Intersecting Black Attractors in 8D N=1 Supergravity
R. Ahl Laamara, L.B Drissi, F.Z Hassani, E.H Saidi, A.A Soumail
aa r X i v : . [ h e p - t h ] N ov LPHE-MS-10-03 / CPM-1002
Intersecting Black Attractors in N = R. Ahl Laamara , , L.B Drissi , F.Z Hassani , E.H Saidi , A.A Soumail ,
1. LPHE-Modeling and Simulation, Facult´e des Sciences, Rabat, Morocco2. Centre of Physics and Mathematics, CPM, CNESTEN, Morocco3. Inanotech-MAScIR, Institute of Nanomaterials and Nanotechnology, Rabat, Morocco
September 7, 2018
Abstract
We study intersecting extremal black attractors in non chiral N = 1 su-pergravity with moduli space SO (2 ,N ) SO (2) × SO ( N ) × SO (1 ,
1) and work out explicitly theattractor mechanism for various black p-brane configurations with the typical nearhorizon geometries
AdS p +2 × S m × T − p − m . We also give the classification ofthe solutions of the attractor equations in terms of the SO ( N − k ) subgroups of SO (2) × SO ( N ) symmetry of the moduli space as well as their interpretations interms of both heterotic string on 2-torus and its type IIA dual. Other featuressuch as non trivial SO(1 ,
7) central charges Z µ ...µ p in N = 1 supergravity andtheir connections to p-form gauge fields are also given. Key Words : Supergravity, Superstring compactifications, Attractor Mecha-nism, Intersecting Attractors.
PACS numbers : 04.70.-s, 11.25.-w, 04.65.+e, 04.70.-s, 04.50.+h, 04.70.Dy
During the last decade, black attractor solutions in supergravity theories have been asubject of big interest; especially in connection with low energy superstring and
M-theory compactifications [1]-[18]. Because of their specific properties [19]-[38],static, asymptotically flat and spherically symmetric extremal (vanishing temperaturefor non-zero entropy) black attractors have been investigated for supergravities in diverse1pace time dimensions; with various numbers of conserved supersymmetries [39]-[56].Guided by the new solutions on extremal BPS and non BPS black attractors in higherdimensional supergravity; in particular those on intersecting attractors obtained first byFerrara et al. in [57], see also [58]; we focus in this paper on non chiral N = 1 su-pergravity with moduli space SO (2 ,N ) SO (2) × SO ( N ) × SO (1 ,
1) and study explicitly the attractormechanism for various configurations of extremal black p-branes with the typical nearhorizon geometries
AdS p +2 × S m × T − p − m where p = 0 , , , , m = 2 , , , , maximal N = 2 supergravity in ; it also gives new solutions, along the line of [57], classifiedby SO ( N − k ) subgroups of the SO (2) × SO ( N ) symmetry of the moduli space of thenon chiral N = 1 supergravity.The interest into this study is also motivated from the two following features: firstbecause of its conserved supersymmetries, extremal black attractors in this su-pergravity may be viewed as the ancestor of an interesting class of black holes in and supergravities [1, 5]; in particular in N = 2 and N = 4 resultingfrom adequate compactifications of the space time down to . It is also interestingfrom the view of higher dimensions since non chiral N = 1 supergravity may ariseas low energy of heterotic string on T and type IIA string on a real compact surfaceΣ that preserves half of the conserved supercharges of the type II superstrings.Black attractor solutions in offers therefore a framework to explicitly check specificfeatures of the heterotic/type IIA duality in [59].The paper is organized as follows. In section 2, we review the N = 1 supersymmetryalgebra with central charges Z µ ...µ p . We derive the various SO (1 ,
7) charges of these Z µ ...µ p ’s and give their connection with p-branes. It is also shown why the 3-form gaugefield in N = 1 theory should vanish. In section 3, we develop the study of the nonchiral N = 1 supergravity in and its links with the low energy limit of the heteroticsuperstring on T and its type IIA superstring dual. The various charges of the blackattractors are also given . In section 4, we first give the effective potential and the at-tractor eqs; then we derive their solutions together with their classification in terms of SO ( N − k ) subgroups of the SO (2) × SO ( N ) symmetry of the moduli space. In section5, we study the intersecting attractors along the line of the approach of [57, 58] and insection 6, we give a conclusion. N = 1 supersymmetry In this section, we identify the full set of the bosonic ”central charges” Z µ ...µ p involved inthe generalized non chiral N = 1 superalgebra and give their links to black p-brane2ttractors by using group theoretical methods.To start it is interesting to recall that like in space time, supersymmetry with sixteensupercharges may also live in other space times. In eight dimensions; this is precisely N = 1 supersymmetry given by a graded superalgebra with both commutators andanticommutators; it exchanges space time bosons into space time fermions. Inaddition to the twenty eight M [ µν ] symmetry generators of SO (1 , N = 1 non chiral supersymmetry is moreover generated by the energy mo-mentum vector operator P µ and the fermionic generators Q α and ¯ Q ˙ α transforming asWeyl spinors under SO (1 , , we give below some useful details. N = 1 superalgebras in We begin by recalling the group theoretical nature of the fermionic generators in N = 1 non chiral supersymmetry; these are SO (1 ,
