multicenter black holes with superpotentials Jan Perz Katholieke - - PowerPoint PPT Presentation

Non-supersymmetric extremal

multicenter black holes with superpotentials

Jan Perz Katholieke Universiteit Leuven

Non-supersymmetric extremal

multicenter black holes with superpotentials

Jan Perz Katholieke Universiteit Leuven

Based on: P . Galli, J. Perz arXiv:0909.???? [hep-th]

Single-center vs multicenter solutions

• superposition holds for linear systems
• typically not possible for black holes in GR
• but: (Weyl)–Majumdar–Papapetrou solutions

in Einstein–Maxwell theory —arbitrary distribution

• f extremally charged dust

—static (as in Newtonian approximation) —described by harmonic functions

Single-center vs multicenter solutions

• superposition holds for linear systems
• typically not possible for black holes in GR
• but: (Weyl)–Majumdar–Papapetrou solutions

in Einstein–Maxwell theory —arbitrary distribution

• f extremally charged dust

—static (as in Newtonian approximation) —described by harmonic functions

Single-center vs multicenter solutions

• superposition holds for linear systems
• typically not possible for black holes in GR
• but: (Weyl)–Majumdar–Papapetrou solutions

in Einstein–Maxwell theory —arbitrary distribution

• f extremally charged dust

—static (as in Newtonian approximation) —described by harmonic functions

Supersymmetric black hole composites

• extremal multi-RN solutions are susy
• susy (hence extremal) multicenter solutions in

4d supergravity with vector multiplets

• with identical charges [Behrndt, Lüst, Sabra]

N = 2

[Gibbons, Hull]

Supersymmetric black hole composites

• extremal multi-RN solutions are susy
• susy (hence extremal) multicenter solutions in

4d supergravity with vector multiplets

• with identical charges [Behrndt, Lüst, Sabra]
• with arbitrary charges [Denef]

N = 2

[Gibbons, Hull]

Supersymmetric black hole composites

• extremal multi-RN solutions are susy
• susy (hence extremal) multicenter solutions in

4d supergravity with vector multiplets

• with identical charges [Behrndt, Lüst, Sabra]
• with arbitrary charges [Denef]

—relative positions of centers constrained

N = 2

[Gibbons, Hull]

Supersymmetric black hole composites

• extremal multi-RN solutions are susy
• susy (hence extremal) multicenter solutions in

4d supergravity with vector multiplets

• with identical charges [Behrndt, Lüst, Sabra]
• with arbitrary charges [Denef]

—relative positions of centers constrained —single-center solution may not exist, where a multicenter can

N = 2

[Gibbons, Hull]

Non-susy extremal composites

Non-susy extremal composites

• via timelike dimensional reduction [Gaiotto, Li, Padi]
• generate solutions (both susy and non-susy)

as geodesics on augmented scalar manifold

[Breitenlohner, Maison, Gibbons]

Non-susy extremal composites

• via timelike dimensional reduction [Gaiotto, Li, Padi]
• generate solutions (both susy and non-susy)

as geodesics on augmented scalar manifold

[Breitenlohner, Maison, Gibbons]

• almost-susy [Goldstein, Katmadas]
• reverse orientation of base space

in 5D susy solutions

Non-susy extremal composites

• via timelike dimensional reduction [Gaiotto, Li, Padi]
• generate solutions (both susy and non-susy)

as geodesics on augmented scalar manifold

[Breitenlohner, Maison, Gibbons]

