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Extremal black holes fromnilpotent orbits
Guillaume Bossard
AEI, Max-Planck-Institut fur Gravitationsphysik
Penn StateSeptember 2010
Outline
[ G. Bossard, H. Nicolai and K. S. Stelle, 0902.4438, 0809.5218 ]
[ G. Bossard and H. Nicolai, 0906.1987 ]
[ G. Bossard, 1001.3157, 0906.1988 ]
[ G. Bossard, Y. Michel and B. Pioline, 0908.1742 ]
Time-like dimensional reductionCharacteristic equationFake superpotentialStationary composites
Conclusion and outlook
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Black holes
Four-dimensional black hole solutionsSemi-classical string theory
Microscopic interpretation of Bekenstein–Hawkingentropy via microstates counting
Non-perturbative symmetries of string theory /M theory
SL(2,Z) × SO(6, 6)(Z) → E7(7)(Z) → E10(?)
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Pure gravity
For stationary solutions (space-time M ∼ R× V )
ds2 = −e2U(
dt + ωµdxµ)2
+ e−2U γµνdxµdxν
Coset representative V =„
eU e−U σ0 e−U
«
∈ SL(2,R)/SO(2)
defined on V
dσ = −e4U ⋆γ dω
For which the equations of motion are
Rµν(γ) =1
2Tr PµPν d ⋆γ VPV−1 = 0
with P ≡ 12
(
V−1dV + (V−1dV)t)
.
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Conserved charges
In four dimensions, the Komar mass and its dual aredefined from K ≡ dg(κ) (with κ being the time-like Killing vector)
m ≡1
8π
∫
∂V
s∗ ⋆ K n ≡1
8π
∫
∂V
s∗K
where s defines a patch of local sections of M+ over anatlas of V .
In term of the SL(2,R) Noether charge
C ≡1
4π
∫
Σ⋆VPV−1 =
(
m n
n −m
)
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Supergravity
The vierbein field ea
Electromagnetic fields AΛ in l4
Scalar fields φA parameterising a symmetric spaceG4/H4
1
2εabcde
a∧eb
∧Rcd + GAB(φ)dφA∧ ⋆ dφB
+ NΛΞ(φ)FΛ∧ ⋆ FΞ + MΛΞ(φ)FΛ
∧FΞ
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Time-like dimensional reduction
Kaluza–Klein Ansatz
The metric
ds2 = −e2U(
dt + ωµdxµ)2
+ e−2U γµνdxµdxν
where γ is the metric on V and ωµdxµ the Kaluza–Kleinvector.
And the abelian 1-form fields
AΛ = ζΛ(
dt + ωµdxµ)
+ ζΛµ dxµ
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Duality symmetry
The equations of motion permit to dualize ωµ σ
E ≡ e2U + iσAs well as ζΛ ζΛ
ΦIJ ≡(
ζΛ, ζΛ
)
and G4 is enlarged to G
g ∼= sl(2,R) ⊕ g4 ⊕ 2 ⊗ l4 ∼= 1(−2) ⊕ l
(−1)
4 ⊕(
gl1 ⊕ g4
)(0)⊕ l
(1)
4 ⊕ 1(2)
h∗ ∼= so(2) ⊕ h4 ⊕ l4 ∼= 1(−2) − 1
(2) ⊕ l(−1)
4 + l(1)
4 ⊕ h(0)
4
And(
E ,ΦIJ , v)
V ∈ G/H∗
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Duality symmetryG-invariant equations of motion
Rµν =1
kgTr PµPν d ⋆ VPV−1 = 0
in function of P ≡ V−1dV −(
V−1dV)
|h∗.
