Table Of ContentNon-Supersymmetric Attractors in Symmetric
Coset Spaces
Wei Li
8 Jefferson Physical Laboratory, Harvard University, Cambridge MA 02138, USA
0 [email protected]
0
2
n 1 Introduction
a
J
The attractor mechanism for supersymmetric (BPS) black holes was discov-
7
eredin1995[1]:atthehorizonofasupersymmetricblackhole,themoduliare
1
completely determined by the charges of the black hole, independent of their
] asymptoticvalues.In2005,Senshowedthatallextremalblackholes,bothsu-
h
persymmetricandnon-supersymmetric(non-BPS),exhibitattractorbehavior
t
- [2]: it is a result of the near-horizon geometry of extremal black holes, rather
p
than supersymmetry. Since then, non-BPS attractors have been a very active
e
h field of research (see for instance [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]).
[ In particular, a microstate counting for certain non-BPS black holes was pro-
posed in [16]. Moreover, a new extension of topological string theory was
2
v suggested to generalize the Ooguri-Strominger-Vafa (OSV) formula so that it
6 also applies to non-supersymmetric black holes [17].
3 Both BPS and non-BPS attractor points are simply determined as the
5
critical points of the black hole potential V [18, 7]. However, it is much
2 BH
easier to solve the full BPS attractor flow equations than to solve the non-
.
1
BPS ones: the supersymmetry condition reduces the second-order equations
0
of motion to first-order ones. Once the BPS attractor moduli are known in
8
0 terms of D-brane charges, the full BPS attractor flow can be generated via
: a harmonic function procedure, i.e., by replacing the charges in the attractor
v
moduli with corresponding harmonic functions:
i
X
r tBPS(x)=t∗BPS(pI →HI(x),qI →HI(x)) (1)
a
Inparticular,whentheharmonicfunctions(HI(x),H (x))aremulti-centered,
I
this procedure generates multi-centered BPS solutions [19].
The existence of multi-centered BPS bound states is crucial in under-
standing the microscopic entropy counting of BPS black holes and the exact
formulation of OSV formula [20]. One can imagine that a similarly important
role could be played by multi-centered non-BPS solutions in understanding
2 Wei Li
non-BPS black holes microscopically. However, the multi-centered non-BPS
attractor solutions have not been constructed until [21], on which this talk is
based. In fact, even their existence has been in question.
In the BPS case, the construction of multi-centered attractor solutions
is a simple generalization of the full attractor flows of single-centered black
holes: one needs simply to replace the single-centered harmonic functions in
a single-centered BPS flow with multi-centered harmonic functions. However,
thefullattractorflowofagenericsingle-centerednon-BPSblackholehasnot
beensolvedanalytically,duetothedifficultyofsolvingsecond-orderequations
of motion. Ceresole et al. obtained an equivalent first-order equation for non-
BPSattractorsintermsofa“fakesuperpotential,”butthefakesuperpotential
can only be explicitly constructed for special charges and asymptotic moduli
[22, 23]. Similarly, the harmonic function procedure was only shown to apply
to a special subclass of non-BPS black holes, but has not been proven for
generic cases [11].
In this talk, we will develop a method of constructing generic black hole
attractor solutions, both BPS and non-BPS, single-centered as well as multi-
centered, in a large class of 4D N = 2 supergravities coupled to vector-
multiplets with cubic prepotentials. The method is applicable to models for
which the 3D moduli spaces obtained via c∗-map are symmetric coset spaces.
All attractor solutions in such a 3D moduli space can be constructed alge-
braically in a unified way. Then the 3D attractor solutions are mapped back
into four dimensions to give 4D extremal black holes.
The outline of the talk is as follows. Section 2 lays out the framework and
presents our solution generating procedures; section 3 focuses on the theory
of 4D N = 2 supergravity coupled to one vector-multiplet, and shows in
detail how to determine the attractor flow generators; section 4 then uses
these generators to construct single-centered attractors, both BPS and non-
BPS, and proves that generic non-BPS solutions cannot be generated via the
harmonic function procedure; section 5 constructs multi-centered solutions,
and shows the great contrasts between BPS and non-BPS ones. We end with
a discussion on various future directions.
2 Framework
2.1 3D Moduli Space M
3D
The technique of studying stationary configurations of 4D supergravities by
dimensionally reducing the 4D theories to 3D non-linear σ-models coupled to
gravity was described in the pioneering work [24]. The 3D moduli space for
4D N = 2 supergravity coupled to n vector-multiplets is well-studied, for
V
example in [25, 26, 27, 28]. Here we briefly review the essential points.