• almost-susy [Goldstein, Katmadas]
• reverse orientation of base space

in 5D susy solutions

• here: superpotential approach

supergravity in 4 dimensions

• bosonic action with vector multiplets
• target space geometry: (very) special

N = 2

nv

+ Im NIJ(z)F I ∧ ⋆ F J + Re NIJ(z)F I ∧ F J

I4D ∝ R ⋆ 1 − 2ga¯

b(z)dza ∧ ⋆ d¯

z

¯ b

supergravity in 4 dimensions

• bosonic action with vector multiplets
• target space geometry: (very) special

N = 2

nv

I = (0, a) a = 1, . . . , nv

+ Im NIJ(z)F I ∧ ⋆ F J + Re NIJ(z)F I ∧ F J

I4D ∝ R ⋆ 1 − 2ga¯

b(z)dza ∧ ⋆ d¯

z

¯ b

supergravity in 4 dimensions

• bosonic action with vector multiplets
• target space geometry: (very) special

F = −1 6 Dabc XaXbXc X0

N = 2

nv

I = (0, a) a = 1, . . . , nv

+ Im NIJ(z)F I ∧ ⋆ F J + Re NIJ(z)F I ∧ F J

za = Xa X0 I4D ∝ R ⋆ 1 − 2ga¯

b(z)dza ∧ ⋆ d¯

z

¯ b

supergravity in 4 dimensions

• bosonic action with vector multiplets
• target space geometry: (very) special

F = −1 6 Dabc XaXbXc X0

N = 2

nv

I = (0, a) a = 1, . . . , nv

+ Im NIJ(z)F I ∧ ⋆ F J + Re NIJ(z)F I ∧ F J

za = Xa X0 I4D ∝ R ⋆ 1 − 2ga¯

b(z)dza ∧ ⋆ d¯

z

¯ b

ga¯

b = ∂za∂¯ z¯

bK

supergravity in 4 dimensions

• bosonic action with vector multiplets
• target space geometry: (very) special

F = −1 6 Dabc XaXbXc X0

N = 2

nv

I = (0, a) a = 1, . . . , nv

+ Im NIJ(z)F I ∧ ⋆ F J + Re NIJ(z)F I ∧ F J

za = Xa X0 I4D ∝ R ⋆ 1 − 2ga¯

b(z)dza ∧ ⋆ d¯

z

¯ b

K = − ln

• i
• XI

∂IF

−1

1 XI ∂IF

• ga¯

b = ∂za∂¯ z¯

bK

Black holes in 4d supergravity

• bosonic action with vector multiplets
• static, spherically symmetric ansatz (1 center)
• charged solution

pI ∝

• S2

∞

F I

• pI

qI

• =: Q

N = 2

nv

+ Im NIJ(z)F I ∧ ⋆ F J + Re NIJ(z)F I ∧ F J

I4D ∝ R ⋆ 1 − 2ga¯

b(z)dza ∧ ⋆ d¯

z

¯ b

τ = 1

|x|

ds2 = −e2U(τ)dt2 + e−2U(τ)δijdxidxj qI ∝

• S2

∞

∂L ∂F I

Black hole potential (single-center)

+ Im NIJ(z)F I ∧ ⋆ F J + Re NIJ(z)F I ∧ F J

I4D ∝ R ⋆ 1 − 2ga¯

b(z)dza ∧ ⋆ d¯

z

¯ b

Black hole potential (single-center)

+ Im NIJ(z)F I ∧ ⋆ F J + Re NIJ(z)F I ∧ F J − 2ga¯

b(z)dza ∧ ⋆ d¯

z

¯ b

Ieff ∝

• dτ

˙ U2

˙ = d dτ

Black hole potential (single-center)

+ ga¯

b ˙

za ˙ ¯ z

¯ b

+ Im NIJ(z)F I ∧ ⋆ F J + Re NIJ(z)F I ∧ F J

Ieff ∝

• dτ

˙ U2

˙ = d dτ

Black hole potential (single-center)

• action with effective potential [Ferrara, Gibbons, Kallosh]

+ ga¯

b ˙

za ˙ ¯ z

¯ b

VBH = |Z|2 + 4ga¯

b ∂a|Z|∂¯ b|Z|

Ieff ∝

• dτ

˙ U2

˙ = d dτ

+ e2UVBH

Black hole potential (single-center)

• action with effective potential [Ferrara, Gibbons, Kallosh]

+ ga¯

b ˙

za ˙ ¯ z

¯ b

VBH = |Z|2 + 4ga¯

b ∂a|Z|∂¯ b|Z|

Ieff ∝

• dτ

˙ U2 Z = eK/2 pI qI

−1

1 XI ∂IF

• ˙ = d

dτ

+ e2UVBH

Black hole potential (single-center)

• action with effective potential [Ferrara, Gibbons, Kallosh]
• rewriting not unique [Ceresole, Dall’Agata]

+ ga¯

b ˙

za ˙ ¯ z

¯ b

VBH = |Z|2 + 4ga¯

b ∂a|Z|∂¯ b|Z|

VBH = QTMQ = QTSTMSQ STMS = M Ieff ∝

• dτ

˙ U2

˙ = d dτ

+ e2UVBH

Black hole potential (single-center)

• action with effective potential [Ferrara, Gibbons, Kallosh]
• rewriting not unique [Ceresole, Dall’Agata]
• ‘superpotential’ not necessarily equal to

