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Problemi Attuali di Fisica Teorica 2010 - Vietri Sul Mare (SA)

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- 1. Alessandra Gnecchi Dip. Fisica “G. Galilei” - Padua University Duality properties of extremalblack holes in N=8 SupergravityVietri sul mare, SA - 27 Marzo 2010 - Based on A. Ceresole, S. Ferrara, A.G. and A. Marrani, Phys. Rev. D 80, 045020 A. Ceresole, S. Ferrara, A.G, Phys. Rev. D 80, 125033
- 2. Extremal black holes in supergravityPeculiar properties of these conﬁgurations link them to String theory black holes microstates counting D-branes systems Stability of BPS-states and ﬁrst order ﬂows Walls of marginal stability and split attractor ﬂows Phase transition from the BPS to the non-BPS branch Flat directions and properties of non-perturbative spectrum in the UV completion of the theories of supergravity 2
- 3. OutlineIntroduction: Black holes in SupergravityAttractors in N=8Reissner-Nordstrom and Kaluza-Klein solutionsBlack hole potential formulationCentral charge at the critical points 3
- 4. Extremal black holes in maximal Supergravity Black holes are solutions in Supergravity spectrum Thermic radiation instability A solitonic description of black holes is allowed iff T=0 !! Extremal Black HolesWe consider STATIC and SPHERICALLY SYMMETRIC solutions inasymptotically ﬂat space. “In general, an extremal black hole attractor is associated to a critical point of a suitably deﬁned black hole effective potential, and it describes a scalar conﬁguration stabilized purely in terms of conserved electric and magnetic charges, at the event horizon, regardless the values of the scalar ﬁelds at spatial inﬁnity.” S. Ferrara, R. Kallosh, A. Strominger hep-th/9508072 A. Strominger hep-th/9602111 S. Ferrara, R. Kallosh, hep-th/9602136 S. Ferrara, R. Kallosh, hep-th/9603090 S. Ferrara, G. W. Gibbons, R. Kallosh, hep-th/9702103 4
- 5. Electric-Magnetic dualityDuality transformations in a theory of gravity coupled to Maxwell ﬁeld F µν = (cos α + j sin α)F µν , α∈Rleave equations of motions invariant ∂µ F µν = 0 1 Rµν − Rgµν = −8πGTµν 2For a Lagrangian L = L(F , χ a i , χµ ) i aone can deﬁne a ﬁeld strength dual to Fµν as ˜ µν 1 ∂L Ga = µνρσ G aρσ ≡ 2 a µν 2 ∂F 5
- 6. Electric-Magnetic dualityThe generalized duality transformations are deﬁned as F A B F δ = , G C D G δχi = ξ i (χ) , ∂ξ i δ(∂µ χi ) = ∂µ ξ i = ∂µ χj j . ∂χand brings to the variation of the Lagrangian ∂L j ∂ ∂ δL = ξ i j + χµ i + (F A + G B ) b L , c bc c bc ∂χ ∂χµ ∂FThe most general group of transformations leaving the equations of motionsinvariant is the symplectic group A B ∈ sp(2n, R) C D 6