7) spinors with 2 = 16 complex com-ponents that transform in the reducible 8 s ⊕ c representation of the Lorentz grouprespectively given by the eight component Weyl spinors Q + α and ¯ Q − ˙ α . These fermionicgenerators carry also charges under the SO (2) ∼ U R (1) R-symmetry of the supersym-metric algebra.Using general properties of tensor products of SO (1 , × U R (1) representations, we learnthat one may distinguish three kinds of N = 1 anticommutation relations in ; twochiral (complex) relations and a vector like (real) one:(1 , chiral relations in 8D These are complex relations involving only the fermionic generator Q + α , (cid:8) Q + α , Q + β (cid:9) = Z ++( αβ ) , (cid:2) Q + α , Q + β (cid:3) = Z ++[ αβ ] , (2.1)where the symmetric Z ++( αβ ) ’s should be thought of as operators carrying charges of blackbranes in . The antisymmetric term Z ++[ αβ ] ’s, which may be expanded as σ µν [ αβ ] M µν maybe interpreted in terms of SO (1 ,
7) rotations in the Weyl representation.(0 , antichiral relations in 8D These are the complex conjugate of (2.1); they involve the ¯ Q ˙ α spinor n ¯ Q − ˙ α , ¯ Q − ˙ β o = ¯ Z −− ( ˙ α ˙ β ) , h ¯ Q − ˙ α , ¯ Q − ˙ β i = ¯ Z −− [ ˙ α ˙ β ] , (2.2)where ¯ Z −− ( ˙ α ˙ β ) and ¯ Z −− [ ˙ α ˙ β ] are the complex conjugate of Z ++( αβ ) and Z ++[ αβ ] . N = (1 , superalgebra in 8D This is a vector like superalgebra with fermionic generators Q + α and ¯ Q − ˙ α obeying the3ollowing anticommutation relations, (cid:8) Q + α , Q + β (cid:9) = Z ++( αβ ) , n Q + α , ¯ Q − ˙ β o = Z α ˙ β , n ¯ Q − ˙ α , ¯ Q − ˙ β o = ¯ Z −− ( ˙ α ˙ β ) , (2.3)where the bosonic operators Z ++( αβ ) , ¯ Z −− ( ˙ α ˙ β ) are as before and where Z α ˙ β contains the usualenergy momentum vector P µ generating space time translations. N = 1 supersymmetry The bosonic operators Z ++( αβ ) , ¯ Z −− ( ˙ α ˙ β ), Z α ˙ β capture several irreducible SO (1 ,
7) space timerepresentations. To get their irreducible components, we use the correspondence Q + α ∼ (8 s , +1) , ¯ Q − ˙ α ∼ (8 c , −
1) (2.4)and tensor product properties of the SO (1 , × U R (1) representations; in particular(8 s , +1) × (8 s , +1) = (1 , +2) + (28 , +2) + (35 s , +2) ,(8 s , +1) × (8 c , −
1) = (8 v ,
0) + (56 v ,
0) ,(8 c , − × (8 c , −
1) = (1 , −
2) + (28 , −
2) + (35 c , −
2) . (2.5)The symmetry property of the anticommutators of eqs(2.3) allows to read the grouptheoretical structure of the Z ++( αβ ) , ¯ Z −− ( ˙ α ˙ β ) and Z α ˙ β ; we have: Z ++( αβ ) ∼ (1 , +2) ⊕ (35 s , +2) , Z α ˙ β ∼ (8 v , ⊕ (56 v ,
0) ,¯ Z −− ( ˙ α ˙ β ) ∼ (1 , − ⊕ (35 c , −
2) . (2.6)Notice that the sub-index i = v, s, c refer to the triality property of the SO (1 ,
7) sym-metry which have three kinds of fundamental representations with same dimension.Moreover, using the SO (1 ,
7) Dynkin labels ( l l l l ), the three eight dimensional basicrepresentations read as follows,8 v = (1000) , s = (0001) , c = (0010) . (2.7)With these basic representations, one can build the higher dimensional ones by takingtensor products. For the example of the leading lower dimensional representations, wehave 8 i × i = 1 + 28 + 35 i ,8 i × j = 8 k + 56 k , (2.8) viewed from , this corresponds to N = 4 supersymmetry with fermionic generators Q Iα in the(2 s ,
4) representation of SO (1 , × SU (4). i, j, k cyclic and where35 v = (2000) , s = (0002) , c = (0020) , v = (0011) , s = (1010) , c = (1001) . (2.9)Notice also that besides the real energy momentum vector P µ ∼ v and complex sin-glets Z ++0 = T r (cid:0) Z ++ αβ (cid:1) ∼
1, the bosonic operators Z ++( αβ ) , ¯ Z −− ( ˙ α ˙ β ), Z α ˙ β capture moreover SO (1 ,
7) higher dimensional representations namely the 35 s , 35 c , 56 v .In terms of SO (1 ,
7) vector indices, these representations may be decomposed by using antisymmetric products of the 8 × µ - matrices as follows Z ++( αβ ) = δ αβ Z ++0 + Γ µνρσ ( αβ ) Z ++[ µνρσ ] ,¯ Z −− ( ˙ α ˙ β ) = δ ˙ α ˙ β ¯ Z −− + Γ µνρσ ( αβ ) ˜ Z −− [ µνρσ ] , Z α ˙ β = Γ µα ˙ β Z µ + Γ µνρα ˙ β Z µνρ ] , (2.10)where antisymmetrization with respect to space time indices is understood. Notice thatan antisymmetric rank 4-tensor type Z [ µνρσ ] has in general × degrees of freedom; butthe 4- forms Z ++[ µνρσ ] and ˜ Z −− [ µνρσ ] involved in (2.10) capture each 35 degrees of freedomassociated with the self dual and anti-self dual antisymmetric 4-rank tensors in spacetime, Z ++ µ µ µ µ = ε µ ....µ Z ++ µ µ µ µ ,˜ Z −− µ µ µ µ = − ε µ ....µ ˜ Z −− µ µ µ µ . (2.11)From this group theoretical analysis, it follows amongst others the two following features:( ) the simplest form of the non chiral N = 1 supersymmetric algebra reads asfollows, n Q + α , Q − ˙ β o ∼ Γ µα ˙ β P µ , (cid:8) Q + α , Q + β (cid:9) ∼ n Q − ˙ α , Q − ˙ β o = 0 , (2.12)and corresponds to switching off the p-forms Z ++0 , Z µ , Z µνρ ] and Z ±± [ µνρσ ] , ( ) there are no Z ±± [ µν ] components in eq(2.10); this means that in non chiral N = 1supergravity we should have Z ++[ µν ] = 0 , Z −− [ µν ] = 0 , (2.13)showing in turn that the supergravity multiplet has 1-form and 2-form gauge fields; butno 3-form gauge field.Below, we switch on these charges and study extremal black attractors in non chiral N = 1 supergravity arising from superstring compactifications.5 .3 Central charges and branes From the above analysis, we learn that the bosonic Z - generators appearing in the gener-alized supersymmetric algebra (2.3) exhibit a set of remarkable properties; in particularthe three following ones:( ) to the bosonic operators Z µ ...µ p , which are charged under SO (1 , × U R (1), weassociate a space time p-form operator density Z p = p ! dx µ ∧ ... ∧ dx µ p Z µ ...µ p , (2.14)together with the charge, J p = R M p Z p , (2.15)where M p is a p-dimensional space time submanifold which may be thought of as theworld volume of a p-brane.( ) The Z µ ...µ p operators have an interpretation in terms of fluxes of gauge fields in nonchiral supergravity. By using the usual relations m = π R S F and e = π R S ˜ F giving the magnetic and electric charges of particles coupled to Maxwell gauge fieldsand thinking about the Z p ’s in the same manner, we end with the following relations Z ∼ R S F , Z ∼ R S F , Z ∼ R S F (2.16)as well as their duals. In these relations, the F p ’s stand for the gauge invariant p-forms, F = d A , F = d A , F = d A , (2.17)with Hodge duals ˜ F = ( ⋆ F ) , ˜ F = ( ⋆ F ) , ˜ F = ( ⋆ F ) , (2.18)from which we learn J p = R M p × S F p +2 , (2.19)teaching us that the Z p ’s describe precisely charges of p-branes that couple to the supergravity ( p + 1)- form gauge fields A p +1 with the field strengths F p +2 and theirmagnetic duals ˜ F − p .