1001.0003.txt

arXiv:1001.0003v3  [hep-th]  10 May 2010Preprint typeset in JHEP style - HYPER VERSION KUL-TF-09/28
HD-THEP-09-31
A landscape of non-supersymmetric AdS vacua on
coset manifolds
Paul Koerber∗
Instituut voor Theoretische Fysica, Katholieke Universit eit Leuven, Celestijnenlaan
200D, B-3001 Leuven, Belgium
Email:koerber atitf.fys.kuleuven.be
Simon K¨ ors
Institut f¨ ur Theoretische Physik, Universit¨ at Heidelbe rg, Philosophenweg 16-19, D-69120
Heidelberg, Germany
Email:s.koers atthphys.uni-heidelberg.de
Abstract: We construct new families of non-supersymmetric sourceles s type IIA AdS 4
vacua on those coset manifolds that also admit supersymmetr ic solutions. We investigate
the spectrum of left-invariant modes and find that most, but n ot all, of the vacua are stable
under these fluctuations. Generically, there are also no mas sless moduli.
∗Postdoctoral Fellow FWO – Vlaanderen.Contents
1. Introduction 1
2. Ansatz 3
3. Solutions 6
4. Stability analysis 11
5. Conclusions 15
A. SU(3)-structure 15
B. Type II supergravity 16
1. Introduction
The reasons for studying AdS 4vacua of type IIA supergravity are twofold: first they are
examples of flux compactifications away from the Calabi-Yau r egime, where all the moduli
can be stabilized at the classical level. Secondly, they can serve as a gravity dual in the
AdS4/CFT3-correspondence, which became the focus of attention due to recent progress
in the understanding of the CFT-side as a Chern-Simons-matt er theory describing the
world-volume of coinciding M2-branes [1].
Itismucheasiertofindsupersymmetricsolutionsofsupergr avityasthesupersymmetry
conditions are simpler than the full equations of motion, wh ile at the same time there
are general theorems stating that the former – supplemented with the Bianchi identities
of the form fields – imply the latter [2, 3, 4, 5]. Although spec ial type IIA solutions
that came from the reduction of supersymmetric M-theory vac ua were already known (see
e.g. [6, 7, 8]), it was only in [3] that the supersymmetry cond itions for type IIA vacua with
SU(3)-structure were first worked out in general. It was disc overed that there are natural
solutions to these equations on the four coset manifolds G/Hthat have a nearly-K¨ ahler
limit [9, 10, 11, 12, 13, 14] (solutions on other manifolds ca n be found in e.g. [3, 15, 16]).1
To be precise these are the manifolds SU(2) ×SU(2),G2
SU(3),Sp(2)
S(U(2)×U(1))andSU(3)
U(1)×U(1).2
These solutions are particularly simple in the sense that bo th the SU(3)-structure, which
determines the metric, as well as all the form fluxes can be exp anded in terms of forms
which are left-invariant under the action of the group G. The supersymmetry equations
1For an early appearance of these coset manifolds in the strin g literature see e.g. [17].
2See [18] for a review and a proof that these are the only homoge neous manifolds admitting a nearly-
K¨ ahler geometry.
– 1 –of [3] then reduce to purely algebraic equations and can be ex plicitly solved. Nevertheless,
these solutions still have non-trivial geometric fluxes as o pposed to the Calabi-Yau or torus
orientifolds of [15, 16]. Similarly to those papers it is pos sible to classically stabilize all
left-invariant moduli [14]. Inspired by the AdS 4/CFT3correspondence more complicated
type IIA solutions have in the meantime been proposed. The so lutions have a more generic
form for the supersymmetry generators, called SU(3) ×SU(3)-structure [19], and are not
left-invariant anymore [20, 21, 22, 23] (see also [24]). Sup ersymmetric AdS 4vacua in type
IIB with SU(2)-structure have also been studied in [25, 26, 2 7, 28] and in particular it has
been shown in [28] that also in this setup classical moduli st abilization is possible.
