Non-supersymmetric extremal

multicenter black holeswith superpotentials

Jan Perz Katholieke Universiteit Leuven

Non-supersymmetric extremal

multicenter black holeswith 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 of 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 of 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 of 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

+ ImNI J(z)F I ! !F J + ReNI J(z)F I !F J!

I4D !! "

R ! 1! 2gab(z)dza " ! dzb

supergravity in 4 dimensions‣ bosonic action with vector multiplets

‣ target space geometry: (very) special

N = 2

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

+ ImNI J(z)F I ! !F J + ReNI J(z)F I !F J!

I4D !! "

R ! 1! 2gab(z)dza " ! dzb

supergravity in 4 dimensions‣ bosonic action with vector multiplets

‣ target space geometry: (very) special

F = !16

DabcXaXbXc

X0

N = 2

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

+ ImNI J(z)F I ! !F J + ReNI J(z)F I !F J!

za =Xa

X0

I4D !! "

R ! 1! 2gab(z)dza " ! dzb

supergravity in 4 dimensions‣ bosonic action with vector multiplets

‣ target space geometry: (very) special

F = !16

DabcXaXbXc

X0

N = 2

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

+ ImNI J(z)F I ! !F J + ReNI J(z)F I !F J!

za =Xa

X0

I4D !! "

R ! 1! 2gab(z)dza " ! dzb

gab = !za !zb K

supergravity in 4 dimensions‣ bosonic action with vector multiplets

‣ target space geometry: (very) special

F = !16

DabcXaXbXc

X0

N = 2

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

+ ImNI J(z)F I ! !F J + ReNI J(z)F I !F J!

za =Xa

X0

I4D !! "

R ! 1! 2gab(z)dza " ! dzb

K = ! ln

!i"XI !I F

# $0 !11 0

% $XI

!I F

%&gab = !za !zb K

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

+ ImNI J(z)F I ! !F J + ReNI J(z)F I !F J!

I4D !! "

R ! 1! 2gab(z)dza " ! dzb

! =1

|x|ds2 = !e2U(!)dt2 + e!2U(!)"ijdxidxj

qI !!

S2"

!L!F I

Black hole potential (single-center)

+ ImNI J(z)F I ! !F J + ReNI J(z)F I !F J!

I4D !! "

R ! 1! 2gab(z)dza " ! dzb

Black hole potential (single-center)

+ ImNI J(z)F I ! !F J + ReNI J(z)F I !F J!

! 2gab(z)dza " ! dzbIeff !!

d!"U2 ˙ =

dd!

Black hole potential (single-center)

+ gabza ˙zb

+ ImNI J(z)F I ! !F J + ReNI J(z)F I !F J!

Ieff !!

d!"U2 ˙ =

dd!

Black hole potential (single-center)‣ action with effective potential [Ferrara, Gibbons, Kallosh]

+ gabza ˙zb

VBH = |Z|2 + 4gab !a|Z|!b|Z|

Ieff !!

d!"U2 ˙ =

dd!+ e2UVBH

!

Black hole potential (single-center)‣ action with effective potential [Ferrara, Gibbons, Kallosh]

+ gabza ˙zb

VBH = |Z|2 + 4gab !a|Z|!b|Z|

Ieff !!

d!"U2

Z = eK/2 !pI qI

" #0 !11 0

$ #XI

!I F

$

˙ =d

d!+ e2UVBH!

Black hole potential (single-center)‣ action with effective potential [Ferrara, Gibbons, Kallosh]

‣ rewriting not unique [Ceresole, Dall’Agata]

+ gabza ˙zb

VBH = |Z|2 + 4gab !a|Z|!b|Z|

VBH = QTMQ = QTSTMSQ STMS =M

Ieff !!

d!"U2 ˙ =

dd!+ e2UVBH

!

Black hole potential (single-center)‣ action with effective potential [Ferrara, Gibbons, Kallosh]

‣ rewriting not unique [Ceresole, Dall’Agata]

‣ ‘superpotential’ not necessarily equal to

+ gabza ˙zb

VBH = |Z|2 + 4gab !a|Z|!b|Z|

VBH = QTMQ = QTSTMSQ STMS =MW |Z|

VBH = W2 + 4gab !aW!bW

Ieff !!

d!"U2 ˙ =

dd!+ e2UVBH

!

Flow equations‣ effective Lagrangian as a sum of squares

‣ first-order gradient flow, equivalent to EOM

‣ when : non-supersymmetric

Leff ! U2 + gabza ˙zb + e2U(W2 + 4gab!aW!bW)

Flow equations‣ effective Lagrangian as a sum of squares

‣ first-order gradient flow, equivalent to EOM

‣ when : non-supersymmetric

Leff ! U2 + gabza ˙zb + e2U(W2 + 4gab!aW!bW)

!!

U + eUW"2

+###za + 2eU gab!bW

###2

Flow equations‣ effective Lagrangian as a sum of squares

‣ first-order gradient flow, equivalent to EOM

Leff ! U2 + gabza ˙zb + e2U(W2 + 4gab!aW!bW)

U = !eUW

!!

U + eUW"2

+###za + 2eU gab!bW

###2

za = !2eU gab!bW

Flow equations‣ effective Lagrangian as a sum of squares

‣ first-order gradient flow, equivalent to EOM

‣ when : non-supersymmetricW != |Z|

Leff ! U2 + gabza ˙zb + e2U(W2 + 4gab!aW!bW)

U = !eUW

!!