+ ga¯

b ˙

za ˙ ¯ z

¯ b

VBH = |Z|2 + 4ga¯

b ∂a|Z|∂¯ b|Z|

VBH = QTMQ = QTSTMSQ STMS = M W

|Z|

VBH = W2 + 4ga¯

b ∂aW∂¯ bW

Ieff ∝

• dτ

˙ U2

˙ = d dτ

+ e2UVBH

Flow equations

• effective Lagrangian as a sum of squares
• first-order gradient flow, equivalent to EOM
• when : non-supersymmetric

Leff ∝ ˙

U2 + ga¯

b ˙

za ˙ ¯ z

¯ b + e2U(W2 + 4ga¯ b∂aW∂¯ bW)

Flow equations

• effective Lagrangian as a sum of squares
• first-order gradient flow, equivalent to EOM
• when : non-supersymmetric

Leff ∝ ˙

U2 + ga¯

b ˙

za ˙ ¯ z

¯ b + e2U(W2 + 4ga¯ b∂aW∂¯ bW)

∝

• ˙

U + eUW 2

+

• ˙

za + 2eUga¯

b∂¯ bW

• 2

Flow equations

• effective Lagrangian as a sum of squares
• first-order gradient flow, equivalent to EOM

Leff ∝ ˙

U2 + ga¯

b ˙

za ˙ ¯ z

¯ b + e2U(W2 + 4ga¯ b∂aW∂¯ bW)

˙ U = −eUW ∝

• ˙

U + eUW 2

+

• ˙

za + 2eUga¯

b∂¯ bW

• 2

˙ za = −2eUga¯

b∂¯ bW

Flow equations

• effective Lagrangian as a sum of squares
• first-order gradient flow, equivalent to EOM
• when : non-supersymmetric

W = |Z|

Leff ∝ ˙

U2 + ga¯

b ˙

za ˙ ¯ z

¯ b + e2U(W2 + 4ga¯ b∂aW∂¯ bW)

˙ U = −eUW ∝

• ˙

U + eUW 2

+

• ˙

za + 2eUga¯

b∂¯ bW

• 2

˙ za = −2eUga¯

b∂¯ bW

Geometrical perspective

• IIA string theory compactified on a CY 3-fold
• scalars: in the normalized period vector
• charges: branes wrapping even cycles of
• central charge

Dabc =

• X Da ∧ Db ∧ Dc

X Da : basis of H2(X, Z)

Geometrical perspective

• IIA string theory compactified on a CY 3-fold
• scalars: in the normalized period vector
• charges: branes wrapping even cycles of
• central charge

Dabc =

• X Da ∧ Db ∧ Dc

X F Da : basis of H2(X, Z)

Geometrical perspective

• IIA string theory compactified on a CY 3-fold
• scalars: in the normalized period vector

Dabc =

• X Da ∧ Db ∧ Dc

X z2

a = Dabczbzc

Ω = eK/2

• −1 − zaDa − z2

aDa

2

− z3

6 dV

• Da : basis of H2(X, Z)

Da : dual basis of H4(X, Z) z3 = Dabczazbzc

Geometrical perspective

• IIA string theory compactified on a CY 3-fold
• scalars: in the normalized period vector

Dabc =

• X Da ∧ Db ∧ Dc

X z2

a = Dabczbzc

eK/2

• XI

∂IF

• Ω = eK/2
• −1 − zaDa − z2

aDa

2

− z3

6 dV

• Da : basis of H2(X, Z)

Da : dual basis of H4(X, Z) z3 = Dabczazbzc

Geometrical perspective

• IIA string theory compactified on a CY 3-fold
• scalars: in the normalized period vector
• charges: branes wrapping even cycles of

Dabc =

• X Da ∧ Db ∧ Dc

X X Ω = eK/2

• −1 − zaDa − z2

aDa

2

− z3

6 dV

• Γ = p0 + paDa + qaDa + q0dV ∈ H2∗(X, Z)

Da : basis of H2(X, Z)

Geometrical perspective

• IIA string theory compactified on a CY 3-fold
• scalars: in the normalized period vector
• charges: branes wrapping even cycles of

Dabc =

• X Da ∧ Db ∧ Dc

X X Q Ω = eK/2

• −1 − zaDa − z2

aDa

2

− z3

6 dV

• Γ = p0 + paDa + qaDa + q0dV ∈ H2∗(X, Z)

Da : basis of H2(X, Z)