At some point, however, supersymmetry has to be broken and we have to leave
the safe haven of the supersymmetry conditions. In this pape r we construct new non-
supersymmetric AdS 4vacua without source terms. This means that the more complic ated
equations of motion of supergravity should be tackled direc tly3. In order to simplify the
equations we use a specific ansatz: we start from a supersymme tric AdS 4solution and scan
for non-supersymmetric solutions with the samegeometry (and thus SU(3)-structure), but
withdifferent NSNS- and RR-fluxes. Moreover, we expand these form fields in t erms of the
SU(3)-structure and its torsion classes. This may seem rest rictive at first, but it works for
11D supergravity, where solutions like this have been found and are known as Englert-type
solutions [31, 32, 33] (see [34] for a review). To be specific, for each supersymmetric M-
theory solution of Freund-Rubin type (which means the M-the ory four-form flux has only
legs along the external AdS 4space, i.e.F4=fvol4wherefis called the Freund-Rubin
parameter) it is possible to construct a non-supersymmetri c solution with the same inter-
nal geometry but with a different four-form flux. The modified fo ur-form of the Englert
solution has then a non-zero internal part: ˆF4∝η†γm1m2m3m4ηdxm1m2m3m4, whereηis
the 7D supersymmetry generator, and a different Freund-Rubin parameterfE=−(2/3)f.
Also the Ricci scalar of the AdS 4space, and thus the effective 4D cosmological constant,
differs:R4D,E= (5/6)R4D. In type IIA with non-zero Romans mass (so that there is no lif t
to M-theory) non-supersymmetric solutions of this form hav e been found as well: for the
nearly-K¨ ahler geometry in [35, 29, 36] and for the K¨ ahler- Einstein geometry in [35, 20, 37].
In this paper we show that this type of solutions is not restri cted to these limits and sys-
tematically scan for them. Applying our ansatz to the coset m anifolds with nearly-K¨ ahler
limit, mentioned above, we find that the most interesting man ifolds areSp(2)
S(U(2)×U(1))and
SU(3)
U(1)×U(1), on which we find several families of non-supersymmetric AdS 4solutions. We
also find some non-supersymmetric solutions in regimes of th e geometry that do not allow
for a supersymmetric solution.
These non-supersymmetric solutions are not necessarily st able. For instance, it is
known that if there is more than one Killing spinor on the inte rnal manifold (which holds
in particular for S7, the M-theory lift of CP3=Sp(2)
S(U(2)×U(1))), the Englert-type solution is
unstable [38]. We investigate stability of our solutions ag ainst left-invariant fluctuations.
This means we calculate the spectrum of left-invariant mode s, and check for each mode
3Anotherroute would be tofindsome alternative first-ordereq uations, which extendthe supersymmetry
conditions in that they still automatically imply the full e quations of motion in certain non-supersymmetric
cases, see e.g. [29, 30].
– 2 –whether the mass-squared is above the Breitenlohner-Freed man bound [39, 40]. This is not
a complete stability analysis in that there could still be no n-left-invariant modes that are
unstable. We do believe it provides a good first indication. I n particular, we find for the
type IIA reduction of the Englert solution on S7that the unstable mode of [38] is among
our left-invariant fluctuations and we find the exact same mas s-squared.
These non-supersymmetric AdS 4vacua are interesting, because, provided they are
stable, they should have a CFT-dual. For instance in [20] the CFT-dual for a non-
supersymmetric K¨ ahler-Einstein solution on CP3was proposed. Furthermore, for phe-
nomenologically more realistic vacua, supersymmetry-bre aking is essential. Really, one
would like to construct classical solutions with a dS 4-factor, which are necessarily non-
supersymmetric. Because of a series of no-go theorems – from very general to more specific:
[41, 42, 43, 44, 45] – this is a very non-trivial task. For pape rs nevertheless addressing this
problemsee[46,47,45,48,49,28]. Inthiscontext thelands capeofthenon-supersymmetric
AdS4vacua of this paper can be considered as a playground to gain e xperience before try-
ing to construct dS 4-vacua. In fact, in [48] an ansatz very similar to the one used in this
paper was proposed in order to construct dS 4-vacua. Applied to the coset manifolds above,
it did however not yield any solutions, in agreement with the no-go theorem of [45].
In section 2 we explain our ansatz in full detail, while in sec tion 3 we present the
explicit solutions we found on the coset manifolds. In secti on 4 we analyse the stability
against left-invariant fluctuations before ending with som e short conclusions. We provide
an appendix with some useful formulae involving SU(3)-stru ctures and an appendix on our
supergravity conventions.
Thenon-supersymmetricsolutions of this paperappearedbe forein thesecond author’s
PhD thesis [50].