- 7. Radial evolution and black hole dynamicsThe action of the bosonic sector of d=4, N-extended supergravity isgiven by √ 1 1 S = −g d x − R + ImNΛΓ Fµν F Γ, 4 Λ µν + √ ReNΛΓ µνρσ Fµν Fρσ + Λ Γ 2 2 −g 1 + grs (Φ)∂µ Φr ∂ µ Φs . 2The equations for the scalar ﬁelds are geodesic equations d2 φ(τ ) dφj dφk dφi dφj 2 + Γjk (φ) i =0, Gij (φ) = 2c2 , dτ dτ dτ dτ dτwhere dτ f −2 (r) = − c2 = 4S 2 T 2 drBy integration of the action on angular variables the Lagrangian becomes 2 dU dφa dφb L= + Gab + e2U VBH − c2 dτ dτ dτ 7
- 8. Radial evolution and black hole dynamics We can write the black hole potential in a manifestly symplectic way 1 TΛ VBH = Q MΛΣ QΣ , 2 where µ + νµ−1 ν νµ−1 pΛ M= . QΛ = , µ−1 ν µ−1 qΛ d2 U = 2e2U VBH (φ, p, q), dτ 2Equations of motions are Dφa ∂VBH 2 = e2U a , Dτ ∂φIn order for the dφascalar ﬁelds Gij ∂m φi ∂n φj γ mn < ∞ =0dynamics to be dω 1regular we need ω = log ρ , ρ=− , τfrom the expansion of scalars near the horizon we get the criticalcondition on the effective potential 2π ∂VBH ∂VBH a φ ≈ φa H + log τ =0 A ∂φa ∂φa hor 8
- 9. Extremal BPS black holes in maximal SupergravityBlack holes which are BPS states of four or ﬁve dimensional N=8 Supergravityobtained from String and M-theory compactiﬁcations can be invariantly classiﬁedin terms of orbits of the fundamental representations of E₆₍₆₎ and E₇₍₇₎.Bekenstein Hawking entropy is given by Sd=5 = I3 Sd=4 = I4 1 IJK I3 (sI ) = d sI sJ sK 3! † 2 1 2 I4 = T r ZZ − T r ZZ † + 8ReP f (Z) 4Among the BPS states, depending on the amount of supersymmetry preserved,three orbits exist in the maximal supegravity 1 − BP S 1 4 Large BH − BP S Small BH 8 1 − BP S 2 9
- 10. Attractors in Maximal Supergravity Ferrara - Kallosh ‘06In order to get the attractor equations we impose the critical condition on thepotential ¯ VBH ( φ, Q) = ZAB Z ∗AB = Q, VAB Q, V AB A, B = 1, . . . , 8the covariant derivative on the central charge is deﬁned by Maurer-Cartanequations for the coset space 1 Di ZAB = Pi,[ABCD] (φ)Z ∗CD (φ, Q) 2yielding to the minimum condition 1 1 ∂i VBH = Pi,[ABCD] Z ∗[CD ∗AB] Z + CDABEF GH ZEF ZGH = 0 4 4!From this one can get an algebraic expression, once we work in the canonicalbasis, so that the skew eigenvalues at the minimum satisfy the equations z1 z2 + z z∗3 ∗4 =0 z1 z3 + z z ∗2 ∗4 =0 z2 z3 + z z∗1 ∗4 =0 10
- 11. Attractors in Maximal SupergravityWe can act with an SU(8) rotation on the central charge matrix, so choosing zi = ρi eiϕ/4 i = 1, 2, 3, 4thus we are left with only 5 independent parameters, has it is expected for a 4dblack hole, ρ1 0 0 0 0 0 0 ZAB = ρ2 ⊗ 0 1 eiϕ/4 0 0 ρ3 0 −1 0 0 0 0 ρ4• 1/8-BPS solution z1 = ρBP S eiϕ1 = 0 z2 = z3 = z4 = 0 I4 S = ρ4 S > 0 BP BP SBP S (Q) = I4 S (Q) = ρ2 S BP BP π• non-BPS solution iπ zi = ρ e 4 nonBP S I4 = −16ρ4 nonBP S SnonBP S (Q) = non−BP S −I4 (Q) = 4ρ2 nonBP S π 11
- 12. RN and KK black holes - the entropyExtremal solutions may or may not preserve some supersymmetry.The ﬁve dimensional BPS orbit descends to the four dimensions in twobranches, since in this case a non-BPS solutions appears.Reissner-Nordstrom and Kaluza-Klein black holes correspond to solutionswith non vanishing horizon SRN = π(e + m ) 2 2 SKK = π|p q| ⅛-BPS non-BPStheir entropy is the square root of the modulus of the invariants of theduality group associated to the orbit E7(7) E7(7) O1/8BP S : , I4 > 0 OnonBP S : , I4 < 0 E6(2) E6(6) 12