Geometrical perspective

• IIA string theory compactified on a CY 3-fold
• scalars: in the normalized period vector
• charges: branes wrapping even cycles of
• central charge

Dabc =

• X Da ∧ Db ∧ Dc

Z(Γ) = Γ, Ω X X Ω = eK/2

• −1 − zaDa − z2

aDa

2

− z3

6 dV

• Γ = p0 + paDa + qaDa + q0dV ∈ H2∗(X, Z)

Da : basis of H2(X, Z)

Geometrical perspective

• IIA string theory compactified on a CY 3-fold
• scalars: in the normalized period vector
• charges: branes wrapping even cycles of
• central charge

Dabc =

• X Da ∧ Db ∧ Dc

Z(Γ) = Γ, Ω X X Ω = eK/2

• −1 − zaDa − z2

aDa

2

− z3

6 dV

• Γ = p0 + paDa + qaDa + q0dV ∈ H2∗(X, Z)

Z = eK/2 pI qI

−1

1 XI ∂IF

• Da : basis of H2(X, Z)

Integration of flow equations

• yet another rewriting (susy case) [Denef]

L ∝ e2U

• 2 Im
• (∂τ + i Im(∂aK ˙

za) + i˙ α) (e−Ue−iαΩ)

• + Γ
• 2

α = arg Z

Integration of flow equations

• yet another rewriting (susy case) [Denef]
• equations can be directly integrated

L ∝ e2U

• 2 Im
• (∂τ + i Im(∂aK ˙

za) + i˙ α) (e−Ue−iαΩ)

• + Γ
• 2

2∂τ Im(e−Ue−iαΩ) = −Γ

α = arg Z

Integration of flow equations

• yet another rewriting (susy case) [Denef]
• equations can be directly integrated

L ∝ e2U

• 2 Im
• (∂τ + i Im(∂aK ˙

za) + i˙ α) (e−Ue−iαΩ)

• + Γ
• 2

2∂τ Im(e−Ue−iαΩ) = −Γ 2 Im

• e−Ue−iαΩ

= −H H = Γτ − 2 Im[e−iαΩ]τ=0

α = arg Z

Integration of flow equations

• yet another rewriting (susy case) [Denef]
• equations can be directly integrated
• solutions for scalars implicit, but can be

inverted explicitly, also for multiple centers

[Bates & Denef]

L ∝ e2U

• 2 Im
• (∂τ + i Im(∂aK ˙

za) + i˙ α) (e−Ue−iαΩ)

• + Γ
• 2

2∂τ Im(e−Ue−iαΩ) = −Γ 2 Im

• e−Ue−iαΩ

= −H H = Γτ − 2 Im[e−iαΩ]τ=0

α = arg Z

Multicenter generalization

• metric
• multicenter harmonic function

τn =

1

|x−xn|

[Denef]

ds2 = −e2U(dt + ωidxi)2 + e−2Uδijdxidxj

H =

N

∑

n=1

Γnτn − 2 Im[e−iαΩ]τ=0

Multicenter generalization

• metric
• multicenter harmonic function
• constraints on positions

τn =

1

|x−xn|

[Denef]

ds2 = −e2U(dt + ωidxi)2 + e−2Uδijdxidxj

H =

N

∑

n=1

Γnτn − 2 Im[e−iαΩ]τ=0

N

∑

m=1

Γn, Γm |xn − xm| = 2 Im[e−iαZ(Γn)]τ=0

Multicenter generalization

• metric
• multicenter harmonic function
• constraints on positions
• angular momentum

J = 1 2 ∑

m<n

Γm, Γn xm − xn |xm − xn| τn =

1

|x−xn|

[Denef]

ds2 = −e2U(dt + ωidxi)2 + e−2Uδijdxidxj

H =

N

∑

n=1

Γnτn − 2 Im[e−iαΩ]τ=0

N

∑

m=1

Γn, Γm |xn − xm| = 2 Im[e−iαZ(Γn)]τ=0

Extension to non-susy solutions

• Denef’s formalism involves a change of basis
• analogously, in our generalization:
• non-susy solutions

Γ = p0 · 1 + paDa + qaDa + q0dV

Extension to non-susy solutions

• Denef’s formalism involves a change of basis
• analogously, in our generalization:
• non-susy solutions