2. Ansatz
In this section we explain the ansatz for our non-supersymme tric solutions. The reader
interested in the details might want to check out our SU(3)-s tructure conventions in ap-
pendix A, while towards the end of the section we need the type II supergravity equations
of motion outlined in appendix B.
We start with a supersymmetric SU(3)-structure solution of type IIA supergravity.
The SU(3)-structure is defined by a real two-form Jand a complex decomposable three-
form Ω satisfying (A.1). Moreover, Jand Ω together determine the metric as in (A.2). In
order for the solution to preserve at least one supersymmetr y (N= 1) [3] one finds that
the warp factor Aand the dilaton Φ should be constant, the torsion classes W1,W2purely
imaginary and all other torsion classes zero (for the definit ion of the torsion classes see
(A.3)). This implies
dJ=3
2W1ReΩ, (2.1a)
dReΩ = 0, (2.1b)
dImΩ =W1J∧J+W2∧J, (2.1c)
– 3 –where we defined W1≡ −iW1andW2≡ −iW2. The fluxes can then be expressed in terms
of Ω,Jand the torsion classes and are given by
eΦˆF0=f1, (2.2a)
eΦˆF2=f2J+f3ˆW2, (2.2b)
eΦˆF4=f4J∧J+f5ˆW2∧J, (2.2c)
eΦˆF6=f6vol6, (2.2d)
H=f7ReΩ, (2.2e)
where for the supersymmetric solution
f1=eΦm, f 2=−W1
4, f3=−w2, f4=3eΦm
10,
f5= 0, f 6=9W1
4, f7=2eΦm
5.(2.3)
Using the duality relation f=˜F0=−⋆6ˆF6=−e−Φf6(see (B.6)) we find that f6is
proportional to the Freund-Rubin parameter f, whilef1is proportional to the Romans
massm. Furthermore, we introduced here a normalized version of W2, enabling us later
on to use (2.2) as an ansatz for the fluxes also in the limit W2→0:
ˆW2=W2
w2,withw2=±/radicalbig
(W2)2, (2.4)
where one can choose a convenient sign in the last expression .
The Bianchi identity for ˆF2imposes dW2∝ReΩ. Working out the proportionality
constant [3] we find
dW2=−1
4(W2)2ReΩ. (2.5)
Furthermore, using the values for the fluxes (2.3) it fixes the Romans mass:
e2Φm2=5
16/parenleftbig
3(W1)2−2(W2)2/parenrightbig
. (2.6)
We now want to construct non-supersymmetric AdS solutions o n the manifolds men-
tioned in the introduction with the samegeometry as in the supersymmetric solution, and
thus the same SU(3)-structure ( J,Ω), but with different fluxes. We make the ansatz that
the fluxes can still be expanded in terms of J,Ω and the torsion class ˆW2as in (2.2), but
with different values for the coefficients fi. To this end we plug the ansatz for the geometry
(J,Ω) — eqs. (2.1) — and the ansatz for the fluxes — eqs. (2.2) — into the equations of
motion (B.7) and solve for the fi. We will make one more assumption, namely that
ˆW2∧ˆW2=cJ∧J+pˆW2∧J, (2.7)
withc,psome parameters. This is an extra constraint only for theSU(3)
U(1)×U(1)coset and
we will discuss its relaxation later.4Wedging with Jwe find then immediately c=−1/6.
4With the ansatz (2.2) the constraint is forced upon us. Indee d, suppose that instead ˆW2∧ˆW2=
−1/6J∧J+pˆW2∧J+P∧J, wherePis a non-zero simple (1,1)-form independent of ˆW2. We find then
from the equation of motion for Hand the internal part of the Einstein equation respectively f5f3= 0 and
(f3)2−(f5)2−(w2)2= 0. So the only possibility is then f5= 0 and f3=±w2, which leads in the end to
the supersymmetric solution. They way out is to also include Pas an expansion form in (2.2).
– 4 –Furthermore we need expressions for the Ricci scalar and ten sor, which for a manifold with
SU(3)-structure can be expressed in terms of the torsion cla sses [51]. Taking into account
that onlyW1,2are non-zero we find:
R6D=15(W1)2
2−(W2)2
2, (2.8a)
Rmn=1
6gmnR6D+W1
4W2(m·Jn)+1
2[W2m·W2n]0+1
2Re/bracketleftbig
dW2|(2,1)m·¯Ωn/bracketrightbig
,(2.8b)
where (P)2andPm·Pnfor a form Pare defined in (B.2) and |0indicates taking the
traceless part. From eq. (2.5) follows that for our purposes dW2|2,1= 0 so that the last
term in (2.8b) vanishes. Moreover, using (2.7) [ W2m·W2n]0can be expressed in terms of
W2(m·Jn).