- 13. RN and KK black holes - OrbitsTo choose a representative vector for the orbit we look at thedecomposition of the vector ﬁelds representations with respect to thesubgroups E₆₍₆₎ and E₆₍₂₎ E7(7) → E6(2) × U (1) ; RN O1/8−BP S : 56 → (27, 1) + (1, 3) + 27, −1 + (1, −3) ; E7(7) → E6(6) × SO (1, 1) ; KK Onon−BP S : 56 → (27, 1) + (1, 3) + (27 , −1) + (1 , −3) ,The two extremal conﬁgurations determining the embedding in N=8, d=4supergravity are given by the two singlets in the above decompositionThey are associated to two different parametrization of the real symmetricscalar manifold E7(7) M= φijkl 70 of SU(8) SU (8) 13
- 14. RN and KK black holes - Scalar sector The scalar ﬁelds conﬁguration at the horizon of the RN black hole is RN φijkl,H =0 E7(7)which corresponds to the origin of SU (8)the solution has a residual 40-dim moduli space determined by the ﬂatdirections of the black hole potential E6(2) M1/8−BP S = SU (6) × SU (2) The KK black hole solution has a stabilized scalar ﬁeld q 3 rKK,H ≡ VH ≡ e 6ϕH =4 I aH =0 pthe moduli space is the scalar manifold of the 5d theory, which leaves 42scalars undetermined E6(6) Mnon−BP S = U Sp(8) 14
- 15. Covariant expressions of I₄ and truncation for the bare charges SU(8)-invariant form SBH = |I4 | Z = ZAB (φ) e + im 0 0 0 0 1 0 0 0 0ZAB (φ = 0) ≡ QAB QAB = −1 0 ⊗ 0 0 0 0 0 0 0 0 4 2 2 I4 = |z| = e + m2 BPS solutionE₆₍₆₎-invariant form, expressed in terms of the bare charges 2 I4 = − p q + p qii + 4 qI3 pi − pI3 (qi ) + I3 pi , I3 (qi ) KK solution corresponds to the pi = 0 = q i truncation of the ﬂuxes: 2 giving I4 = − (pq) non-BPS solution 15
- 16. Symplectic frames and symmetries of the theoryIn the de Wit-Nicolai formulation of N=8, d=4 Supergravity the Lagrangianhas SL(8,ℝ) maximal non-compact symmetry thus SO(8) is the maximalcompact symmetry group, which is also a subgroup of the stabilizer of thescalar manifold E₇₍₇₎⧸SU(8).However, the dimensionally reduced Lagrangian has as maximalnoncompact symmetry E₆₍₆₎⨂ SO(1,1) ⨀ T₂₇ , and the maximal compactsymmetry now contains the stabilizer of the 5 dimensional manifold,E₆₍₆₎⧸USp(8). 16
- 17. Symplectic frames and symmetries of the theory The ﬁelds in the fundamental representation of E₇₍₇₎ decompose as original formulation E7(7) −→ SL(8, R) −→ SL(6, R) × SL(2, R) × SO(1, 1) de Wit - Nicolai (15, 1, 1) + (6, 2, −1) + (1, 1, −3) + 56 → 28 + 28 → + (15 , 1, −1) + (6 , 2, 1) + (1, 1, 3) . dimensionally - E7(7) −→ E6(6) −→ SL(6, R) × SL(2, R) × SO(1, 1) reduced action Sezgin - Van Nieuwenhuizen E6(6) → SL(6, R) × SL (2, R) 27 → (15, 1) + (6 , 2) 1 → (1, 1) (15, 1, 1) + (6 , 2, 1) + (1, 1, 3) +56 → (27, 1) + (1, 3) + (27 , −1) + (1, −3) → + (15 , 1, −1) + (6, 2, −1) + (1, 1, −3) the ﬁnal decomposition admits indeed a unique embedding into E₇₍₇₎ 17