( ) Using the relation (2.13) and eqs(2.15-2.19) it follows that J = 0 and R M × S F = R ( ∂M ) × S C = 0 , (2.20)showing that, in non chiral N = 1 supergravity, there is no magnetic nor electric chargesassociated with the dyonic 4-form gauge invariant field strength F = d C . In otherwords there is no D2- brane in the type IIA set up of non chiral N = 1 supergravity.6 Fluxes of black attractors in In this section we study the non chiral N = 1 supergravity arising from low energycompactifications of superstring that preserve sixteen supersymmetric charges.We first study the case of N =1 supergravity embedded in heterotic string on T with moduli space M N =18 D - Het/T = SO (2 ,r +2) SO (2) × SO ( r +2) × SO (1 ,
1) , r ≥ ( r )2 . In thiscase, we will focus on the class of real surfaces given by the following union of irreducible2-cycles (2-spheres) C I Σ ( r )2 = C ∪ r − [ I =1 C I ! (3.2)with intersection matrix C I .C I = − K IJ (3.3)coinciding with the Cartan matrix of the of simply laced ADE Lie algebras. The modulispace of this theory is M N =18 D − IIA/ Σ = SO (2 ,r +1) SO (2) × SO ( r +1) × SO (1 ,
1) , r ≥ ( r )2 corresponds obviously to taking r = 0 and its singular limitgiven by vol ( C ) → SU (2) gauge symmetry.The similarity between M N =18 D - Het/T and M N =18 D − IIA/ Σ shows precisely the duality betweenthe two constructions; for details see [59]. First recall that the massless bosonic fields of the heterotic string belong to tworepresentations of the supersymmetric algebras; these are G DMN , B DMN , Φ Ddil of thesupergravity multiplet and the typical gauge fields A IM belonging to the Yang Mills mul-tiplets. As we are interested in this study into black attractor solutions, we will restrictbelow to the abelian sector and think about A IM as Maxwell gauge fields associated withthe Cartan subalgebra of a given rank r gauge group; i.e: I = 1 , ..., r. Under compactification of these bosonic fields on the two torus T , we get the following8 D ones G µν , B µν , σ, A Iµ , (3.5)together with the four 8 D gauge fields G iµ , B iµ , (3.6)7s well as the (4 + 2 r ) scalars G ( ij ) , B [ ij ] = ε ij B, A iI . (3.7)These fields combine into two N = 1 supermultiplets namely: • the gravity multiplet with bosonic content G µν , B µν , C iµ , σ (3.8)containing the graviton G µν , the B µν antisymmetric field, two gauge fields C iµ = (cid:0) C µ , C µ (cid:1) transforming as a real 2-vector under SO (2) R-symmetry; and the dilaton σ .The total number of the degrees of freedom of this gravity multiplet is ; theother superpartners come from the gravitino Ψ µα and a photino χ α carryingrespectively and fermionic degrees of freedom. • the Maxwell multiplets whose bosonic fields are given by A iµ , φ ij , A Iµ , φ iI . (3.9)These N = 1 supermultiplets contain (2 + r ) Maxwell gauge fields ( A iµ , A Iµ )which we denote collectively as A aµ with a = 1 , ...r + 2; and 2 ( r + 2) real scalars φ ia ≡ (cid:0) φ ij , φ iI (cid:1) . Together with these bosons, we also have r + 2 gauginos λ aα givenby pseudo-Majorana spinors in .The moduli space of this N = 1 supergravity that is embedded heterotic superstringon T reads as follows M N =18 D - Het/T = SO (2 ,r +2) SO (2) × SO ( r +2) × SO (1 ,
1) (3.10)where the extra factor SO (1 ,
1) refers to the dilaton σ and SO (2 ,r +2) SO (2) × SO ( r +2) for φ ia . Thisreal space has (2 r + 5) dimensions; it reduces for the particular case r = 0, to the fivedimensional one SO (2 , SO (2) × SO (2) × SO (1 ,
1) (3.11)In addition to the dilaton, the four scalars φ ij have geometric and stringy interpretations;three of them are given by the Kahler and complex structure of the 2-torus; the fourthis given by the value of the B NS field on T .The field strengths associated with the various gauge fields of the supergravity aregiven by the gauge invariant forms F i = d C i , F a = d A a , F = d B , (3.12)8or later use, we give below the magnetic and electric charges associated with these fieldstrengths as well as the brane interpretation; more details will be given when we considerthe type IIA dual derivation. We have: • a black hole and its 4-brane dual associated with the two graviphotons C iµ withmagnetic and electric charges as follows g i = R S F i , e i = R ˜ S ˜ F | i (3.13) • a black hole and its 4-brane dual associated with the ( r + 2) Maxwell fields A aµ ;their magnetic and electric charges are given by p a = R S F a , q a = R ˜ S ˜ F | a (3.14)these two kinds of magnetic and electric charges of the black hole/black 4-branecombine into SO (2 , r + 2) vector charges as given below P Λ = ( g i , p a ) , Q Λ = ( e i , q a ) (3.15) • a black string and its 3-brane dual associated with the B µν -field; the correspondingcharges are given by p = R S F , q = R ˜ S ˜ F (3.16)All these electric and magnetic charges are linked by the usual Dirac quantization rela-tion; they determine the effective potential V heteff = V heteff (cid:0) P, Q ; p , q , ... (cid:1) (3.17)of the black attractors to be considered later.Notice that one of the remarkable features of this analysis is the absence of the 4-formfield strengths F , ˜ F as predicted from the group theory view. In what follows, weexplore this issue by studying the type IIA dual compactification down to . Σ ( r )2 In this subsection, we study the embedding of non chiral N = 1 supergravity in typeIIA superstring on Σ ( r )2 . To that purpose, we first study the compactification of type IIAon the 2-sphere S corresponding to Σ (0)2 . Then, we extend the analysis to the surfaceΣ ( r )2 given by eq(3.2). 9n type ten dimensional type IIA superstring with supercharges, the massless bosonicparticles of the perturbative spectrum, describing the low energy type IIA super-gravity, is given by bosons : G MN , B MN , Φ dil , A M , C MNK , (3.18)with indices M, N, K = 0 , . . . , SO (1 ,