The Noether charge defined on any 2-cycle Σ of V
C ≡1
4π
∫
Σ⋆VPV−1
For Σ ≈ ∂V then
C
g ⊖ h∗∼=
(
m, n)
sl(2,R) ⊖ so(2)⊕
(
qΛ , pΛ
)
l4⊕
ΣIJKL
g4 ⊖ h4
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Duality symmetryThe Noether charge defined on any 2-cycle Σ of V
C ≡1
4π
∫
Σ⋆VPV−1
For Σ ≈ ∂V then
C
g ⊖ h∗∼=
(
m, n)
sl(2,R) ⊖ so(2)⊕
(
qΛ , pΛ
)
l4⊕
ΣIJKL
g4 ⊖ h4
where ΣIJKL is defined from the G4 current 3-form
ΣIJKL ≡1
4π
∫
Σs∗iκJIJKL
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Duality symmetry
The Noether charge defined on any 2-cycle Σ of V
C ≡1
4πV0
−1
∫
Σ⋆VPV−1 V0
For Σ ≈ ∂V then
C
g ⊖ h∗∼=
(
m, n)
sl(2,R) ⊖ so(2)⊕
Zij
l4⊕
Σijkl
g4 ⊖ h4
where Σijkl is defined from the G4 current 3-form
Σijkl ≡1
4πv0
−1ij
IJ
∫
Σs∗iκJIJKL v0
KLij
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Characteristic equation
Breitenlohner, Maison and Gibbons theorem:
If G is simple, all the non-extremal single-black holesolutions are in the H∗-orbit of a Kerr solution.
Σijkl is not a conserved charge, and C is constrained.
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Characteristic equation
Breitenlohner, Maison and Gibbons theorem:
If G is simple, all the non-extremal single-black holesolutions are in the H∗-orbit of a Kerr solution.
Σijkl is not a conserved charge, and C is constrained.
Five-graded decomposition of g with respect with theSchwarzschild Noether charge C = mh
g ∼= 1(−2) ⊕ l
(−1)
4 ⊕ gl1 ⊕ g(0)
4 ⊕ l(1)
4 ⊕ 1(2)
Three-graded decomposition of the fundamentalrepresentation R
R ∼= r(−1)
4 ⊕ R(0)
4 ⊕ r(1)
4
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Characteristic equation
Breitenlohner, Maison and Gibbons theorem:
If G is simple, all the non-extremal single-black holesolutions are in the H∗-orbit of a Kerr solution.
Σijkl is not a conserved charge, and C is constrained.
Generically, one has
C3 =
1
kgTr C
2 · C
and for N = 8 supergravity (E8)
C5 =
5
64Tr C
2 · C 3 −1
1024Tr2
C2 · C
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Characteristic equation
Breitenlohner, Maison and Gibbons theorem:
If G is simple, all the non-extremal single-black holesolutions are in the H∗-orbit of a Kerr solution.
Σijkl is not a conserved charge, and C is constrained.
Generically, one has
C3 =
1
kgTr C
2 · C
(
m, n , qΛ , pΛ
)
transform all together in a
non-linear representation of H∗.
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Spherically symmetric black holes
C ≡1
4π
∫
∂V
⋆VPV−1 =1
4π
∫
H⋆VPV−1
Tr C 2 only depends on the field U defining the metric.
A κ = 4π
√
1
kgTr C 2
The Noether charge is nilpotent for extremalspherically symmetric black holes.
C3 = 0 and C
5 = 0 for N = 8
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
H∗ non-semi-simple
C3 =
1
kgTr C
2 · C
reduces to a quadratic holomorphic equation in thecomplex parameters
W ≡ m + in Zij ∼ qΛ + ipΛ Σijkl
and can be solved explicitly as
Σijkl =Z[ijZkl]
2WΣA
ij =ZijZ
A
W
For Pure N ≤ 5-extended supergravity, C is a Spin∗(2N )
Cartan pure spinor.Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
H∗ non-semi-simple
Then1
kgTr C
2 =
(
|W |2 − |z1|2)(
|W |2 − |z2|2)
|W |2
Extremal non-rotating black holes have “Bogomolnisaturated” electromagnetic charges.
For Pure N ≤ 5-extended supergravity, all the extremal
non-rotating black holes are BPS.
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
H∗ semi-simple
C3 =
1
kgTr C
2 · C
is not holomorphic in the complex parameters
W ≡ m + in Zij ∼ qΛ + ipΛ Σijkl
and Σ is an irrational function of Zij and W that can notbe written in closed form.
For Pure N ≥ 6-extended supergravity, there are non-BPS
extremal non-rotating black holes.
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Nilpotent orbits
Nilpotent adjoint orbits of semi-simple Lie groups
C ∈ g | Cn = 0 G · C ∼= G/JC
have been classified by mathematicians. They admit asymplectic form
ω(x, y)|C ≡ Tr C [x, y]
If G/JC ∩ g ⊖ h∗ 6= ∅, the corresponding H∗-orbit, H∗/IC
C ∈ g ⊖ h∗ | Cn = 0 H∗ · C ∼= H∗/IC
is a Lagrangian submanifold of G/JC with respect with ω.
