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

1. Alessandra Gnecchi
Dip. Fisica “G. Galilei” - Padua University
Duality properties of extremal
black holes in N=8 Supergravity
Vietri 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 supergravity
Peculiar 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. Outline
Introduction: Black holes in Supergravity
Attractors in N=8
Reissner-Nordstrom and Kaluza-Klein solutions
Black hole potential formulation
Central 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 Holes
We consider STATIC and SPHERICALLY SYMMETRIC solutions in
asymptotically 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 duality
Duality transformations in a theory of gravity coupled to Maxwell field
F µν
= (cos α + j sin α)F µν , α∈R
leave equations of motions invariant
∂µ F µν = 0
1
Rµν − Rgµν = −8πGTµν
2
For a Lagrangian
L = L(F , χ
a i
, χµ )
i
a
one 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 duality
The 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 .
∂χ
and 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 ∂F
The most general group of transformations leaving the equations of motions
invariant is the symplectic group
A B
∈ sp(2n, R)
C D
6

7. Radial evolution and black hole dynamics
The action of the bosonic sector of d=4, N-extended supergravity is
given 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 .
2
The 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 dτ
where dτ
f −2
(r) = − c2 = 4S 2 T 2
dr
By 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 2
Equations of motions are
Dφa ∂VBH
2
= e2U a
,
Dτ ∂φ
In order for the
dφa
scalar fields Gij ∂m φi ∂n φj γ mn < ∞ =0
dynamics to be dω
1
regular we need ω = log ρ , ρ=− ,
τ
from the expansion of scalars near the horizon we get the critical
condition 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 Supergravity
Black holes which are BPS states of four or five dimensional N=8 Supergravity
obtained from String and M-theory compactifications can be invariantly classified
in 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)
4
Among 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 ‘06
In order to get the attractor equations we impose the critical condition on the
potential
¯
VBH ( φ, Q) = ZAB Z ∗AB = Q, VAB Q, V AB A, B = 1, . . . , 8
the covariant derivative on the central charge is defined by Maurer-Cartan
equations for the coset space
1
Di ZAB = Pi,[ABCD] (φ)Z ∗CD (φ, Q)
2
yielding 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 canonical
basis, 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 Supergravity
We can act with an SU(8) rotation on the central charge matrix, so choosing
zi = ρi eiϕ/4 i = 1, 2, 3, 4
thus we are left with only 5 independent parameters, has it is expected for a 4d
black 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 entropy
Extremal solutions may or may not preserve some supersymmetry.
The five dimensional BPS orbit descends to the four dimensions in two
branches, since in this case a non-BPS solutions appears.
Reissner-Nordstrom and Kaluza-Klein black holes correspond to solutions
with non vanishing horizon
SRN = π(e + m ) 2 2 SKK = π|p q|
⅛-BPS non-BPS
their entropy is the square root of the modulus of the invariants of the
duality 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 - Orbits
To choose a representative vector for the orbit we look at the
decomposition of the vector fields representations with respect to the
subgroups 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=4
supergravity are given by the two singlets in the above decomposition
They are associated to two different parametrization of the real symmetric
scalar 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 flat
directions 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
p
the moduli space is the scalar manifold of the 5d theory, which leaves 42
scalars 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 0
ZAB (φ = 0) ≡ QAB QAB =
−1 0
⊗
 0 0 0 0

0 0 0 0
4 2 2
I4 = |z| = e + m2
BPS solution
E₆₍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 theory
In the de Wit-Nicolai formulation of N=8, d=4 Supergravity the Lagrangian
has SL(8,ℝ) maximal non-compact symmetry thus SO(8) is the maximal
compact symmetry group, which is also a subgroup of the stabilizer of the
scalar manifold E₇₍₇₎⧸SU(8).
However, the dimensionally reduced Lagrangian has as maximal
noncompact symmetry E₆₍₆₎⨂ SO(1,1) ⨀ T₂₇ , and the maximal compact
symmetry 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 fields
RN charges are the singlet that survives after branching with respect to
either 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 + 1
The presence of a compact U(1) factor in the first maximal decomposition,
according to the electric magnetic duality invariant construction, requires
scalar fields to vanish in the RN case; the singlet in the latter one allows for
the stabilization of a scalar, the five dimension radius, at the horizon of the
KK black hole.
19