9) vectors.These fields capture on shell degrees of freedom partitioned as follows128 = 64 + 64 = (35 + 28 + 1) + (8 + 56) , (3.19)with the first coming from NS-NS sector ( G MN , B MN , Φ dil ) and the other fromRR-sector; i.e ( A M , C MNK ).We also have a non perturbative sector with p-branes namelyF1 string, NS 5-brane ; D0, D2, D4, D6. (3.20)Some of these branes are the source of the gauge fields involved in the Maxwell sectorof non chiral N = 1 supergravity. The fields are mainly similar to those given byeq(3.9); but here they should be thought of as the gauge fields associated with a D2-brane wrapped the irreducible 2-cycles of Σ ( r )2 . S After the space time compactification R , → R , × S where the local coordinates( x , ..., x ) get split as ( x , ..., x ) and y = ( z, ¯ z ) with z = x + ix parameterizing the2-sphere, the bosonic fields of the spectrum (3.18) reduces to: ( G µν φ , ( B µν φ , σ ; A µ , C µνρ , C µ , (3.21)with φ = G z ¯ z , φ = B z ¯ z , C µ = C z ¯ zµ (3.22)respectively describing the Kahler modulus of the 2-sphere, the B NS field on S and thegauge particle associated with a D2-brane wrapping S .This field spectrum has bosonic degrees of freedom; but only of them combinewith the gravitino Ψ µ = (cid:0) Ψ αµ , ¯Ψ µ ˙ α (cid:1) and the graviphotino χ = ( χ α , ¯ χ ˙ α ) to form thenon chiral N = 1 supergravity multiplet G µν , B µν , σ, A iµ ; Ψ µ , χ (3.23)10here A iµ stands for the SO (2) doublet ( A µ , C µ ). The on shell degrees of freedom arepartitioned as 48 bose = (20 + 15 + 1) + 2 × fermi = 40 + 8. We also have thefollowing branes, F1 string, ( NS 5-brane / S ) ; D0, (cid:0)
D6/S (cid:1) , (cid:0) D2/S (cid:1) , (cid:0) D4/S (cid:1) . (3.24)satisfying the usual electric magnetic duality relation between electric q-brane andits magnetic p-brane dual with the integers p and q constrained as p + q = 4. Σ ( r )2 The compactification of field content of the type IIA superstring on Σ ( r )2 extends the caseof the 2-sphere; it leads to the following:( ) a gravity multiplet; the same as in eq(3.23)( ) r Maxwell multiplets given by A aµ , φ ia , a = 1 , ..., r (3.25)where the φ a ’s are the Kahler parameters of the irreducible 2-cycles C a (2-spheres S a )involved in Σ ( r )2 and the φ a ’s stand for the values of the B NS fields on these S a ’s.The gauge fields A aµ are associated with the wrapping of D2-brane on the S a ’s of thecompact surface Σ ( r )2 ; i.e: A aµ : (D2/ S a ) (3.26)The moduli space M N =18 D - IIA/ Σ r of this N = 1 supergravity is parameterized by the(2 r + 1) scalars namely the dilaton σ and the SO (2) doublets φ ia ; it is given by M N =18 D - IIA/ Σ r = SO (2 ,r ) SO (2) × SO ( r ) × SO (1 ,
1) (3.27)This space is comparable to M N =18 D -het /T eq(3.10); this is due to the string-string dualitybetween the heterotic on T and type IIA superstring on Σ ( r )2 that follows from the wellknown duality relation in space time, Het/T ↔ Type IIA/K3 (3.28)The field strengths F p +1 = d A p , associated with the various gauge fields of the N = 1supergravity, are given by F i = d A i , F a = d A a , F = d B (3.29)11hese gauge invariant fields transform under the SO (2) × SO ( r ) group as follows F i F a F SO (2) × SO ( r ) (2 ,
1) (1 , r ) (1 ,
1) (3.30)The corresponding magnetic and electric charges are as follows:( a ) the string and its dual 3-brane p = R S F , q = R ˜ S ˜ F (3.31)the string is magnetically charged while the 3-brane is electrically charged.( b ) the black hole magnetic charges ( g i , p a ) and the electric duals ( e i , q a ) given by g i = R S F i , e i = R ˜ S ˜ F | i , p a = R S a F , q a = R ˜ S a ˜ F | a . (3.32)These are respectively SO (2) and SO ( r ) vectors.We end this section by noting that one may write down the lagrangian densities of thevarious gauge fields. For the bosonic sector we have, in addition to the Einstein-Hilbertterm πG R M √−GR D , two other contributions; the first one is given by the 1-formgauge fields A Λ µ = (cid:0) A iµ , A aµ (cid:1) L = πG R M √−G (cid:2) N ab F aµν F µνb + N ij F iµν F µνj (cid:3) , (3.33)where the field metric N ΛΓ = N ΛΓ ( φ, σ ) reads as e σ L i Λ ( φ ) δ ij L j Γ ( φ ) − e σ L a Λ ( φ ) η ab L b Γ ( φ )with the field matrix L ΛΥ parameterizing the SO (2 , N ) group [see eqs(4.4-4.5)] and φ ≡ (cid:8) φ ia (cid:9) being the free moduli that parameterize the moduli space SO (2 ,N ) SO (2) × SO ( N ) . Thesecond contribution comes from the B gauge field; it reads as follows L = πG R M √−G N ( σ ) F µνρ F µνρ , (3.34)where now N ( σ ) is the metric associated with the SO (1 ,
1) factor parameterized by thedilaton σ with no dependence in the φ ia s. We first describe the effective potential of the black branes in non chiral N = 1supergravity. Then, we study the attractor eqs and their solutions.12 .1 Effective potential The effective potential V eff of the black attractors has contributions coming from thevarious gauge fields of the non chiral N = 1 supergravity. As there is no contributioncoming from the D2-brane we have: V eff = ( V BH + V B ) + ( V string + V B ) , (4.1)with: ( a ) V BH is the effective potential of the black hole associated with the chargesof the two graviphotons A iµ and the Maxwell gauge fields A aµ . It is given by the usualWeinhold relation whose expression, in the flat coordinate frame, reads as: V BH = X i,j =1 δ ij X i X j + N X a,b =1 δ ab Y a Y b (4.2)In this relation, X i and Y a , which respectively transform as SO (2) and SO ( N ) vectors,are the dressed central charges related to the magnetic bare P Λ = ( g i , p a ) of eqs(3.32)like, X i = e σ L i Λ P Λ , Y a = e σ L a Λ P Λ . (4.3)In these relations, e σ and L ΛΥ parameterize respectively the SO (1 ,