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
D– okovic classification
E8(8)-orbits of nilpotent elements of e8(8), C3875
5 = 0
E6(2)
E8(8) − E7(7) − Spin(6, 7) − SL(2) × F4(4)
E6(6)
Spin∗(16)-orbits of nilpotent elements of e8(8) ⊖ so∗(16)
SU(2) × SU(6)
Spin∗(16) − SU∗(8) − SU∗(4) × Spin(1, 6) − SU(2) × Sp(3)
Sp(4)
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
D– okovic classification
E8(−24)-orbits of nilpotent elements of e8(−24), C3875
5 = 0
SO(2, 11) SL(2) × F4(−52) − E6(−78)
E8(−24) − E7(−25) E6(−25)
SO(3, 10) SL(2) × F4(−20) − E6(−14)
SL(2,R) × E7(−25)-orbits of nilpotent elements ofe8(−25) ⊖
(
sl2 ⊕ e7(−25)
)
.
SO(1, 10) F4(−52) − E6(−78)
SL(2,R) × E7(−25) − E6(−26) F4(−52)
SO(2, 9) SO(9) − SO(10)
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
D– okovic classification
SO(8, 2 + n)-orbits of nilpotent elements of so(8, 2 + n),CS
3 = 0
SO(6, 1+n) SL(2) × SO(4, n-1) − SO(4, n)
− SL(2) × SO(6, n) Sp(4,R) × SO(4, n-2) SO(5, n-1)
SO(7, n) SL(2) × SO(5, n-2) − SO(6, n-2)
SO(6, 2) × SO(2, n)-orbits of nilpotent elements ofso(8, 2 + n) ⊖
(
so(6, 2) ⊕ so(2, n))
.
SO(5, 1) × SO(1, n) SO(4) × SO(n-1) − SO(4) × SO(n)
− SO(5, 1) × SO(1, n-1) SL(2) × SO(4) × SO(n-2) SO(5) × SO(n-1)
SO(6, 1) × SO(1, n-1) SO(5) × SO(n-2) − SO(6) × SO(n-2)
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Normal triplets
For a nilpotent orbit G · e ∼= G/Je of representative e ∈ g
An sl2 triplet f , h, e [h, e] = 2e
h lies in a Cartan subalgebra of g.
For a complex Lie algebra, h determines the orbit.
For an orbit H∗ · e ∼= H∗/Ie of representative e ∈ g ⊖ h∗
An sl2 triplet f , h, e [h, e] = 2e
h lies in a Cartan subalgebra of h∗.
h determines the orbit.
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Supersymmetry ‘Dirac equation’
For N -extended supergravity theories,H∗ ∼= Spin∗(2N )c × H0, and the Noether charge C
transforms as a chiral Weyl spinor of Spin∗(2N ) valued ina representation of H0.
In a harmonic oscillator basis
|C 〉 =
(
W + Zijaiaj + Σijkla
iajakal + · · ·
)
|0〉
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Supersymmetry ‘Dirac equation’
In a harmonic oscillator basis
|C 〉 =
(
W + Zijaiaj + Σijkla
iajakal + · · ·
)
|0〉
The asymptotic behaviour of the supersymmetry
variation of the dilatini translates the BPS condition into
the ‘Dirac equation’,
(
ǫiαai + εαβǫ
βi a
i)
|C 〉 = 0
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Supersymmetry ‘Dirac equation’nN BPS black holes are left invariant by thesupersymmetry transformations of parameter satisfying
ǫAα + εαβΩABǫβ
B = 0 ǫAα = 0
for a symplectic form ΩAB of C2n satisfying ΩACΩBC = δBA .