Plugging the ansatz for the fluxes (2.2) into the equations of motion (B.7) and using
eqs. (2.1), (2.5), (A.5), (2.7), (B.5), (B.6) and (2.8b) we fi nd:
BianchiF2: 0 =3
2W1f2−1
4w2f3+f1f7,
eomF4: 0 = 3W1f4+1
4w2f5−f6f7,
eomH: 0 = 6W1f7−3f1f2−12f4f2−6f4f6−f3f5,
0 =w2f7+f1f3+f2f5−2f3f4−f5f6+pf3f5, (2.9)
dilaton eom : 0 = R4D+R6D−2f2
7,
Einstein ext. : 0 = R4D+(f1)2+3(f2)2+12(f4)2+(f6)2+(f3)2+(f5)2,
Einstein int. : 0 = R6D−6(f7)2+1
2/bracketleftbig
3(f1)2+3(f2)2−12(f4)2−3(f6)2+(f3)2−(f5)2/bracketrightbig
,
0 = 4(f2f3+2f4f5)−w2W1−p/bracketleftbig
(f3)2−(f5)2−(w2)2/bracketrightbig
.
In the equation of motion for Hwe get separate conditions from the coefficients of J∧J
andˆW2∧Jrespectively. In the internal Einstein equation we find like wise a separate
condition from the trace and the coefficient of W2(m·Jn). In the next section we find
explicit solutions to these equations for the coset manifol ds with nearly-K¨ ahler limit, the
stability of which we investigate in section 4.
Flipping signs
The Einstein and dilaton equation are quadratic in the form fl uxes and thus insensitive to
flipping the signs of these fluxes. Taking into account also th e flux equations of motion
and Bianchi identities, we find that for each solution to the s upergravity equations, we
automatically obtain new ones by making the following sign fl ips:
H→ −H,ˆF0→ −ˆF0,ˆF2→ˆF2,ˆF4→ −ˆF4,ˆF6→ˆF6,
H→ −H,ˆF0→ˆF0,ˆF2→ −ˆF2,ˆF4→ˆF4,ˆF6→ −ˆF6,
H→H,ˆF0→ −ˆF0,ˆF2→ −ˆF2,ˆF4→ −ˆF4,ˆF6→ −ˆF6.(2.10)
In particular, these sign flips will transform a supersymmet ric solution into another super-
symmetric solution (as can be verified using the conditions ( 2.1),(2.3) allowing for suitable
– 5 –sign flips of J, ReΩ and ImΩ compatible with the metric). If some fluxes are ze ro, more
sign flips are possible. For instance for ˆF0=ˆF4= 0 we find the following extra sign-flip,
known as skew-whiffing in the M-theory compactification literature [52] (see also t he review
[34])
H→ ±H,ˆF2→ˆF2,ˆF6→ −ˆF6, (2.11)
which transforms a supersymmetric solution into a non-supersymmetric one. When dis-
cussing different solutions, we will from now on implicitly co nsider each solution together
with its signed-flipped counterparts.
3. Solutions
Let us now solve the equations obtained in the previous secti on for the coset manifolds that
admit sourceless supersymmetric solutions, namelyG2
SU(3), SU(2)×SU(2),Sp(2)
S(U(2)×U(1))and
SU(3)
U(1)×U(1). For the supersymmetricsolutions on these manifolds we wil l use the conventions
and presentation of [13, 14]. For moredetails, includingin particular ourchoice of structure
constants for the relevant algebras, we refer to these paper s.
On a coset manifold G/Hone can define a coframe emthrough the decomposition of
the Lie-valued one-form L−1dL=emKm+ωaHain terms of the algebras of GandH. Here
Lis a coset representative, the Haspan the algebra of Hand theKmspan the complement
of this algebra within the algebra of G. The exterior derivative on the emis then given
in terms of the structure constants through the Maurer-Cart an relation. Furthermore,
the forms that are left-invariant under the action of Gare precisely those forms that are
constant in the basis spanned by emand for which the exterior derivative is also constant
in this basis. For these forms the exterior derivative can th en be expressed solely in terms
of the structure constants only involving the Km. We refer to [53, 54] for a review on coset
technology or to the above papers for a quick explanation.