1) and SO (2 , N )group factors of the moduli space of the non chiral N = 1 supergravity. Notice thatthe L ΛΥ matrix is a real (2 + N ) × (2 + N ) matrix L ΓΛ = L ji L bi L ja L ba ! , (4.4)satisfying the usual orthogonality relation L T ηL = η which explicitly reads like L ΓΛ η ΓΣ L ΣΥ = η ΛΥ , (4.5)with η ΛΥ = diag (+ , + , − , · · · , − ). A priori L ΛΓ has (2 + N ) parameters; but thisrelation may be viewed as a constraint relation that reduces this number down to ( N + 3 N + 2). Furthermore subtracting the ( N − N + 2) gauge degrees of free-dom captured by the SO (2) × SO ( N ) symmetry of the moduli space, we end with 2 N moduli parameterizing SO (2 ,N ) SO (2) × SO ( N ) .( b ) V B is the effective potential associated with the black 4D- branes dual the blackholes, V B = X i,j =1 δ ij ˜ X i ˜ X j + N X a,b =1 δ ab ˜ Y a ˜ Y b (4.6)The ˜ X i and ˜ Y a are dressed central charges related to the electric Q Λ = ( e i , q a ) as follows˜ X i = Q Λ ( L − ) Λ i e − σ , ˜ Y a = Q Λ ( L − ) Λ a e − σ (4.7)13 c ) the term ( V string + V B ) is the effective potential of the black string and its 3- brane(NS 5-brane/S ) dual; it is given by V string = e σ p , V B = e − σ q (4.8)where the magnetic charge p and the electric q one are as in eq(3.31).Adding all terms, we get the total effective potential of the black attractors in non chiral N = 1 supergravity V eff = + X i,j =1 (cid:16) X i δ ij X j + ˜ X i δ ij ˜ X j (cid:17) + ( e σ p + e − σ q )+ N X a,b =1 (cid:16) Y a δ ab Y b + ˜ Y a δ ab ˜ Y b (cid:17) (4.9)It is manifestly invariant under the SO (2) × SO ( N ) symmetry of the moduli space.Substituting the dressed central charges X i , ˜ X i , Y a , ˜ Y a by their explicit expressions interms of the field moduli, we end with a function depending on the electric and magneticcharges as well as on the scalars σ and L Λ M , V eff = V eff (cid:0) σ, L Λ M ; P Λ , Q Λ ; p , q (cid:1) . (4.10)More explicitly, we have V eff = + X i,j =1 (cid:16) e σ P Λ L i Λ δ ij L j Υ P Υ + e − σ Q Λ ( L − ) Λ i δ ij ( L − ) Υ j Q Υ (cid:17) + N X a,b =1 (cid:16) e σ P Λ L a Λ δ ab L b Υ P Υ + e − σ Q Λ ( L − ) Λ a δ ab ( L − ) Υ b Q Υ (cid:17) + ( e σ p + e − σ q ) (4.11)with L ΛΥ belonging to SO (2 , N ) as given by eqs(4.5).Notice that invariance of the effective potential V eff under the electric/magnetic dualitysymmetry between the charges of the black branes and their duals is captured by therelation ( M, σ ) → ( E, − σ ) with M standing form the magnetic charges and E for electricones. These are given as usual by minimizing the effective potential with respect to the fieldmoduli σ and L ΛΥ ; ∂ V eff ∂σ = 0 , ∂ V eff ∂L ΛΥ = 0 (4.12)14y taking into account the constraint relation L T ηL = η . This constraint relation maybe implemented in the effective potential by using the Lagrange multiplier method; fortechnical details see [47] developed for the case of black attractors in supergravity.We also need to compute the Hessian matrix ∂ V eff ∂σ = 0 , ∂ V eff ∂σ∂L ΛΥ = 0 , ∂ V eff ∂L ΛΥ ∂L ΓΣ = 0 (4.13)which needs to be positive definite for stable solutions. Computing ∂ V eff /∂σ = 0Now, using the fact that X i , ˜ X i , Y a , ˜ Y a are eigenvectors of ∂∂σ ; i.e ∂X i ∂σ = X i , ∂Y a ∂σ = Y a , ∂ ˜ X i ∂σ = − ˜ X i , ∂ ˜ Y a ∂σ = − ˜ Y a , the extremization with respect to the dilaton σ gives,0 = + X i,j =1 (cid:16) X i δ ij X j − ˜ X i δ ij ˜ X j (cid:17) + N X a,b =1 (cid:16) Y a δ ab Y b − ˜ Y a δ ab ˜ Y b (cid:17) +2 ( e σ p − e − σ q ) . (4.14)There are different ways to solve this attractor eq; one of them is to cast it as follows X i δ ij X j − ˜ X i δ ij ˜ X j = 0 , Y a δ ab Y b − ˜ Y a δ ab ˜ Y b = 0 , e σ p − e − σ q = 0 , (4.15)where summation on repeated indices is understood. An other way is to compensate theterms with X i , ˜ X i with the terms with Y b , ˜ Y a as follows, X i δ ij X j + Y a δ ab Y b = 0 ,˜ X i δ ij ˜ X j + ˜ Y a δ ab ˜ Y b = 0 , e σ p − e − σ q = 0 , (4.16)or like X i δ ij X j − ˜ Y a δ ab ˜ Y b = 0 ,˜ X i δ ij ˜ X j − Y a δ ab Y b = 0 , e σ p − e − σ q = 0 . (4.17)Further solutions are obtained by compensating X i , ˜ X i , Y b , ˜ Y a with e σ p and e − σ q ;for instance as follows: X i δ ij X j = e − σ q , e σ p = ˜ X i δ ij ˜ X j , Y a δ ab Y b = ˜ Y a δ ab ˜ Y b . (4.18)15e will give some explicit examples later on.Substituting the dressed central charges by their field expressions back into (4.15), weget the following attractor eqs e σ P Λ L i Λ δ ij L j Υ P Υ − e − σ Q Λ ( L − ) Λ i δ ij ( L − ) Υ j Q Υ = 0 , e σ P Λ L a Λ δ ab L b Υ P Υ − e − σ Q Λ ( L − ) Λ a δ ab ( L − ) Υ b Q Υ = 0 ,( e σ p − e − σ q ) = 0 . (4.19)Similar attractor eqs may be written down for the other cases given above. Computing δ L V eff = 0The extremization of the effective potential of the black attractors with respect to thefield matrix L ΛΥ is some how lengthy. Below, we give the main steps by using theexpression of V eff in terms of X i , ˜ X i , Y a , ˜ Y a . First, we have δ L ΛΥ V eff = +2 X i,j =1 h ( δ L ΛΥ X i ) δ ij X j + (cid:16) δ L ΛΥ ˜ X i (cid:17) δ ij ˜ X j i +2 N X a,b =1 h ( δ L ΛΥ Y a ) δ ab Y b + (cid:16) δ L ΛΥ ˜ Y a (cid:17) δ ab ˜ Y b i (4.20)where δ L ΛΥ X i and so on are the variation of the dressed central charges with respect tothe field matrix L ΛΥ . These variations may be nicely expressed in terms of the Maurer-Cartan 1-form Ω = dLL − = − L ( dL − ) , (4.21)of the orthogonal group SO (2 , N ). Indeed, denoting X i = e σ L i Λ P Λ in a condensed man-ner as X i = e σ ( L.P ) i and similarly for the other dressed central charges, the