20. Symplectic formulation
Consider 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 † )−1
u and v are related to the normalized symplectic sections i(f † h − h† f ) = 1
1 1 hT f − f T h = 0
by f = √ (u + v) ih = √ (u − v)
2 2
and 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 field
strengths
˜
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 symplectic
product dressing the bare charges with symplectic sections
Zij = Q, V kl
= fij qkl − hij|kl p ,
kl pΛ fΛ
Ω Q= ,V =
qΛ hΛ
1
the effective black hole potential is defined as VBH = Zij Z ij
2
from the exact dependence on the scalar fields one can study the
expansion around the origin of the coset manifold
1 ¯ ¯
¯ ij + (Qij φijkl Qkl − Qij φijkl Qkl ) + ...
¯
VBH (φ) ∼ Qij Q
4
The 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 attractor
point for the RN solution.
21

22. Black hole solution from attractor equations (II)
The geometry of the five dimensional theory determines the four
dimensional symplectic structure. Following the decomposition of the
scalar fields from the compactification structure one can recover the period
matrix 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 T
the effective black hole potential can be written as VBH = − Q M(N )Q
2
Focusing 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 =0
one 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 2
it gets extremized at the horizon, once the volume is fixed to e6φ = q0 ,
p0
to the value
VBH (q0 , p ) aI =0
∗ 0
= |q0 p | = π SBH (p0 , q )
0 0
the axions being zero, we are E6(6)
left with the 42 dimensional Mnon−BP S =
U Sp(8)
moduli space
spanned by the scalar fields of the five dimensions, that arrange in the 70
of 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
3/4
which 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 crit
by comparison with symmetric d-geometries we can relate the effective
black 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 purely
cubic supergravities, for which it correctly holds that
SBH
= |I4 |
π
24

25. RN and KK black holes
From KK to RN entropies - an interesting remark
We notice that, performing an analytic continuation on the charges
p → p + iq ,
0
q0 → p − iq
one 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 two
orbits are disjoint, since the transformation that links one to the other does
not lie inside the duality group.
25

26. Four dimensional theory in relation to five dimensions
From the form of the black hole potential obtained as
1 T
VBH = − Q M(N )Q
2
which shows the explicit dependence on scalar fields, we can recast it in
the 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 2
thus recognizing the complex central charges
1 1
Z0 ≡ √ (Z0 + iZm ) ,
e 0
Za ≡ √ (Za + iZm )
e a
2 2
where we have introduced the real vielbein in order to work with flat indices
for the central charges components
Za = ZI (a−1/2 )I ,
e e
a Zm = Zm (a1/2 )a
a I
I
the black hole potential can now be written as
¯
VBH = |Z0 |2 + Za Za ,
26

27. Four dimensional theory in relation to five dimensions
The central charge components for N=8 supergravity, in a basis following
from 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 pJ
One can still perform a unitary transformation on the symplectic sections,
not affecting the potential nor their orthonormality relations
h → hM , f → fM MM† = 1
and we use this to relate the components of the central charge to the
vector of the symplectic geometry, (Z, DI Z), in order to describe the
¯
embedding of N=2 configurations in the maximal supergravity; by
comparison 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 geometries
Kähler 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 5
which 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 5
in 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 I3
the critical values of the fields at the horizon are fixed as λIcrit =
qI
28

29. Attractors from five dimensions for cubic geometries
Central charge skew-eigenvalues are extremized, as functions of scalar
fields, 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 −3I3
the 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 the
four 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 I3
Electric 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 2
the 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 pI
Magnetic 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 2
the 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 geometries
The case of the singlet p⁰, q₀ is obtained by setting the five dimensional
charges 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 −3φ
which is given by the critical value of Z0 = √ (e q0 + ie3φ p0 )
2
q0
after one stabilizes the five dimensional volume
6φ
e |crit. = 0
p
This shows how the choice of the sign in the charges does not affect the
solution, 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 =
F4
the 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 of
the 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 but
we 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)
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34. Comparing N=8 and N=2 attractive orbits from 5 dim theory
Moreover, while E6(−26) contains both F4 and F4(−20) so that one expects two
orbits and two classes of solution, in the N = 8 case E6(6) contains only the
non 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 q3
However, electric and magnetic configurations embedded in the octonionic
model, a new non-BPS orbit (Z=0, ∂Z≠0) is generated in d=4, depending on how
the (+++) and the (-++) charges are combined with the sign of the KK charge, in
particular
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.
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35. Conclusions
•Extremal black holes solutions are determined by the geometrical
structure of the particular supergravity theory under consideration.
•If a solution of a truncated theory is supported by a suitable symplectic
frame, it can be embedded in the maximal theory. Its supersymmetric
properties are determined by this embedding.
•Attractor mechanism precisely takes into account this embedding, thus
allowing one to recover different solutions.
•The branching of fields representations is manifest in the reduction of
extended supergravity from 5 to 4 dimensions.
•In both the cases of N=2 and N=8 dimensionally reduced thoery, one can
go from the supersymmetric to non supersymmetric branche acting on
charge configuration by flipping some signs; these transformations are not
included in the duality group.
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