G2
SU(3)and SU(2) ×SU(2)
We start from the supersymmetric nearly-K¨ ahler solution o nG2
SU(3). The SU(3)-structure
is given by
J=a(e12−e34+e56),
Ω =a3/2/bracketleftbig
(e245−e236−e146−e135)+i(e246+e235+e145−e136)/bracketrightbig
,(3.1)
whereais the overall scale.
Since this SU(3)-structure corresponds to a nearly-K¨ ahle r geometry the torsion class
W2is zero. Furthermore we find
W1=−2√
3a−1/2, w2=p= 0. (3.2)
– 6 –Plugging this into the equations (2.9) we find exactly three s olutions for ( f1,...,f7) (up
to the sign flips (2.10)):
a−1/2(√
5
2,1
2√
3,0,3
4√
5,0,−9
2√
3,1√
5),
a−1/2(/radicalbigg
5
3,0,0,0,0,5√
3,0),
a−1/2(1,1√
3,0,−1
2,0,√
3,1).(3.3)
The first is the supersymmetric solution, while the last two a re non-supersymmetric solu-
tions, which were already found in [35, 29, 36]. Truncating t o the 4D effective theory it
was shown in [30] that a generalization of this family of solu tions is quite universal as it
appears in a large class of N= 2 gauged supergravities.
On the SU(2) ×SU(2) manifold, requiring the same geometry as the supersym metric
solution and not allowing for source terms will restrict us t o the nearly-K¨ ahler point. The
analysis is then basically the same as forG2
SU(3)above.
Sp(2)
S(U(2)×U(1))
The family of supersymmetric solutions on this manifold has , next to the overall scale,
an extra parameter determining the shape of the solutions. I t is then possible to turn on
the torsion class W2and venture away from the nearly-K¨ ahler geometry. This mak es this
class much richer and enables us this time to find new non-supe rsymmetric solutions. The
SU(3)-structure is given by [12, 13, 14]
J=a(e12+e34−σe56),
Ω =a3/2σ1/2/bracketleftbig
(e245−e236−e146−e135)+i(e246+e235+e145−e136)/bracketrightbig
,(3.4)
whereais the overall scale and σis the shape parameter. We find for the torsion classes
and the parameter p:
W1= (aσ)−1/22+σ
3,
(W2)2= (aσ)−18(1−σ)2
3⇒w2= (aσ)−1/22√
2(1−σ)√
3,
ˆW2=−1√
3/parenleftbig
e12+e34+2σe56/parenrightbig
,
p=−/radicalbig
2/3.(3.5)
We easily read off that σ= 1 corresponds to the nearly-K¨ ahler geometry. Note that ev en
thoughW2→0 forσ→1,ˆW2is well-defined and non-zero in this limit so that we can
still use it as an expansion form for the fluxes. The points σ= 2 andσ= 2/5 are also
special, since eq. (2.6) then implies that the supersymmetr ic solution has zero Romans
mass and, in particular, can be lifted to M-theory. Moreover , these are the endpoints of
the interval where supersymmetric solutions exist (since o utside this interval we would find
from eq. (2.6) that m2<0). They are indicated as vertical dashed lines in the plots.
– 7 –Figure 1:Sp(2)
S(U(2)×U(1))-model: plot of aR4Dfor the supersymmetric solutions (light green) and
the new non-supersymmetric solutions (other colors) in terms of t he shape parameter σ. Unstable
solutions are indicated in red.
Pluggingeqs.(3.5) intothesupergravityequationsofmoti on(2.9)wefindnumericallya
rich spectrumofsolutions, whicharedisplayed infigures1a nd2. Note that thedependence
on the overall scale can be easily extracted from all plotted quantities by multiplying by
ato a suitable power. We plotted the value of the 4D Ricci scala rR4Dof the AdS-space
against the shape parameter σin figure 1. Note that R4Dis inversely proportional to the
AdS-radius squared and related to the effective 4D cosmologic al constant and the vev of
the 4D scalar potential Vas follows
Λ =∝angb∇acketleftV∝angb∇acket∇ight=R4D/4. (3.6)
The supersymmetric solutions are plotted in light green, wh ile red is used for the non-
supersymmetric solutions found to be unstable in section 4. For completeness of the pre-
sentation of our numeric results, we provide the values of ea ch of the coefficients fiof the
ansatz (2.2) in figure 2.