The bosonic part of the 4D action is:
Non-Supersymmetric Attractors in Symmetric Coset Spaces 3
S =−161π (cid:90) d4x(cid:112)−g(4)(cid:104)R−2Gi¯jdti∧∗4dt¯¯j −FI ∧GI(cid:105) (2)
where I = 0,1...n , and G = (ReN) FJ +(ImN) ∗FJ. For a theory
V I IJ IJ
endowed with a prepotential F(X), N = F +2i(ImF·X)I(ImF·X)J where
IJ IJ X·ImF·X
F =∂ ∂ F(X) [28]. We will consider generic stationary solutions, allowing
IJ I J
non-zero angular momentum. The ansatz for the metric and gauge fields are:
ds2 =−e2U(dt+ω)2+e−2Ug dxadxb (3)
ab
AI =AI(dt+ω)+AI (4)
0
where g is the 3D space metric and bold fonts denote three-dimensional
ab
fields and operators. The variables are 3n + 2 scalars {U,ti,t¯¯i,AI}, and
V 0
n +2 vectors {ω,AI}.
V
The existence of a time-like isometry allows us to reduce the 4D theory to
a 3D non-linear σ-model on this isometry. Dualizing the vectors {ω,AI} to
the scalars {σ,B }, and renaming AI as AI, we arrive at the 3D Lagrangian,
I 0
which is a non-linear σ-model minimally coupled to 3D gravity:1
1√ 1
L= g(− R+∂ φm∂aφng ) (5)
2 κ a mn
where φn are the 4(n +1) moduli fields {U,ti,t¯¯i,σ,AI,B }, and g is the
V I mn
metric of the 3D moduli space M , whose line element is:
3D
1
ds2 =dU2+ 4e−4U(dσ+AIdBI −BIdAI)2+gi¯j(t,t¯)dti·dt¯¯j
1
+ e−2U[(ImN−1)IJ(dB +N dAK)·(dB +N dAL)] (6)
2 I IK J JL
The resulting M is a para-quaternionic-K¨ahler manifold, with special
3D
holonomySp(2,R)×Sp(2n +2,R)[29].Itistheanalyticalcontinuationofthe
V
quaternionic-K¨ahlermanifoldwithspecialholonomyUSp(2,R)×USp(2n +
V
2,R)studiedin[26].Thusthevielbeinhastwoindices(α,A),transformingun-
derSp(2,R)andSp(2n +2,R),respectively.Thepara-quaternionicvielbein
V
is the analytical continuation of the quaternionic vielbein computed in [26].
This procedure is called the c∗-map [29], as it is the analytical continuation
of the c-map in [25, 26]
TheisometriesoftheM descendsfromthesymmetryofthe4Dsystem.
3D
In particular, the gauge symmetries in 4D give the shift isometries of M ,
3D
whose associated conserved charges are:
q dτ =J =P −B P , pIdτ =J =P +AIP , adτ =J =P
I AI AI I σ BI BI σ σ σ
(7)
1 Notethattheblackholepotentialtermin4Dbreaksdownintokinetictermsof
the 3D moduli, thus there is no potential term for the 3D moduli.
4 Wei Li
where the {P ,P ,P } are the momenta. Here τ is the affine parameter
σ AI BI
defined as dτ ≡ −∗ sinθdθdφ. (pI,q ) are the D-brane charges, and a the
3 I
NUT charge. A non-zero a gives rise to closed time-like curves, so we will set
a=0 from now on.
2.2 Attractor Flow Equations
The E.O.M. of 3D gravity is Einstein’s equation:
1 1
R − g R=κT =κ(∂ φm∂ φng − g ∂ φm∂cφng ) (8)
ab 2 ab ab a b mn 2 ab c mn
and the E.O.M. of the 3D moduli are the geodesic equations in M :
3D
∇ ∇aφn+Γn ∂ φm∂aφp =0 (9)
a mp a
It is not easy to solve a non-linear σ-model that couples to gravity. How-
ever, the theory greatly simplifies when the 3D spatial slice is flat: the dy-
namics of the moduli are decoupled from that of 3D gravity:
T =0=∂ φm∂ φng and ∂ ∂aφn+Γn ∂ φm∂aφp =0 (10)
ab a b mn a mp a
In particular, a single-centered attractor flow then corresponds to a null
geodesics in M : ds2 =dφmdφng =0.