U + eUW"2

+###za + 2eU gab!bW

###2

za = !2eU gab!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 =!

XDa ! 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 =!

XDa ! 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 =!

XDa ! Db ! Dc

X

z2a = Dabczbzc

! = eK/2!!1! zaDa !

z2aDa

2! z3

6dV

"

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 =!

XDa ! Db ! Dc

X

z2a = Dabczbzc

eK/2!

XI

!I F

"

! = eK/2!!1! zaDa !

z2aDa

2! z3

6dV

"

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 =!

XDa ! Db ! Dc

X

X

! = eK/2!!1! zaDa !

z2aDa

2! z3

6dV

"

! = 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 =!

XDa ! Db ! Dc

X

X

Q

! = eK/2!!1! zaDa !

z2aDa

2! z3

6dV

"

! = 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 =!

XDa ! Db ! Dc

Z(!) = !!, ""

X

X

! = eK/2!!1! zaDa !

z2aDa

2! z3

6dV

"

! = 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 =!

XDa ! Db ! Dc

Z(!) = !!, ""

X

X

! = eK/2!!1! zaDa !

z2aDa

2! z3

6dV

"

! = p0 + paDa + qaDa + q0dV ! H2"(X, Z)

Z = eK/2 !pI qI

" #0 !11 0

$ #XI

!I F

$

Da : basis of H2(X, Z)

Integration of flow equations‣ yet another rewriting (susy case) [Denef]

L ! e2U!!!2 Im

"(!" + i Im(!aKza) + i#) (e!Ue!i#")

#+ #

!!!2

! = arg Z

Integration of flow equations‣ yet another rewriting (susy case) [Denef]

‣ equations can be directly integrated

L ! e2U!!!2 Im

"(!" + i Im(!aKza) + i#) (e!Ue!i#")

#+ #

!!!2

2!" Im(e!Ue!i#!) = !"

! = arg Z

Integration of flow equations‣ yet another rewriting (susy case) [Denef]

‣ equations can be directly integrated

L ! e2U!!!2 Im

"(!" + i Im(!aKza) + i#) (e!Ue!i#")

#+ #

!!!2

2!" Im(e!Ue!i#!) = !"2 Im

!e!Ue!i!!

"= !H

H = !! ! 2 Im[e!i""]!=0

! = 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

"(!" + i Im(!aKza) + i#) (e!Ue!i#")

#+ #

!!!2

2!" Im(e!Ue!i#!) = !"2 Im

!e!Ue!i!!

"= !H

H = !! ! 2 Im[e!i""]!=0

! = arg Z

Multicenter generalization‣ metric

‣ multicenter harmonic function

!n = 1|x!xn |

[Denef]

ds2 = !e2U(dt + !idxi)2 + e!2U"ijdxidxj

H =N

!n=1

"n!n ! 2 Im[e!i"#]!=0

Multicenter generalization‣ metric

‣ multicenter harmonic function

‣ constraints on positions

!n = 1|x!xn |

[Denef]

ds2 = !e2U(dt + !idxi)2 + e!2U"ijdxidxj

H =N

!n=1

"n!n ! 2 Im[e!i"#]!=0

N

!m=1

!"n, "m"|xn # xm| = 2 Im[e#i!Z("n)]"=0

Multicenter generalization‣ metric

‣ multicenter harmonic function

‣ constraints on positions

‣ angular momentum

J =12 !

m<n!"m, "n"

xm # xn|xm # xn|

!n = 1|x!xn |

[Denef]

ds2 = !e2U(dt + !idxi)2 + e!2U"ijdxidxj

H =N

!n=1

"n!n ! 2 Im[e!i"#]!=0

N

!m=1

!"n, "m"|xn # xm| = 2 Im[e#i!Z("n)]"=0

Extension to non-susy solutions‣ Denef’s formalism involves a change of basis

‣ analogously, in our generalization:

‣ non-susy solutions

! = 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

! = iZ"! igabDa ZDb" + igabDaZDb"! iZ"

Da! = !a! +12

!aK !

Extension to non-susy solutions‣ Denef’s formalism involves a change of basis

‣ analogously, in our generalization:

‣ non-susy solutions

! = 2 Im!Z(!)"! gabDa Z(!)Db"

"

Da! = !a! +12

!aK !

Extension to non-susy solutions‣ Denef’s formalism involves a change of basis

‣ analogously, in our generalization:! = 2 Im

!Z(!)"! gabDa Z(!)Db"

"

! = 2 Im!Z(!)"! gabDa Z(!)Db"

"

W = |Z(!)| = |!!, ""| = |!!(SQ), ""|

Extension to non-susy solutions‣ Denef’s formalism involves a change of basis

‣ analogously, in our generalization:

‣ non-susy solutions

! = 2 Im!Z(!)"! gabDa Z(!)Db"

"

! = 2 Im!Z(!)"! gabDa Z(!)Db"

"

W = |Z(!)| = |!!, ""| = |!!(SQ), ""|

H(x) =N

!n=1

"n!n ! 2 Im[e!i"#]!=0

2 Im(e!Ue!i!!) = !H ! = arg Z(!)

!n = !(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

!!m, !n" = 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

!!m, !n" = 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

SiSiS

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?