- 18. Symplectic frames - the vector ﬁeldsRN charges are the singlet that survives after branching with respect toeither one of the two maximal subgroup of the duality group, E7(7) −→ SU (8) ↓ ↓ E6(2) × U (1) −→ SU (6) × SU (2) × U (1)Both yield to the same decomposition, after dualizing 15 of the 28 vectors: (15, 1, 1) + (6, 2, −1) + (1, 1, −3) + 56 → 28 + 28 → + 15, 1, −1 + 6, 2, 1 + (1, 1, 3) (15, 1, 1) + 6, 2, 1 + (1, 1, 3) + 56 → (27, 1) + 27, −1 + (1, 3) + (1, −3) → + 15, 1, −1 + (6, 2, −1) + (1, 1, −3) 18
- 19. Symplectic frames - the scalar sector E7(7) The coordinate system of M= SU (8) are the 70 scalars φijkl in the 70 of SU(8), (with indices i=1,..,8). the embedding of RN black hole is described by the scalar decomposition SU (8) → SU (6) × SU (2) × U (1) 70 → (20, 2, 0) + (15, 1, −2) + 15, 1, 2 while the KK conﬁguration is supported by SU (8) → U Sp(8) 70 → 42 + 27 + 1The presence of a compact U(1) factor in the ﬁrst maximal decomposition,according to the electric magnetic duality invariant construction, requiresscalar ﬁelds to vanish in the RN case; the singlet in the latter one allows forthe stabilization of a scalar, the ﬁve dimension radius, at the horizon of theKK black hole. 19
- 20. Symplectic formulationConsider the coset representative of E₇₍₇₎⧸SU(8) uIJ vijKL u = ( 1 − Y Y † )−1 V = ij v klIJ ukl KL v = −Y † ( 1 − Y Y † )−1u and v are related to the normalized symplectic sections i(f † h − h† f ) = 1 1 1 hT f − f T h = 0by f = √ (u + v) ih = √ (u − v) 2 2and in terms of the scalar ﬁelds we have 1 1 i 1 f= √ [1 − Y † ] √ , h=− √ [1 + Y † ] √ 2 1−YY † 2 1−YY†The symplectic matrix describing the scalar coupling to vector ﬁeldstrengths ˜ LV = ImN ΛΣ F Λ F Σ + ReN ΛΣ F Λ F Σ 1+Y†is explicitly given by N = hf −1 = −i 1−Y† 20
- 21. Black hole solution from attractor equations (I)The central charge of the supergravity theory is deﬁned as a symplecticproduct dressing the bare charges with symplectic sections Zij = Q, V kl = fij qkl − hij|kl p , kl pΛ fΛ Ω Q= ,V = qΛ hΛ 1the effective black hole potential is deﬁned as VBH = Zij Z ij 2from the exact dependence on the scalar ﬁelds one can study theexpansion around the origin of the coset manifold 1 ¯ ¯ ¯ ij + (Qij φijkl Qkl − Qij φijkl Qkl ) + ... ¯ VBH (φ) ∼ Qij Q 4The conﬁguration of the charges QAB in the singlet of SU(6)xSU(2)satisﬁes the minimum condition and thus implies φ=0 to be an attractorpoint for the RN solution. 21
- 22. Black hole solution from attractor equations (II)The geometry of the ﬁve dimensional theory determines the fourdimensional symplectic structure. Following the decomposition of thescalar ﬁelds from the compactiﬁcation structure one can recover the periodmatrix as 1 dIJK aI aJ aK − i e2φ aIJ aI aJ + e6φ − 1 dIJK aI aK + ie2φ aKJ aK 3 2 NΛ Σ = − 1 dIKL aK aL + ie2φ aIK aK dIJK aK − ie2φ aIJ 2 1 Tthe effective black hole potential can be written as VBH = − Q M(N )Q 2Focusing on axion-free solutions, the attractor equation for VBH implies ∂VBH = −e2φ p0 pK aKI − e−2φ qJ pK dILK aJL + q0 qI e−6φ = 0 ∂aI aJ =0one solution is clearly supported by KK charges ( p⁰, q₀ ). 22