3D mn
The condition of the 3D spatial slice being flat is guaranteed for BPS
attractors, both single-centered and multi-centered, by supersymmetry. Fur-
thermore, for single-centered attractors, both BPS and non-BPS, extremality
condition ensures the flatness of the 3D spatial slice. In this paper, we will
impose this flat 3D spatial slice condition on all multi-centered non-BPS at-
tractors we are looking for. They correspond to the multi-centered solutions
that are directly “assembled” by single-centered attractors, and have prop-
erties similar to their single-centered constituents: they live in certain null
totally geodesic sub-manifolds of M . We will discuss the relaxation of this
3D
condition at the end of the paper.
Tosummarize,theproblemoffinding4Dsingle-centeredblackholeattrac-
tors can be translated into finding appropriate null geodesics in M , and
3D
that of finding 4D multi-centered black hole bound states into finding cor-
responding 3D multi-centered solutions living in certain null totally geodesic
sub-manifold of M .
3D
Thenullgeodesicthatcorrespondstoa4Dblackholeattractorisonethat
terminates at a point on the U → −∞ boundary and in the interior region
with respect to all other coordinates of the moduli space M . However, it
3D
is difficult to find such geodesics since a generic null geodesic flows to the
boundary of M . For BPS attractors, the termination of the null geodesic
3D
at its attractor point is guaranteed by the constraints imposed by supersym-
metry. For non-BPS attractor, one need to find the constraints without the
aid of supersymmetry. We will show that this can be done for models with
M that are symmetric coset spaces. Moreover, the method can be easily
3D
generalized to find the multi-centered non-BPS attractor solutions.
Non-Supersymmetric Attractors in Symmetric Coset Spaces 5
2.3 Models with M Being Symmetric Coset Spaces
3D
A homogeneous space M is a manifold on which its isometry group G acts
transitively. It is isomorphic to the coset space G/H, with G being the isom-
etry group and H the isotropy group. For M = G/H, H is the maximal
3D
compact subgroup of G when one compactifies on a spatial isometry down to
(1,2) space, or the analytical continuation of the maximal compact subgroup
when one compactifies on the time isometry down to (0,3) space.
The Lie algebra g has Cartan decomposition: g=h⊕k where
[h,h]=h [h,k]=k (11)
When G is semi-simple, the coset space G/H is symmetric, meaning:
[k,k]=h (12)
The building block of the non-linear σ-model with symmetric coset space
M as target space is the coset representative M, from which the left-
3D
invariant current is constructed:
J =M−1dM =J +J (13)
k h
where J is the projection of J onto the coset algebra k. The lagrangian
k
density of the σ-model with target space G/H is then given by J as:
k
L=Tr(J ∧∗ J ) (14)
k 3 k
The symmetric coset space has the nice property that its geodesics M(τ)
are simply generated by exponentiation of the coset algebra k:
M(τ)=M ekτ/2 with k ∈k (15)
0
where M parameterizes the initial point of the geodesic, and the factor 1 in
0 2
theexponentisforlaterconvenience.Anullgeodesiccorrespondsto|k|2 =0.
Therefore, in the symmetric coset space M , the problem of finding the
3D
nullgeodesicsthatterminateatattractorpointsistranslatedintofindingthe
appropriate constraints on the null elements of the coset algebra k.
Thetheorieswith3DmodulispacesM thataresymmetriccosetspaces
3D
include:D-dimensionalgravitytoroidallycompactifiedtofourdimensions,all
4D N > 2 extended supergravities, and certain 4D N = 2 supergravities
coupled to vector-multiplets with cubic prepotentials. The entropies in the
last two classes are U-duality invariant. In this talk, we will focus on the last
class. The discussion on the first class can be found in [21].
Parametrization of M
3D
The symmetric coset space M =G/H can be parameterized by exponenti-
3D
ation of the solvable subalgebra solv of g:
6 Wei Li
M =G/H=esolv with g=h⊕solv (16)
3D
The solvable subalgebra solv is determined via Iwasawa decomposition of g.
Being semi-simple, g has Iwasawa decomposition: g = h⊕a⊕n, where a
is the maximal abelian subspace of k, and n the nilpotent subspace of the
positiverootspaceΣ+ ofa.Thesolvablesubalgebrasolv =a⊕n.Eachpoint
φn in M corresponds to a solvable element Σ(φ)=esolv, thus the solvable
3D
elements can serve as coset representatives.
We briefly explain how to extract the values of moduli from the coset
representative M. Since M is defined up to the action of the isotropy group
H, we need to construct from M an entity that encodes the values of moduli
in an H-independent way. The symmetric matrix S defined as:
S ≡MS MT (17)
0
has such a property, where S is the signature matrix.2 Moreover, as the
0
isometrygroupGactstransitivelyonthespaceofmatriceswithsignatureS ,
0
thespaceofpossibleSisthesameasthesymmetriccosetspaceM =G/H.