It term of which the ‘Dirac equation’ reads
(
aA − ΩABaB)
|C 〉 = 0 h =1
n
(
ΩABaAaB − ΩABaAaB
)
and has as solution
|C 〉 = e12ΩABaAaB
(
W + ZABaAaB
)
|0〉
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Supersymmetry ‘Dirac equation’
For maximal supergravity, this implies that 14 BPS black
holes have two saturated eigen values of Zij ,
|z1|2 = |z2|
2 ≤ |z3|2 = |z4 |
2 = |W |2
and moreover that the E7(7) quartic invariant vanishes
♦(W− 12 Z) = 0
such that the horizon area vanishes.For 1
2-BPS solutions, C is a Majorana–Weyl pure spinor
Σijkl =1
2WZ[ijZkl] =
1
48WεijklmnpqZ
mnZpq
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Decomposition of the1-form P
In N = 8 supergravity, the 128 1-form P reads
|P 〉 = (1 + E)
„
dU −i
2e2U
⋆ dω + e−U
(vt −1
dΦ)ij aia
j+
1
24
`
Dvv−1´
ijkla
ia
ja
ka
l
«
|0〉
It is convenient to consider the dual fields F IJ = dAIJ
e−Uvt−1(dΦ) =−eU
1−e4U ωµωµ
(
⋆ −e4Uω∧ ⋆ ω − Je2U ⋆ ω⋆)
v(F )
In particular, for static solutions ωµ = 0 and
F IJ = ⋆dHIJ e−Uvt−1(dΦ) = −eUv(dH)
for HIJ Harmonic functions.
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Fake superpotentiel
In the symmetric gauge V = v0 exp(
− C /r)
P ≡ V−1dV −(
V−1dV)
|h∗ = C
In the parabolic gauge associated to the Kaluza–KleinAnsatz
P is in the H∗-orbit of C .
|P 〉 = (1+E)
(
− U + eUZ(v)ij aiaj −1
24
(
vv−1)
ijklaiajakal
)
|0〉
h(Z)|P 〉 = 2|P 〉
U = −eUW φijkl = −eU G−1ijkl,mnpq
(
∂W∂φ
)
mnpq
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
1/8 BPS fake superpotential
In term of the standard diagonalization of the centralcharge
RkiR
ljZkl =
1
2eiϕ
(
0 1
−1 0
)
⊗
ρ0 0 0 0
0 ρ1 0 0
0 0 ρ2 0
0 0 0 ρ3
the 1/8 BPS fake superpotential is
W = ρ0
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
BPS first order system
BPS equations
eµaσa
αβ
(
ǫiβ ai + εβγǫγ
i ai
)
|Pµ〉 = 0 , ∇B ǫiα = 0
Small anisotropy approximation(
ǫiα ai + εαβǫβ
i ai
)
|Pµ〉 = 0 Rµν = 0
Cayley first order system (Ω[ijΩkl] = 0 , ΩijΩij = 2)
(
Ωijaiaj − Ωijaiaj
)
|Pµ〉 = 2 |Pµ〉 , DΩij+i
2e2U⋆dω Ωij = 0
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
BPS first order systemConsistency condition
(
Ωijaiaj − Ωijaiaj
)2|P 〉 = 4|P 〉
implies (Iji ≡ ΩikΩ
jk) 8 ∼= 2 ⊕ 6
2Ik[iv
t −1(dΦ)j]k + ΩijΩklvt −1(dΦ)kl = 0
2Ip[i
(
Dv v−1)
jkl]p+ 3Ω[ijΩ
pq(
Dv v−1)
kl]pq= 0
That is
28 ∼= C⊕ 15 ⊕ 2 ⊗ 6 70 ∼= 15 ⊕ (2 ⊗ 20)R
DIji = 0 dIj
i = 0
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
1/8 BPS Denef’s sum of squares
Positive definite quadratic form (e4Uωµωµ < a < 1)
(
G,F)
=e2U
1−e4U ωµωµ2ℜe
(
v(G)ij(
⋆ −e4Uω∧ ⋆ ω − ie2U ⋆ ω⋆)
v(F )ij
)
Defining
G ≡ F −1
2⋆ d(
e−Uv−1Ω)
−1
2d(
eUJvtωΩ)
the Langragian density reads
L = dU ⋆ dU −1
4e4U
dω ⋆ dω +`
F, F´
+1
12
`
Dvv−1´ijkl`
Dvv−1´
ijkl
=`
G, G´
+1
12
`
Dvv−1´ijkl
„
`
Dvv−1´
ijkl− 6Ωij
`
Dvv−1´
klpqΩ
pq
«
+ d(· · · )
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Solution
Ωijuij
IJ =1
2 4√
♦(H)
(
∂√
♦(H)
∂HIJ+ 2HIJ
)
ΩijvijIJ =1
2 4√
♦(H)
(
∂√
♦(H)
∂HIJ− 2HIJ
)
ΦIJ =1
2♦(H)
∂♦(H)
∂HIJ
e−2U =√
♦(H) dω = 2i ⋆(
HIJdHIJ −HIJdHIJ)
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Regularity
