extremal black holes from nilpotent orbits

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(?)


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)

.


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

)


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


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


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∗


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


Duality symmetryThe Noether charge defined on any 2-cycle Σ of V

C ≡1

4π

∫

Σ⋆VPV−1


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


Duality symmetry

The Noether charge defined on any 2-cycle Σ of V

C ≡1

4πV0

−1

∫

Σ⋆VPV−1 V0


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


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






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






Generically, one has

C3 =

1

kgTr C

2 · C

(

m, n , qΛ , pΛ

)

transform all together in a

non-linear representation of H∗.


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


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.


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.


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.


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)



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)



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)


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.


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〉



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


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〉



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


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.


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


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


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


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


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(· · · )


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)


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)


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)


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)


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 )


The 1/2 BPS charge


|C 〉 = iNe−2iα e12eiαΩija

iaj

|0〉

The generator h 12≡

14

`


´ 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)


The non-BPS charge


|C 〉 = (1 + E)

(

1 +1

4eiαΩija

iaj

)

(

e−2iαM + e−iαΞijaiaj)

|0〉

The generator h ≡12

`


´ 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)


The non-BPS charge


|C 〉 = (1 + E)

(

1 +1

4eiαΩija

iaj

)

(

e−2iαM + e−iαΞijaiaj)

|0〉

The generator h ≡12

`


´ 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)


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


Conclusion

The Noether charge satisfies a characteristicequation.


Conclusion


It is determined in function of the four-dimensionalconserved charges.


Conclusion


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


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.


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.


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)


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)

⊕ · · ·


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)


extremal black holes from nilpotent orbits

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