3D
Therefore, we can read off the values of moduli from S in an H-independent
way.
The non-linear σ-model with target space M can also be described in
3D
terms of S instead of M. First, the left-invariant current of S is J =S−1dS,
S
which is related to J by:
k
J =S−1dS =2(S MT)−1J (S MT) (18)
S 0 k 0
ThelagrangiandensityintermsofS isthusL= 1Tr(J ∧∗ J ).Theequation
4 S 3 S
of motion is the conservation of current:
∇·J =∇·(S−1∇S)=0 (19)
where we have dropped the subscript S in J , since we will only be dealing
S
with this current from now on.
2.4 Example: n = 1
V
In this talk, we will perform the explicit computation only for the simplest
case:4DN =2supergravitycoupledtoonevector-multiplet.Thegeneraliza-
tion to generic n is straightforward. The 3D moduli space M for n =1
V 3D V
is an eight-dimensional quaternionic k¨ahler manifold, with special holonomy
Sp(2,R)×Sp(4,R). Computing the killing symmetries of the metric (6) with
n =1 shows that it is a coset space G /(SL(2,R)×SL(2,R)) 3. Figure 1
V 2(2)
showstherootdiagramofG initsCartandecomposition.Thesixrootson
2(2)
2 Inallsystemsconsideredinthepresentwork,theisotropygroupHisthemaximal
orthogonalsubgroupofG:HS HT =S ,foranyH ∈H.Therefore,Sisinvariant
0 0
under the H-action M →MH.
3 Other work on this coset space has appeared recently, including [31, 32, 33].
Non-Supersymmetric Attractors in Symmetric Coset Spaces 7
a a
21 11
+
L
v
a a
22 12
L3
v
L- L3 L+
h h h
a a
23 13
-
L
v
a a
24 14
Fig. 1. Root Diagram of G in Cartan Decomposition.
2(2)
the horizontal and vertical axes {L±,L3,L±,L3} generate the isotropy sub-
h h v v
groupH=SL(2,R) ×SL(2,R) .Thetwoverticalcolumnsofeightrootsa
h v αA
generate the coset algebra k, with index α labeling a spin-1/2 representation
of SL(2,R) and index A a spin-3/2 representation of SL(2,R) .
h v
The Iwasawa decomposition, g = h ⊕ solv with solv = a ⊕ n, is
shown in Figure 2. The two Cartan generators {u,y} form a, while n is
spanned by {x,σ,A0,A1,B ,B }. {u,y} generates the rescaling of {u,y},
1 0
where u ≡ e2U, and {x,σ,A0,A1,B ,B } generates the translation of
1 0
{x,σ,A0,A1,B ,B } [27].
1 0
The moduli space M can be parameterized by solvable elements:
3D
Σ(φ)=e(lnu)u/2+(lny)yexx+AIAI+BIBI+σσ (20)
The symmetric matrix S can then be expressed in terms of the eight moduli
φn:
S(φ)=Σ(φ)S Σ(φ)T (21)
0
which shows how to extract the values of moduli from S even when S is not
constructed from the solvable elements, since it is invariant under H-action.
8 Wei Li
`
A0 B0
x
`
A1 B1
y
`
a u a
`
B A1
1
`
x
`
B A0
0
Fig. 2. Root Diagram of G in Iwasawa Decomposition.
2(2)
{u,y,x,σ,A0,A1,B ,B } generates the solvable subgroup Solv.
1 0
3 Generators of Attractor Flows
In this section, we will solve 3D attractor flow generators k as in (15). We
willprovethatextremalityconditionensuresthattheyarenilpotentelements
of the coset algebra k. In particular, for n = 1, both BPS and non-BPS
V
generators are third-degree nilpotent. However, despite this common feature,
k and k differ in many aspects.
BPS NB
3.1 Construction of Attractor Flow Generators
Construction of k
BPS
Sincethe4DBPSattractorsolutionsarealreadyknown,onecaneasilyobtain
the BPS flow generator k in the 3D moduli space M .
BPS 3D
The generator k can be expanded by coset elements a as k =
BPS αA BPS
a CαA, where CαA are conserved along the flow. On the other hand, since
αA
the conserved currents in the homogeneous space are constructed by project-
ingtheone-formvaluedLiealgebrag−1·dgontok,aprocedurethatalsogives
the vielbein: J = g−1dg| = a VαA, the vielbein VαA are also conserved
k k αA
(cid:16) (cid:17)
along the flow: d VαAφ˙n = 0. Since both the expansion coefficients CαA
dτ n
Non-Supersymmetric Attractors in Symmetric Coset Spaces 9
and the vielbein VαA transform as (2,4) of SL(2,R) ×(SL(2,R) and are
h v
conserved along the flow, they are related by:
CαA =VαAφ˙n (22)
n
up to an overall scaling factor.