For
HIJ =∑
a
QIJa
|x − xa|+
1
2v−1
0 (Ω0)IJ
the absence of NUT sources requires
∑
b6=a
QIJa Qb IJ − QIJ
b Qa IJ
|xa − xb|=
1
2
(
Ωij0 v0(Q
a)ij − Ω0 ij v0(Qa)ij)
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Regularity
Transitive action of Spin∗(12)c ⋉p E6(2)
v|xa(Qa)ij = eiαav0(
∑
bQb)ij
and for g ∈ E6(2), solution v′ = h(g, v)vg−1
v′|xa(gQa)ij = eiαav′0(
∑
bQb)ij
All embeddings of N = 2 solutions SO∗(12)/U(6)
N = 8 E7(7)/SUc(8)
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
BPS non-supersymmetricSolutions Z∗ = 0 in exceptional N = 2 supergravity
M4∼= E7(−25)/
`
U(1) × E6(−78)
´
M∗3∼= E8(−24)/
`
SL(2,R) × E7(−25)
´
Generator h(Ω) defined from
tabcΩbΩc = 0 ΩaΩa = 2
DΩa −i
2e2U ⋆ dω Ωa = 0
Solution of the N = 2 truncation
M4 ∼= SU(1, 1)/U(1)×SO(2, 10)/`
U(1) × SO(10)´
M∗
3∼= SO(4, 12)/
`
SO(2, 2) × SO(2, 10)´
ι∗z(ι(H)) = −z(H) , ι∗zI(ι(H)) = zI(H)
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
The 1/2 BPS charge
The charge in the 128 is of the form
ǫiα + e−iαεαβΩijǫβ
j = 0
The generator h 12≡
14
`
eiαΩijaiaj − e−iαΩijaiaj
´ decomposes
so∗(16) ∼= 28(−1)
⊕ gl1 ⊕ su∗(8)(0) ⊕ 28(1)
such that16 ∼= 8
(− 12 )⊕ 8
( 12 )
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
The 1/2 BPS charge
The charge in the 128 is of the form
|C 〉 = iNe−2iα e12eiαΩija
iaj
|0〉
The generator h 12≡
14
`
eiαΩijaiaj − e−iαΩijaiaj
´ decomposes
so∗(16) ∼= 28(−1)
⊕ gl1 ⊕ su∗(8)(0) ⊕ 28(1)
such that
e8(8) ⊖ so∗(16) ∼= 1(−2) ⊕ 28
(−1) ⊕ 70(0) ⊕ 28
(1)⊕ 1
(2)
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
The 1/2 BPS charge
The charge in the 128 is of the form
|C 〉 = iNe−2iα e12eiαΩija
iaj
|0〉
The generator h 12≡
14
`
eiαΩijaiaj − e−iαΩijaiaj
´ decomposes
so∗(16) ∼= 28(−1)
⊕ gl1 ⊕ su∗(8)(0) ⊕ 28(1)
such that
e8(8) ⊖ so∗(16) ∼= 1(−2) ⊕ 28
(−1) ⊕ 70(0) ⊕ 28
(1)⊕ 1
(2)
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
The non-BPS charge
The charge in the 128 is of the form
|C 〉 = (1 + E)
(
1 +1
4eiαΩija
iaj
)
(
e−2iαM + e−iαΞijaiaj)
|0〉
The generator h ≡12
`
eiαΩijaiaj − e−iαΩijaiaj
´ decomposes
so∗(16) ∼= 28(−2)
⊕ gl1 ⊕ su∗(8)(0) ⊕ 28(2)
such that
e8(8) ⊖ so∗(16) ∼= 1(−4) ⊕ 28
(−2) ⊕ 70(0) ⊕ 28
(2)⊕ 1
(4)
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
The non-BPS charge
The charge in the 128 is of the form
|C 〉 = (1 + E)
(
1 +1
4eiαΩija
iaj
)
(
e−2iαM + e−iαΞijaiaj)
|0〉
The generator h ≡12
`
eiαΩijaiaj − e−iαΩijaiaj
´ decomposes
so∗(16) ∼= (1 ⊕ 27)(−2) ⊕ gl1 ⊕ ( sp(4) ⊕27)(0) ⊕ (1⊕ 27 )(2)
such that
e8(8) ⊖ so∗(16) ∼= 1(−4) ⊕
(
1⊕ 27)(−2)
⊕(
1⊕ 27⊕ 42)(0)
⊕ (1 ⊕27)(2) ⊕ 1
(4)
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
non-BPS fake superpotential
In term of the non-standard diagonalization of the centralcharge
Rk
iRljZkl =
eiπ4
2
„
0 1
−1 0
«
⊗
2
6
6
4
„
eiα + ie
−iα sin 2α
«
0
B
B
@
0 0 0
0 0 0
0 0 0
0 0 0
1
C
C
A
+e−iα
0
B
B
@
ξ1+ξ2+ξ3 0 0 0
0 −ξ1 0 0
0 0 −ξ2 0
0 0 0 −ξ3
1
C
C
A
3
7
7
5
the non-BPS fake superpotential is
W = 2
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Conclusion
The Noether charge satisfies a characteristicequation.
