2, d = 5 non-BPS orbit for all magic supergravities, as wellas for the “ generic sequence ” of real special symmetric spaces. This latter is directlyrelated to the d = 6 tensor multiplets’ non-BPS moduli space, which describes a neutraldyonic superstring in d = 6.We also considered N = 4, d = 5 supergravity, and related its -BPS and non-BPSattractors to the ones of (2 , d = 6 theory. Also in this case the moduli space ofnon-BPS attractors is spanned by the d = 6 non-BPS flat directions, studied in [50].We stress that our analysis is purely classical and it does not deal with quantumcorrections to the entropy, so it should apply only to the so-called “large” black objects.We leave the study of the quantum regime to future work. Acknowledgments
We would like to acknowledge enlightening discussions with Massimo Bianchi and Au-gusto Sagnotti.A. M. would like to thank the Department of Physics, Theory Unit Group at CERNfor its kind hospitality and support during the completion of the present paper.The work of L.A., S.F. and M.T. has been supported in part by European CommunityHuman Potential Program under contract MRTN-CT-2004-005104 “Constituents, funda-mental forces and symmetries of the universe”, in which L.A. and M.T. are associated toTorino University, and S.F. is associated to INFN Frascati National Laboratories. Thework of S.F. has also been supported in part by D.O.E. grant DE-FG03-91ER40662, TaskC. 20 ppendix : Relevant Embeddings
Let us first fix the notations to be used in the this appendix. If α is a root of a complexLie algebra g , the normalizations of the corresponding non-compact Cartan generator H α and of the shift generators E ± α will be defined as follows [78]: H α = 2( α · α ) α i H i ; ( H i , H j ) = δ ij ,E − α = ( E α ) † ; ( E α , E − α ) = 2( α · α ) , (A.1)where ( · , · ) is the Killing form. The above normalizations imply the following commuta-tion relations [ H α , E β ] = h β, α i E β ; [ E α , E − α ] = H α , h β, α i = 2( α · α ) β · α . (A.2) J C , d = 5 : the SU(2 , ⊂ F embedding The simple roots of the sl (3 , C ) subalgebra of f over C are defined in terms of the simpleroots of the latter α k ( k = 1 , . . . , α , α being long roots) as follows a = α , a = α , b = α , b = α + 3 α + 4 α + 2 α . (A.3)The real form f contains an sl (2 , R ) subalgebra defined by the following mutuallyorthogonal roots: a , b , c = α + α + α , d = α + α + 2 α + 2 α . (A.4)We can define the roots of f using a Cartan subalgebra h generated by two non-compact H a , H b and two compact i H c , i H d generators, the latter corresponding to the so (2) generators inside sl (2 , R ) c ⊕ sl (2 , R ) d . In terms of the generators of h , we canchoose a basis of Cartan generators for sl (3 , C ) to consist of H a , H b as well as of H a = −
12 ( H a + i H c − H d ) , H b = −
12 ( H a − i H c − i H d ) . (A.5)These generators define the Cartan subalgebra of an su (2 , subalgebra of f . Indeedone can verify that the sl (3 , C ) root system defined by the simultaneous eigenvalues ofthe h generators, is stable with respect to the conjugation σ relative to f , namely that a σ = a ; a σ = − ( a + a ) ; b σ = b ; b σ = − ( b + b ) . (A.6)The su (2 , generators are thus defined by σ –invariant combinations of the sl (3 , C ) shift generators. The fact that this construction defines an su (2 , subalgebra of f and not an sl (3 , R ) algebra is proven by the existence in each factor of a compact Cartansubalgebra, defined by the generators { E a − E − a , i ( H c − H d ) } for the first factor and { E b − E − b , i ( H c + H d ) } for the second. 21 R , d = 5 : the SL(2 , R ) × G ⊂ F embedding Denoting by a the sl (2 , R ) root and by b , b the simple roots of g , the SL(2 , R ) × G generators can be written in terms of the F generators as follows: H b = H α + α + H α ; H b = H α +2 α = H α + H α ; E b = E α + α + E α ,E b = E α +2 α ; E b + b = − E α +2 α +2 α + E α +2 α + α ,E b + b = − E α +2 α +2 α + α + E α +2 α +2 α ; E b + b = − E α +2 α +2 α +2 α ,E b +2 b = E α +3 α +4 α +2 α ,H a = 2 ( H α + α + H α + α + α ) ; E a = √ E α + α + E α + α + α ) . (A.7) Matrix representation of f generators For the sake of completeness, let us give below an explicit realization of the generators H α i , E α i and f , in the fundamental representation. f generators: H α = diag( − , , , , , , , , , − , , , , − , , − , − , , , , , − , , , , , ,H α = diag(0 , , , − , , , , , , − , − , , , , , , , − , − , , , , , − , , ,H α = diag(0 , , − , , − , , , − , , , , − , , , , − , − , , , , − , , − , , − , ,H α = diag(1 , − , , , , , − , , − , , − , , , , − , , − , , − , , , − , , , , − ,E α = I , + I , + I , + I , + I , + I , ,E α = I , + I , + I , + I , + I , + I , ,E α = I , + I , + I , + I , + c I , + c I , + c I , + c I , + I , + I , ++ I , + I , ,E α = I , − I , − I , − I , + c I , + c I , + c I , + c I , − I , − I , −− I , + I , , (A.8)where c = (1 + √ / c = (1 − √ / I I,J ) KL = δ IK δ JL . The Killing form is( M , M ) = Tr( M M ). References [1] S. Ferrara, R. Kallosh and A. Strominger, N = , Phys. Rev. D52 , 5412 (1995), hep-th/9508072 .[2] A. Strominger,
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