- 23. Black hole solution from attractor equations (II)The potential for ( p⁰, q₀ ) black hole charges reduces to 1 −6φ 1 6φ 0 2 VBH (φ, q0 , p ) 0 = e (q0 ) + e (p ) 2 aI =0 2 2it gets extremized at the horizon, once the volume is ﬁxed to e6φ = q0 , p0to the value VBH (q0 , p ) aI =0 ∗ 0 = |q0 p | = π SBH (p0 , q ) 0 0the axions being zero, we are E6(6)left with the 42 dimensional Mnon−BP S = U Sp(8)moduli spacespanned by the scalar ﬁelds of the ﬁve dimensions, that arrange in the 70of SU(8) accordingly to SU (8) → U Sp(8) 70 → 42 + 27 + 1 23
- 24. Black hole solution from attractor equations (III)Other charge conﬁgurations supporting vanishing axions solutions are•Electric solution: Qe = (p⁰,0,0,qI) •Magnetic solution Qm = (0, p¹, q0,0)In order to keep manifest the ﬁve dimensional origin of the conﬁguration, in 3/4which S ∼ V5d |crit ∼ |I3 |1/2 , we can write 3/4 3/4 V5e V5m Vcrit = 2|p0 |1/2 e Vcrit = 2|q 0 |1/2 m 3 crit 3 critby comparison with symmetric d-geometries we can relate the effectiveblack hole potential to a duality invariant expression of the charges as |p0 dIJK qI qJ qK | |q0 dIJK pI pJ pK |VBH crit (qI , p0 ) = 2 e VBH crit (q0 , pI ) = 2 m 3! 3!These critical values are also obtained from the embedding of N=2 purelycubic supergravities, for which it correctly holds that SBH = |I4 | π 24
- 25. RN and KK black holesFrom KK to RN entropies - an interesting remarkWe notice that, performing an analytic continuation on the charges p → p + iq , 0 q0 → p − iqone can transform KK entropy formula to the one of RN conﬁguration SRN = π(e + m ) 2 2 SKK = π|p q|This is indeed expected, since the groups E₆₍₆₎ and E₆₍₂₎ are different non-compact real forms of the same group. However, in the full theory the twoorbits are disjoint, since the transformation that links one to the other doesnot lie inside the duality group. 25
- 26. Four dimensional theory in relation to ﬁve dimensionsFrom the form of the black hole potential obtained as 1 T VBH = − Q M(N )Q 2which shows the explicit dependence on scalar ﬁelds, we can recast it inthe expression 1 e 2 1 0 2 1 e IJ e 1 I VBH = (Z0 ) + Zm + ZI a ZJ + Zm aIJ Zm J 2 2 2 2thus recognizing the complex central charges 1 1 Z0 ≡ √ (Z0 + iZm ) , e 0 Za ≡ √ (Za + iZm ) e a 2 2where we have introduced the real vielbein in order to work with ﬂat indicesfor the central charges components Za = ZI (a−1/2 )I , e e a Zm = Zm (a1/2 )a a I Ithe black hole potential can now be written as ¯ VBH = |Z0 |2 + Za Za , 26