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Conclusion
The Noether charge satisfies a characteristicequation.
It is determined in function of the four-dimensionalconserved charges.
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Conclusion
The Noether charge satisfies a characteristicequation.
It is determined in function of the four-dimensionalconserved charges.
Extremal solutions are classified by nilpotentH∗-orbits in g ⊖ h∗.
which are Lagrangian submanifolds ofthe corresponding nilpotent G-orbits.
are characterised by a Cayley triplet
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Conclusion
For extremal solutions of a given type, the Cayley tripletassociates to the coset 1-form P a non-compactgenerator h of h∗ that determines a first order system ofequations
[h, P ] = 2P
In the spherically symmetric case h = h(Z) and itdetermines the fake superpotential.
In the static case h determines the mutually localcharges.
In the stationary case h defines auxiliary functionsthat permit to render the system first order.
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Conclusion
BPS stationary composites are necessarily 1/2 BPS in anappropriate N = 2 truncation.
This is the case for
1/8 BPS composites of maximal supergravity
1/4 BPS and non-BPS Z∗ = 0 composites of N = 4supergravity
non-BPS Z∗ = 0 composites in N = 2 supergravitywith a symmetric moduli space.
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Outlook
Non-BPS stationary composites in maximal supergravity
h(Ω)|Pµ〉 + εµνσλν |Pσ〉 = 2|Pµ〉
Extremal solutions in higher dimensions from higherorder nilpotent orbits
e8(8)∼= 2
(−3)⊕27
(−2)⊕(2⊗27)(−1)
⊕`
gl1 ⊕ sl2 ⊕ e6(6)´(0)
⊕(2⊗27)(1)⊕27(2)
⊕2(3)
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Outlook
Non-BPS stationary composites in maximal supergravity
h(Ω)|Pµ〉 + εµνσλν |Pσ〉 = 2|Pµ〉
Extremal solutions in higher dimensions from higherorder nilpotent orbits
e8(8)∼= 2
(−3)⊕27
(−2)⊕(2⊗27)(−1)
⊕`
gl1 ⊕ sl2 ⊕ e6(6)´(0)
⊕(2⊗27)(1)⊕27(2)
⊕2(3)
so∗(16) ∼= 1(−3)
⊕ (2⊗6)(−2)⊕ (2⊗6⊕15)(−1)
⊕`
gl1 ⊕ gl1 ⊕ sp(1) ⊕ su∗(6)´(0)
⊕· · ·
128 ∼= 1(−3)
⊕ 15(−2)
⊕ (2 ⊗ 6 ⊕ 15)(−1)⊕
`
1 ⊕ (2 ⊗ 20)R ⊕ 1´(0)
⊕ · · ·
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits
Outlook
Five-dimensional fake superpotential in terms of E7(7)
nilpotent orbits C1333 = 0
F4(4)
E7(7) − SO(6, 6) − SL(2,R) × Spin(4, 5)
F4(4)
in SU∗(8) orbits of the 70
Sp(1) × Sp(3)
SU∗(8) − SU∗(4) × SU∗(4) − Sp(1) × Sp(1) × Sp(2)
Sp(1) × Sp(3)
Guillaume Bossard (AEI) Extremal black holes from nilpotent orbits