PAFT10

310 views

Published on

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

Published in: Technology, Education
0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
310
On SlideShare
0
From Embeds
0
Number of Embeds
3
Actions
Shares
0
Downloads
0
Comments
0
Likes
1
Embeds 0
No embeds

No notes for slide

PAFT10

  1. 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. 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. 3. OutlineIntroduction: Black holes in SupergravityAttractors in N=8Reissner-Nordstrom and Kaluza-Klein solutionsBlack hole potential formulationCentral charge at the critical points 3
  4. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 0ZAB (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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

×