- 27. Four dimensional theory in relation to ﬁve dimensionsThe central charge components for N=8 supergravity, in a basis followingfrom the dimensional reduction, are 1 d 1 −3φ Z0 = √ e−3φ q0 + e−3φ aI qI + e−3φ + ie3φ p0 − e d I pI , 2 6 2 1 1 −φ Za = √ e−φ qI (a−1/2 )I a + e dI (a−1/2 )I a − ieφ aJ (a1/2 )J a p0 + 2 2 − e−φ dIJ (a−1/2 )I a − ieφ (a1/2 )J a pJOne can still perform a unitary transformation on the symplectic sections,not affecting the potential nor their orthonormality relations h → hM , f → fM MM† = 1and we use this to relate the components of the central charge to thevector of the symplectic geometry, (Z, DI Z), in order to describe the ¯embedding of N=2 conﬁgurations in the maximal supergravity; bycomparison one ﬁnds that the matrix M is ˆ 1 1 ∂J K ¯ M = 2 −iλI e−2φ e−2φ δJ + ie−2φ λI ∂J K I ¯ ¯ 27
- 28. Attractors in ﬁve dimensions for cubic geometriesKähler potential is determined by 5d geometry 1 K = − ln(8V) V = dIJK λI λJ λK 3!one can write the cubic invariant of the scalar manifold as I3 = Z1 Z2 Z3 5 5 5which relates it to the central charge matrix 5 Z1 + 5 Z2 − 5 Z3 0 0 0 0 Z1 + Z3 − Z2 5 5 5 0 0 0 1 eab = ⊗ 0 0 Z2 + Z3 − Z1 5 5 5 0 −1 0 0 0 0 −(Z1 + Z2 + Z3 ) 5 5 5in the form giving the potential as a sum of squares 1 5 5 ab V5 = Zab Z V5 = (Z1 )2 + (Z2 )2 + (Z3 )2 5 5 5 2 1/3 ˆ I3the critical values of the ﬁelds at the horizon are ﬁxed as λIcrit = qI 28
- 29. Attractors from ﬁve dimensions for cubic geometriesCentral charge skew-eigenvalues are extremized, as functions of scalarﬁelds, thus yielding for the central charge matrix at the critical point 1/3 I3 0 0 0 1/3 0 I3 0 0 0 1 ecrit = ⊗ ab 0 0 1/3 I3 0 −1 0 1/3 0 0 0 −3I3the solution breaks U Sp(8) → U Sp(6) × U Sp(2)it is the only allowed large solution in the maximal 5d supergravity, with orbit E6(6) Od=5 = F4(4)after dimensional reduction, the theory acquires KK charges p⁰, q₀ , and thefour dimensional central charge at the horizon becomes 1 ZAB = (eAB − iZ 0 Ω) 2 29
- 30. Attractors from ﬁve dimensions for cubic geometries 1/3 ˆ I3Electric conﬁguration Q = (p0 , qi ) λIcrit = qI i 0 i Z0 attr = √ |p q1 q2 q3 | sign(p ) = |I4 |1/4 sign(p0 ) 1/4 0 2 2 1/3 1 −1/12 0 1/4 1 I3 Za attr = √ I3 (p ) qI = |I4 |1/4 2 qI 2the sign of the KK monopole determines the orbit of the solution 0 0 0 0 0 0 0 0 p0 < 0 , Z + Z0 = 0 → ZAB = 0 ⊗ 0 0 0 0 0 0 2Z0 residual SU(6)xSU(2) symmetry Z0 0 0 0 0 Z0 0 0 p0 > 0 , Z = Z0 → ZAB = 0 ⊗ 0 Z0 0 0 0 0 −Z0 residual USp(8) symmetry 30
- 31. Attractors from ﬁve dimensions for cubic geometries ˆ pIMagnetic conﬁguration Q = (pi , q 0 ) λI = 1/3 I3 i i crit Za = √ (I3 ) |q0 | 1/4 1/4 = |I4 |1/4 2 2 1 1 crit Z0 = √ (I3 ) |q0 | sign(q0 ) = |I4 |1/4 sign(q0 ) 1/4 1/4 2 2the sign of the KK charge determines the orbit of the solution 0 0 0 0 iπ/2 0 0 0 0 q0 > 0 Z = Z0 → ZAB = e ⊗ 0 0 0 0 0 0 0 −2Z0 residual SU(6)xSU(2) symmetry −Z0 0 0 0 iπ/2 0 −Z0 0 0 q0 < 0 Z = −Z0 → ZAB =e ⊗ 0 0 −Z0 0 0 0 0 Z0 residual USp(8) symmetry 31