variationwith respect to L reads as δX i = e σ ( δL.P ) i . Now inserting the relation L − L = I , weget δX = e σ ( δL.L − .LP ) i where we recognize the Ω term. Doing the same for the otherdressed central charges, we end with the following result: δX i = Ω ik .X k + Ω ic .Y c , δY a = Ω ak .X k + Ω ac .Y c δ ˜ X i = − Ω ki . ˜ X k − Ω ci . ˜ Y c , δ ˜ Y a = − Ω ka . ˜ X k − Ω ca . ˜ Y c (4.22)Putting back into (4.20), we get the vanishing condition of δ L V eff (cid:16) X Ω X − ˜ X Ω ˜ X (cid:17) + (cid:16) X Ω Y − ˜ X Ω ˜ Y (cid:17) + (cid:16) Y Ω Y − ˜ Y Ω ˜ Y (cid:17) + (cid:16) Y Ω X − ˜ Y Ω ˜ X (cid:17) = 0 (4.23)from which we can learn the associated attractor eqs. In this relation, the condensedterms are as follows X Ω X = + X j Ω jk X k , ˜ X Ω ˜ X = + ˜ X j Ω jk ˜ X k Y Ω Y = − Y b Ω bc Y c , ˜ Y Ω ˜ Y = − ˜ Y b Ω bc ˜ Y c X Ω Y = + X j Ω jc Y c , ˜ X Ω ˜ Y = + ˜ X j Ω jc ˜ Y c Y Ω X = − Y b Ω bk X k , ˜ Y Ω ˜ X = − ˜ Y b Ω bk ˜ X k (4.24)16here the i,j indices are raised and lowered by δ ij and δ ij while the indices a, b are raisedand lowered by − δ ab and − δ ab . Notice also that we haveΩ (2+ N, N )ΛΥ = Ω (2 , ij Ω (2 ,N ) ib Ω ( N, aj Ω ( N,N ) ab ! (4.25)Notice as well that the attractor eqs of the black attractors (4.12) are given by eqs(4.14,4.23);a class of solutions of these eqs are given below. We first solve the attractor eq ∂ V eff /∂σ = 0 (4.14) allowing to fix the dilaton in termsof the electric and magnetic bare charges. Then, we consider the case of the attractoreqs ∂ V eff /∂L ΛΥ = 0. ∂ V eff /∂σ = 0Eq(4.14) may be solved in several ways:( ) no D-brane fluxes: P Λ = 0 , Q Λ = 0This configuration corresponds to X i = 0 , Y a = 0; ˜ X i = 0 , ˜ Y a = 0. Substituting, eq(4.14)reduces to ( e σ p − e − σ q ) = 0 whose solution is σ = − ln p q (4.26)giving the value of the dilaton in terms of the magnetic charge of the string and theelectric charge of the 3-brane. The near horizon geometry of this black attractor is givenby AdS × S and AdS × S depending on the values of the magnetic and electric charges.Notice that for p = 0 , σ → + ∞ while for q = 0 , σ → −∞ . ( ) general solutions These solutions correspond to compensate the contributions coming from the electricand magnetic sectors as in eqs(4.15,4.16,4.17,4.18). As an example, we consider the case X i δ ij X j = ˜ X i δ ij ˜ X j , Y a δ ab Y b = ˜ Y a δ ab ˜ Y b , σ = ln q p . (4.27)which may be solved in four ways by taking the dressed central charges as follows:( i ) : X i = + z i , ˜ X j = + z i δ ij , Y a = + y a , ˜ Y b = + y a δ ab ( ii ) : X i = + z i , ˜ X j = + z i δ ij , Y a = + y a , ˜ Y b = − y a δ ab ( iii ) : X i = − z i , ˜ X j = − z i δ ij , Y a = − y a , ˜ Y b = + y a δ ab ( iv ) : X i = − z i , ˜ X j = − z i δ ij , Y a = − y a , ˜ Y b = − y a δ ab (4.28)17nd σ as before and z i , y a some constants. The above relations may also written asfollows ( L − ) Λ i Q Λ = ± e − σ ( L ) j Υ P Υ δ ij ,( L − ) Λ b Q Λ = ± e − σ ( L ) a Υ P Υ δ ab , L T η SO (2 ,N ) L = η SO (2 ,N ) , (4.29)with e − σ given by eq(4.26) and whose solutions allow to express the field matrix L ΛΥ in terms of Q Λ , P Υ as well as p q .Notice that the moduli space of solutions of (4.28) depends on the arbitrary values z i and y a . For instance taking˜ Y b = y , Y a δ ab = ± y (4.30)the SO (2) × SO ( N ) symmetry of the moduli space get reduced down to SO ( N − Y b = y ... y n , Y a δ ab = ± y ... ± y n (4.31)and have a SO ( N − n ) symmetry. ∂ V eff /∂σ = 0 and ∂ V eff /∂L ΛΥ = 0We give here below two classes of solutions; others solutions classified by the SO ( N − n )symmetries can be also written down. Class I
The first class of solutions of the attractor eqs(4.12,4.14,4.23) is obtained by puttingeqs(4.26) and (4.28) back into eq(4.23); then cast it as follows: X Ω X − ˜ X Ω ˜ X = 0 Y Ω Y − ˜ Y Ω ˜ Y = 0 X Ω Y − ˜ X Ω ˜ Y = 0 Y Ω X − ˜ Y Ω ˜ X = 0 (4.32)18aking into account eqs(4.26, 4.28) solving ∂ V eff /∂σ = 0, it is not difficult to see thatthe solutions of eq(4.32) are classified as given below( i ) : X = + ˜ X = z z ! , Y = + ˜ Y = w ... w N ( ii ) : X = − ˜ X = z ′ z ′ ! , Y = − ˜ Y = w ′ ... w ′ N (4.33)together with σ = ln q p and where z i , w a and z ′ i , w ′ a are some constant numbers.Notice that the terms X Ω X = X i Ω ij X j and Y Ω Y = Y a Ω ab Y b are symmetric quadraticforms; so there no contribution coming from the antisymmetric parts Ω [ ij ] = (Ω ij − Ω) ji and Ω [ ab ] = Ω ab − Ω ba of the Cartan-Maurer forms. This property captures precisely the SO (2) × SO ( N ) symmetry of the moduli space (3.10-3.27) of non chiral N = 1supergravity.Notice also that for arbitrary values of z i , w a and z ′ i , w ′ a the symmetry group SO (2) × SO ( N ) of the effective potential is completely broken. The other possibilities wheresome of the parameters are zero or identical, the SO (2) × SO ( N ) symmetry of themoduli space is broken down to a subgroup G . Class II
This class of solutions corresponds to solving the extremum of V eff (4.23) by compen-sating the X i and ˜ X i factors with the Y a and ˜ Y a as in (4.16-4.17). A way to do it is asfollows: Y Ω Y = ˜ X Ω ˜
X , ˜ Y Ω ˜ Y = X Ω XY Ω X = ˜ X Ω ˜
Y , ˜ Y Ω ˜ X = X Ω Y (4.34)In this solution, the term Y Ω Y (resp ˜ Y Ω ˜ Y ) is compensated by ˜ X Ω ˜ X (resp X Ω X ); thiscorresponds to first breaking the moduli space SO (2) × SO ( N ) subsymmetry as SO (2) × SO ( N ) → SO (2) × SO (2) × SO ( N −
2) , (4.35)then compensate the terms associated with the two SO (2) factor. An explicit solution19s given by: X = z z ! , ˜ X = ˜ z ˜ z ! ,Y = ˜ z ˜ z , ˜ Y = z z (4.36)Other configurations with symmetries SO ( N − m ) with m = 3 , ...N may be also writtendown; one of them is given by X i and ˜ X i as in (4.36) and Y a and ˜ Y a like, Y = ˜ z ˜ z w , ˜ Y = z z w (4.37)with SO ( N −
3) symmetry.