In terms of the vielbein VαA, the supersymmetry condition that gives the
BPSattractorsis:VαA =zαVA [29,30,33].Using(22),weconcludethatthe
3D BPS flow generator k has the expansion
BPS
k =a zαCA (23)
BPS αA
A 4D supersymmetric black hole is labeled by four D-brane charges
(p0,p1,q ,q ).A3Dattractorflowgeneratork hasfiveparameters{CA,z}.
1 0 BPS
Aswillbeshownlater,z dropsoffinthefinalsolutionsofBPSattractorflows,
under the zero NUT charge condition. Thus the geodesics generated by k
BPS
are indeed in a four-parameter family.
k can be obtained by a twisting procedure as follows. First, define
BPS
a k0 which is spanned by the four coset generators with positive charges
BPS
under SL(2,R) :
h
k0 ≡a CA (24)
BPS 1A
then, conjugate k0 with lowering operator L−:
BPS h
kBPS =e−zL−hkB0PSezL−h (25)
Using properties of k0 , it is easy to check that k is null:
BPS BPS
|k |2 =0 (26)
BPS
More importantly, k is found to be third-degree nilpotent:
BPS
k3 =0 (27)
BPS
Anaturalquestionthenarises:Isthenilpotencyconditionofk aresultof
BPS
supersymmetry or extremality? If latter, we can use the nilpotency condition
as a constraint to solve for the non-BPS attractor generators k . We will
NB
prove that this is indeed the case.
Extremality implies nilpotency of flow generators
Wewillnowprovethatallattractorflowgenerators,bothBPSandNon-BPS,
arenilpotentelementsinthecosetalgebrak.Itisaresultofthenear-horizon
geometry of extremal black holes.
The near-horizon geometry of a 4D attractor is AdS ×S2, i.e.
2
(cid:112)
e−U → V | τ as τ →∞ (28)
BH ∗
10 Wei Li
Astheflowgoestothenear-horizon,i.e.,asu=e2U →0,thesolvableelement
M =e(lnu)u/2+··· is a polynomial function of τ:
M(τ)∼u−(cid:96)/2 ∼τ(cid:96) (29)
where −(cid:96) is the lowest eigenvalue of u.
On the other hand, since the geodesic flow is generated by k ∈ k via
M(τ)=M ekτ/2, M(τ) is an exponential function of τ. To reconcile the two
0
statements, the attractor flow generator k must be nilpotent:
k(cid:96)+1 =0 (30)
where the value of (cid:96) depends on the particular moduli space under consid-
eration. In G /SL(2,R)2, by looking at the weights of the fundamental
2(2)
representation, we see that (cid:96)=2, thus
k3 =0 (31)
The nilpotency condition of the flow generators also automatically guar-
antees that they are null:
k3 =0 =⇒ (k2)2 =0 =⇒ Tr(k2)=0 (32)
Construction of k
NB
Toconstructnon-BPSattractorflows,oneneedstofindthird-degreenilpotent
elements in the coset algebra k that are distinct from the BPS ones. In the
real G /SL(2,R)2, there are two third-degree nilpotent orbits in total [35].
2(2)
We have shown that kBPS = e−zL−hkB0PSezL−h, with kB0PS spanned by the
four generators with positive charge under SL(2,R) .
h
Since there are only two SL(2,R)’s inside H, a natural guess for k is
NB
that it can be constructed by the same twisting procedure with SL(2,R)
h
replaced by SL(2,R) :
v
kNB =e−zL−vkN0BezL−v with kN0B ≡aαaCαa, α,a=1,2 (33)
where k0 is spanned by the four generators with positive charge under
NB
SL(2,R) .
v
Using properties of k0 , one can easily show that k defined above is
NB NB
indeed third-degree nilpotent:
k3 =0 (34)
NB
That is, k defined in (33) generates non-BPS attractor flows in M .
NB 3D
A 4D non-BPS extremal black hole is labeled by four D-brane charges
(p0,p1,q ,q ). Similar to the BPS case, the 3D attractor flow generator k
1 0 NB
has five parameters {Cαa,z}. As will be shown later, z can be determined
in terms of {Cαa} using the zero NUT charge condition, thus the geodesics
generated by k are also in a four-parameter family.
NB