The first point to note is that where the supersymmetric solut ions are restricted to
the interval σ∈[2/5,2], there exist non-supersymmetric solutions in the somewh at larger
intervalσ∈[0.39958,2.13327]. Furthermore, there are up to five non-supersymmetri c
solutions for each supersymmetric solution.
We remark that the parameters σand the overall scale are not continuous moduli since
they are determined by the vevs of the fluxes, which in a proper string theory treatment
shouldbequantized. Indeed, inthenextsection wewill show that generically all moduliare
stabilized. We leave the analysis of flux quantization, whic h is complicated by the fact that
– 8 –(a) Plot of a1/2f1(Romans mass)
 (b) Plot of a1/2f2(J-part of ˆF2)
(c) Plot of a1/2f3(ˆW2-part of ˆF2)
 (d) Plot of a1/2f4(J∧J-part of ˆF4)
(e) Plot of a1/2f5(J∧ˆW2-part of ˆF4)
 (f) Plot of a1/2f6(Freund-Rubin parameter)
(g) Plot of a1/2f7(ReΩ part of H)
Figure 2: Plots of the solutions on the cosetSp(2)
S(U(2)×U(1)). Different colors indicate different
solutions. Unstable solutions are indicated in red (see section 4) and the supersymmetric solutions
in light green. By a suitable rescaling of the coefficients the dependen ce on the overall scale ais
taken out.
– 9 –there is non-trivial H-flux (twisting the RR-charges), to further work. The expect ation is
that the continuous line of supergravity solutions is repla ced by discrete solutions.
Let us now take a look at some special values of σ. Forσ= 1 we find five solutions
of which three (including the supersymmetric one) are up to s caling equivalent to the
solutions (3.3) onG2
SU(3)of the previous section [35, 29, 36, 30]. They have f3=f5= 0 and
so the fluxes are completely expressed in terms of J. However, there are also two new non-
supersymmetric solutions (the dark green and the purple one ) which have f3∝negationslash= 0,f5∝negationslash= 0.
Next we turn to the case σ= 2. This point is special in that the metric becomes
the Fubini-Study metric on CP3and the bosonic symmetry of the geometry enhances
from Sp(2) to SU(4). In fact, since the RR-forms of the supers ymmetric solution can be
expanded in terms of the closed K¨ ahler form ˜J= (1/3)J+(2a)1/2W2of the Fubini-Study
metric, the symmetry group of the whole supersymmetric solu tion is SU(4). One can also
show that the supersymmetry enhances from the generic N= 1 toN= 6 [6]. In [37] it
was found that there is an infinite continuous family of non-s upersymmetric solutions and
two discrete separate solutions (see also [35] for an incomp lete early discussion), which all
have SU(4)-symmetry. They are notdisplayed in the plot since they can not be found by
taking a continuous limit σ→2. For these solutions H= 0 (f7= 0) and ˆF2andˆF4are
expanded in terms of ˜J(for more details see [37]).
Instead, in the plot we find apart from the supersymmetric sol ution (which merges
with the dark green solution at σ= 2) two more discrete non-supersymmetric solutions,
which have only Sp(2)-symmetry (since the fluxes cannot be ex pressed in terms of ˜Jonly).
The blue one is new, while the red one turns out to be the reduct ion of the Englert-type
solution. Indeed for the Englert-type solution we expect
f1= 0, no Romans mass ,(3.7a)
f2=f2,susy, f3=f3,susy, same geometry in M ⇒sameˆF2as susy,(3.7b)
f7=−2f4=−(1/3)f6,susy, f5= 0, fromˆF4in M-theory ,(3.7c)
f6= (−2/3)f6,susy, Freund-Rubin parameter changes ,(3.7d)
R4D= (5/6)R4D,susy, 4D Λ changes ,(3.7e)
which agrees with the values displayed in the figures for the r ed curve at σ= 2.
Also forσ= 2/5 we find apart from the supersymmetric solution, the Englert solution
(the purplecurve) andoneextra non-supersymmetricsoluti on (the darkgreen curve). Note
that while the supersymmetric curve joins the olive green cu rve atσ= 2/5, the purple
curve only joins the dark green curve at σ= 0.39958.
