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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?
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