- 32. Attractors from ﬁve dimensions for cubic geometriesThe case of the singlet p⁰, q₀ is obtained by setting the ﬁve dimensionalcharges to zero, then the central charge matrix is |p0 q 0| 0 0 0 i i iπ/4 0 |p0 q0 | 0 0 ZAB = − Z0 Ω = − e 2 2 0 0 |p0 q0 | 0 0 0 0 |p0 q0 | 1 −3φwhich is given by the critical value of Z0 = √ (e q0 + ie3φ p0 ) 2 q0after one stabilizes the ﬁve dimensional volume 6φ e |crit. = 0 pThis shows how the choice of the sign in the charges does not affect thesolution, all the choices representing the same non-BPS orbit. 32
- 33. Comparing N=8 and N=2 attractive orbits from 5 dim theory Consider 5 dimensional N=2 pure supergravity theory which symmetric scalar manifold E6(−26) MN =2 ,d=5 = F4the ﬁve dimensional theory has two orbits E6(−26) E6(−26) N =2 Od=5, BP S = N =2 Od=5, non−BP S = F4 F4(−20)The latter one precisely corresponds to the non supersymmetric solution and to(+ + - ), (- - +) signs of the q1, q2, q3 charges (implying ∂Z = 0). For charges ofthe same sign (+ + +), (- - -) one has the 1/8-BPS solution ( ∂Z = 0 ).In the N=8 theory these solutions just interchange Z1, Z2, Z3 , and Z4 = -3Z3 butwe are left in all cases with a matrix in the normal form Z 0 0 0 0 Z 0 0 Zab = 0 0 Z 0 0 0 0 −3Z which has, as maximal symmetry, U Sp(6) ⊗ U Sp(2) ∈ F4(4) 33
- 34. Comparing N=8 and N=2 attractive orbits from 5 dim theoryMoreover, while E6(−26) contains both F4 and F4(−20) so that one expects twoorbits and two classes of solution, in the N = 8 case E6(6) contains only thenon compact F4(4) , thus only one class of solutions is possible.In studying the axion free solutions to N=8, one ﬁnds that I4 = −4p0 q1 q2 q3However, electric and magnetic conﬁgurations embedded in the octonionicmodel, a new non-BPS orbit (Z=0, ∂Z≠0) is generated in d=4, depending on howthe (+++) and the (-++) charges are combined with the sign of the KK charge, inparticular E7(−25) (+, + + +) is BPS with I4 > 0 , O = , E6 E7(−25) (−, − + +) is non BPS with I4 > 0 , O = , E6(−14) E7(−25) (+, − + +) or (−, + + +) is non BPS with I4 < 0 , O = E6(−26) which comes from the properties of the duality group of the theory under consideration. 34
- 35. Conclusions•Extremal black holes solutions are determined by the geometricalstructure of the particular supergravity theory under consideration.•If a solution of a truncated theory is supported by a suitable symplecticframe, it can be embedded in the maximal theory. Its supersymmetricproperties are determined by this embedding.•Attractor mechanism precisely takes into account this embedding, thusallowing one to recover different solutions.•The branching of ﬁelds representations is manifest in the reduction ofextended supergravity from 5 to 4 dimensions.•In both the cases of N=2 and N=8 dimensionally reduced thoery, one cango from the supersymmetric to non supersymmetric branche acting oncharge conﬁguration by ﬂipping some signs; these transformations are notincluded in the duality group. 35

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