Γ = i ¯ ZΩ − ig¯

ab ¯

D¯

a ¯

ZDbΩ + iga¯

bDaZ ¯

D¯

b ¯

Ω − iZ ¯ Ω

DaΩ = ∂aΩ + 1

2∂aK Ω

Extension to non-susy solutions

• Denef’s formalism involves a change of basis
• analogously, in our generalization:
• non-susy solutions

Γ = 2 Im ¯ Z(Γ)Ω − g¯

ab ¯

D¯

a ¯

Z(Γ)DbΩ

• DaΩ = ∂aΩ + 1

2∂aK Ω

Extension to non-susy solutions

• Denef’s formalism involves a change of basis
• analogously, in our generalization:

˜ Γ = 2 Im ¯ Z(˜ Γ)Ω − g¯

ab ¯

D¯

a ¯

Z(˜ Γ)DbΩ

• Γ = 2 Im

¯ Z(Γ)Ω − g¯

ab ¯

D¯

a ¯

Z(Γ)DbΩ

• W = |Z(˜

Γ)| = |˜ Γ, Ω| = |Γ(SQ), Ω|

Extension to non-susy solutions

• Denef’s formalism involves a change of basis
• analogously, in our generalization:
• non-susy solutions

˜ Γ = 2 Im ¯ Z(˜ Γ)Ω − g¯

ab ¯

D¯

a ¯

Z(˜ Γ)DbΩ

• Γ = 2 Im

¯ Z(Γ)Ω − g¯

ab ¯

D¯

a ¯

Z(Γ)DbΩ

• W = |Z(˜

Γ)| = |˜ Γ, Ω| = |Γ(SQ), Ω| ˜ H(x) =

N

∑

n=1

˜ Γnτn − 2 Im[e−i˜

αΩ]τ=0

2 Im(e−Ue−i˜

αΩ) = − ˜

H

˜ α = arg Z(˜ Γ)

˜ Γn = Γ(SnQn)

Properties of solutions

• formalism works unchanged for constant

(a subclass of superpotentials)

• mutually local ( )

electric or magnetic configurations

• constraints on charges (rather than positions)
• static, marginally bound

S

Q =

N

∑

n=1

Qn SQ =

N

∑

n=1

SnQn

Γm, Γn = 0

Properties of solutions

• formalism works unchanged for constant

(a subclass of superpotentials)

• mutually local ( )

electric or magnetic configurations

• constraints on charges (rather than positions)
• static, marginally bound
• stu: solution agrees with known/conjectured

S

Q =

N

∑

n=1

Qn SQ =

N

∑

n=1

SnQn

Γm, Γn = 0

[Kallosh, Sivanandam, Soroush]

BPS constituent model of non-susy bh

• ADM mass formula for a non-susy stu bh:

suggests 4 primitive susy constituents

[Gimon, Larsen, Simón]

mnon-BPS ∝ p0 + q1 + q2 + q3

BPS constituent model of non-susy bh

• ADM mass formula for a non-susy stu bh:

suggests 4 primitive susy constituents

[Gimon, Larsen, Simón]

• in our context: supersymmetry of each center

unaffected by the nontrivial matrices (nontrivial necessary for consistency with nontrivial of a non-supersymmetric single- center black hole) mnon-BPS ∝ p0 + q1 + q2 + q3 Si Si S

Conclusions

• merger of Denef’s and superpotential approach

Conclusions

• merger of Denef’s and superpotential approach
• limitations compared to the original formulation

Conclusions

• merger of Denef’s and superpotential approach
• limitations compared to the original formulation
• constraints on charges rather than positions

Conclusions

• merger of Denef’s and superpotential approach
• limitations compared to the original formulation
• constraints on charges rather than positions
• solutions static and marginally bound

Conclusions

• merger of Denef’s and superpotential approach
• limitations compared to the original formulation
• constraints on charges rather than positions
• solutions static and marginally bound
• natural questions:

Conclusions

• merger of Denef’s and superpotential approach
• limitations compared to the original formulation
• constraints on charges rather than positions
• solutions static and marginally bound
• natural questions:
• can the restrictions be relaxed?

Conclusions

• merger of Denef’s and superpotential approach
• limitations compared to the original formulation
• constraints on charges rather than positions
• solutions static and marginally bound
• natural questions:
• can the restrictions be relaxed?
• what is the relationship between methods?

Conclusions

• merger of Denef’s and superpotential approach
• limitations compared to the original formulation
• constraints on charges rather than positions
• solutions static and marginally bound
• natural questions:
• can the restrictions be relaxed?
• what is the relationship between methods?
• can they yield all possible solutions?