Following [57, 58], one should distinguish two main classes of black p-brane solutions inhigher dimensional supergravity. In non chiral N = 1 supergravity we are consideringhere, these are:( ) the standard black p- brane solutions based on AdS p × S − p with p = 0 , , , ) the intersecting attractors with the typical near horizon geometries AdS p × S m × M − p − m (5.1)where S m is the real m-sphere and M n stands for some manifolds; essentially a n -torus.Moreover, since there is no D2- brane flux in this theory; these geometries are restrictedto black hole and black string geometries as well as their duals. As such, we have:( a ) AdS × S m × M − m ( b ) AdS × S m × M − m The novelty with these geometries is that they allow the two following features:20 i ) a variety of irreducible sub-manifolds that support various kinds of branes and so arich spectrum of electric and magnetic charges;( ii ) non trivial intersections between p i -/ p j - cycles of (5.1) leading to intersecting (BPSand non BPS) attractors.To illustrate the first point, consider the example of the two compact manifolds S m + n and M m + n = S m × T n . While the sphere S m + n supports only charges of ( m + n − F n + m = g ̟ n + m , g = R S m + n F n + m , (5.2)and no ( m − S m × T n allows however many possibili-ties. It has several irreducible k i - cycles that support, in addition to ( m + n − n types of ( m − g a = R C ( a ) m +1 F m +1 , F m +1 = P a g a ̟ m +1 | a , R C ( a ) m +1 ̟ m +1 | b = δ ab , a = 1 , ..., n , (5.3)with C ( a ) m +1 = n [ a =1 ( S a × S m ) , T n = n O a =1 S a . (5.4)The branes may be imagined as filling the fiber F ( a ) m − of these cycles C ( a ) m +1 thought of interms of the fibration C ( a ) m +1 ∼ F ( a ) m − × S with field strength F m +1 = β S ∧ (cid:18)P a g a β F ( a ) m − (cid:19) (5.5)Using the anzats of [52], we focus below on the study of various examples of thesetypical horizon geometries and work out new and explicit solutions regarding intersectingattractors in the case of non chiral N = 1 supergravity. As the solutions are verytechnical, we will concentrate on drawing their main lines and giving the results. AdS and AdS factors We distinguish several
AdS × S m × M − m and their AdS × S m × M − m duals geometries;in particular:(a) AdS × S × T with volume forms α AdS , β S and β T ,(b) AdS × S × T with volume forms α AdS , β S and β T ,(c) AdS × S × S with volume forms α AdS , β S and β S . Below, we study the two first ones. 21 .1.1
AdS × S × T On the geometry
AdS × S × T there is no irreducible 2-cycle nor irreducible 6-cyclethat support the fluxes emanating from the D0 and D6- branes. As such the blackattractor is given by, p-branes (4 − p )- branes p = 0 F Λ2 = 0 ˜ F | Λ = 0 p = 1 F = p β S ˜ F = q α AdS ∧ β T (5.6)from which we read the following effective potential, V eff = e σ p + e − σ q (5.7)The extremization of this potential with respect to the dilaton leads to e σ p − e − σ q = 0 (5.8)The solving of the above equation is given by σ = ln q p ; it fixes the dilaton σ athorizon in terms of the magnetic charge of the black string and the electric charge of theblack 3-brane. AdS × S × T In this geometry which involve the volume forms α AdS , β S , β T , the non vanishingfield strength charges are given byp-branes (4 − p )- branes p = 0 F Λ2 = P Λ β S ˜ F | Υ = Q Υ (cid:0) α AdS ∧ β T (cid:1) p = 1 F = p α AdS ˜ F = q ( β S ∧ β T ) (5.9)where Λ = ( i, a ) with i = 1 , a = 1 , ..., N. The effective potential V eff of these black attractor configuration reads as follows, V eff = + X i,j =1 (cid:16) X i δ ij X j + ˜ X i δ ij ˜ X j (cid:17) + ( e σ p + e − σ q )+ N X a,b =1 (cid:16) Y a δ ab Y b + ˜ Y a δ ab ˜ Y b (cid:17) (5.10)where the first term, which we write as XX + ˜ X ˜ X, is invariant under SO(2) and theterm Y Y + ˜ Y ˜ Y is invariant under SO ( N ). The extremization of V eff gives, (cid:16) X Ω X − ˜ X Ω ˜ X (cid:17) + (cid:16) X Ω Y − ˜ X Ω ˜ Y (cid:17)(cid:16) Y Ω Y − ˜ Y Ω ˜ Y (cid:17) + (cid:16) Y Ω X − ˜ Y Ω ˜ X (cid:17) = 0 e σ p − e − σ q = 0 (5.11)22here Ω is the Maurer Cartan 1-form of SO (2 , N ) introduced previously.The solutions of these attractor eqs may be realized in various ways; one of them is givenby the following: X = ± ˜ X , Y = ± ˜ Y , σ = − ln p q (5.12)These solutions correspond to diverse intersecting configurations composed of a blackhole, a black 4-brane, a black string,and a black 3-brane.Moreover, using eqs(4.22), we compute the following the Hessian matrix δδ V eff = +16 ( e σ p + e − σ q )+4 (cid:16) X Ω X − ˜ X Ω ˜ X + X Ω Y − ˜ X Ω ˜ Y (cid:17) +4 (cid:16) Y Ω Y − ˜ Y Ω ˜ Y + Y Ω X − ˜ Y Ω ˜ X (cid:17) + h X ( δ Ω) Y − ˜ X ( δ Ω) ˜ Y i + h Y ( δ Ω) Y − ˜ Y ( δ Ω) ˜ Y i + h Y ( δ Ω) X − ˜ Y ( δ Ω) ˜ X i + h X ( δ Ω) X − ˜ X ( δ Ω) ˜ X i (5.13)For the case X = + ˜ X , Y = + ˜ Y and X = − ˜ X , Y = − ˜ Y the Hessian reduces to δδ V eff = +16 ( e σ p + e − σ q ) (5.14)which a positive definite definite quantity; it vanishes for p = q = 0; that is no blackstring nor 3-brane. This is clearly seen by using the identity e σ p = e − σ q and replacing e σ = q p , we get δδ V eff = +32 q p , V eff = q p . (5.15) AdS factor We study the following near horizon geometries.( a ) AdS × S × T with volume forms α AdS , β S and β T ,( b ) AdS × S × T with volume forms α AdS , β S and β T ,( c ) AdS × S × T with volume forms α AdS , β S and β T . AdS × S × T Using the various n-cycles of
AdS × S × T and the corresponding n-forms that couldlive on, the general expressions of the field strengths on this geometry reads as follows,p-branes (4-p)- branes p = 0 F Λ2 = Q Λ α AdS ˜ F | Λ = P Λ ( β S ∧ β T ) p = 1 F = P k =1 q k (cid:16) α AdS ∧ α S k (cid:17) ˜ F = P k =1 p k (cid:16) β S ∧ α S k (cid:17) (5.16)23here now the strings are charged electrically and the 3-branes magnetically.The total effective potential V eff associated with this system is given as usual by thesum of the contribution of each extremal black-brane. The attractor equations followingfrom the extremization of V eff are then given by: e σ ( p + p ) − e − σ ( q + q ) = 0 (5.17)and (cid:16) X Ω X − ˜ X Ω ˜ X (cid:17) + (cid:16) X Ω Y − ˜ X Ω ˜ Y (cid:17)(cid:16) Y Ω Y − ˜ Y Ω ˜ Y (cid:17) + (cid:16) Y Ω X − ˜ Y Ω ˜ X (cid:17) = 0 (5.18)A class of solutions of (5.17-5.18) is given by, σ = ln (cid:16) q + q p + p (cid:17) , X = ± ˜ X, Y = ± ˜ Y . (5.19)Other solutions like those given by eqs(4.36,4.37) may be also written down. Followingthe same method as before, we find in the case of (5.19) the following Hessian matrix atthe horizon δδ V eff = +32 q ( q + q ) ( p + p ) (5.20) AdS × S × T The general form of the field strengths on this geometry reads as,p-branes (4 − p )- branes p = 0 F Λ2 = Q Λ α AdS ˜ F | Λ = P Λ β S ∧ β T p = 1 F = p β S ˜ F = q (cid:16) α AdS ∧ β T (cid:17) (5.21)The total effective potential reads, in terms of the dressed central charges of the blackhole/4-brane, the black string/3-brane, as in (5.10) with typical solutions at horizongiven by X = ± ˜ X , Y = ± ˜ Y , σ = − ln p q . Other solutions of type eqs(4.36,4.37) maybe also written down. Notice also that the solutions with plus signs describe intersectingattractor involving string, 3-brane, D0- brane and D4- brane; those with minus signs areassociated with the string, 3-brane, ant-D0 and anti D4- brane. AdS × S × T The associated field strengths on this geometry read as follows,p-branes (4 − p )- branes p = 0 F Λ2 = P Λ β S ˜ F | Λ = Q Λ ( α Ads ∧ β T ) p = 1 F = X k =1 p k (cid:16) β S ∧ β S k (cid:17) ˜ F = X l =1 q l ε lijk (cid:16) α Ads ∧ β S i ∧ β S j ∧ β S k (cid:17) (5.22)24ollowing the same approach we have been using, the effective potential V eff of theseblack brane configurations is given by, V eff = + X i,j =1 (cid:16) X i δ ij X j + ˜ X i δ ij ˜ X j (cid:17) + X k =1 ( e σ p k + e − σ q k )+ N X a,b =1 (cid:16) Y a δ ab Y b + ˜ Y a δ ab ˜ Y b (cid:17) (5.23)Here also there are various types of solutions describing intersecting attractors with themoduli space SO (2) × SO ( N ) symmetries broken down to subgroups; a class of themreads as: σ = ln (cid:16) q + q + q + q p + p + p + p (cid:17) , X = ± ˜ X , Y = ± ˜ Y , (5.24)they correspond to the case where SO (2) × SO ( N ) is completely broken. Notice alsothat for the particular case X = ± ˜ X = 0 and Y = ± ˜ Y = 0, the moduli space symmetryis reduced to SO (2) and in the case X = ± ˜ X = 0 and Y = ± ˜ Y = 0, it reduces to SO ( N ). In this paper, we have studied the attractor mechanism of intersecting black p-branes innon chiral supergravity with supercharges. Actually, this study completes previ-ous results on black attractors in non chiral supergravity with supersymmetries[58] and agrees with the results on higher D -supergravities obtained in [57].To do so, we have first studied the structure of non chiral N = 1 supersymmetricalgebra with non trivial central charges Z µ ...µ p . Then we have given the link betweenthese Z µ ...µ p s and the fluxes R S F µ ...µ p +2 of p-branes; in particular the D- branes of typeIIA string on a compact real surface Σ given by eq(3.2). Using group theoretic method,we have shown that, besides the F-sting and the D0- brane, only the ( D / Σ)-, ( D / Σ)-and (
N S / Σ)-branes wrapping 2-cycles of Σ which survive under compactification; nofree D2- nor ( D / Σ)-brane are allowed in non chiral N = 1 supergravity. This resulthas been also checked by using a field theoretical method by determining directly thefields content that follows from type II spectrum on Σ.We have also studied the attractor mechanism for both standard extremal black attrac-tors in supergravity with supercharges as well as their intersections along theline of [57, 58]. We have worked out various classes of explicit solutions and shown thatthey are completely classified by the SO ( N − m ) subgroups of the SO (2) × SO ( N )symmetry of the moduli space SO (2 ,N ) SO (2) × SO ( N ) × SO (1 , eferences [1] Anna Ceresole, Sergio Ferrara, Black Holes and Attractors in Supergravity ,arXiv:1009.4175,[2] S. Ferrara, D. Z. Freedman and P. Van Nieuwenhuizen,
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