SU(3)
U(1)×U(1)
For this manifold the SU(3)-structure is given by [13, 14]:
J=a(−e12+ρe34−σe56),
Ω =a3/2(ρσ)1/2/bracketleftbig
(e245+e135+e146−e236)+i(e235+e136+e246−e145)/bracketrightbig
,(3.8)
– 10 –whereρandσare the shape parameters of the model. Furthermore we find for the torsion
classes:
W1=−(aρσ)−1/21+ρ+σ
3,
W2=−(2/3)a1/2(ρσ)−1/2/bracketleftbig
(2−ρ−σ)e12+ρ(1−2ρ+σ)e34−σ(1+ρ−2σ)e56/bracketrightbig
.(3.9)
It turns out that the ansatz (2.7) is only satisfied for
ρ= 1, σ= 1 orρ=σ. (3.10)
In all three of these cases the equations (2.9) forSU(3)
U(1)×U(1)reduce to exactly the same
equations as forSp(2)
S(U(2)×U(1))so that we obtain the same solution space. However, as we
will see in the next section, the stability analysis will be d ifferent since the model on
SU(3)
U(1)×U(1)has two extra left-invariant modes.
In order to find further non-supersymmetric solutions, we sh ould go beyond the ansatz
(2.7). Let us put
ˆW2∧ˆW2= (−1/6)J∧J+p1ˆW2∧J+p2ˆP∧J, (3.11)
whereˆPis a primitive normalized (1,1)-form (so that it is orthogon al toJandˆP2= 1).
Furthermore, we also choose it orthogonal to ˆW2i.e.
ˆW2·ˆP= 0 or equivalently J∧ˆW2∧ˆP= 0. (3.12)
From the last equation one finds, using (2.1c), that d ˆP∧ImΩ = 0, which implies on
SU(3)
U(1)×U(1)that
dˆP= 0. (3.13)
One can now allow the RR-fluxes ˆF2andˆF4to have pieces proportional to ˆPandˆP∧
Jrespectively and adapt the equations (2.9) accordingly to a ccommodate for the new
contributions. Now it is possible to numerically find non-su persymmetric solutions for ρ
andσnot satisfying (3.10). In particular, there are Englert-ty pe solutions on the ellipse of
values for (ρ,σ) where the supersymmetric solution has zero Romans mass. Fr om eq. (2.6)
we find that this ellipse is described by
m2=5
16ρσ/bracketleftbig
−5(ρ2+σ2)+6(ρ+σ+ρσ)−5/bracketrightbig
= 0. (3.14)
We will not go into more detail on these solutions in this pape r.
4. Stability analysis
Inthissectionweinvestigate whetherthenewnon-supersym metricsolutionsonSp(2)
S(U(2)×U(1))
andSU(3)
U(1)×U(1)are stable5. To this end we calculate the spectrum of scalar fluctuations . We
5In [36] it was found that the non-supersymmetric solutions o nG2
SU(3)and the similar solutions on the
nearly-K¨ ahler limits of the other two coset manifolds unde r study are stable. We find exactly the same
spectrum as the authors of that paper, which provides a consi stency check on our approach. We thank
Davide Cassani for providing us with these numbers, which ar e not explicitly given in their paper. We did
not investigate the spectrum of the similar solution on SU(2 )×SU(2), which is more complicated as there
are more modes.
– 11 –use the well-known result of [39, 40] that in an AdS 4vacuum a tachyonic mode does not yet
signal an instability. Only a mode with a mass-squared below the Breitenlohner-Freedman
bound,
M2<−3|Λ|
4, (4.1)
where Λ<0 is the 4D effective cosmological constant, leads to an instab ility. We restrict
ourselves to left-invariant fluctuations, which implies th at even if we do not find any modes
below the Breitenlohner-Freedman bound, the vacuum might s till be unstable, since there
might be fluctuations with sufficiently negative mass-square d that are not left-invariant.
This analysis can however pinpoint many unstable vacua and w e do believe it gives a
valuable first indication for the stability of the others.
Truncatingtotheleft-invariant modesonthecoset manifol dsunderstudyleads toa4D
N= 2 gauged supergravity6. It has been shown in [36] that this truncation is consistent .
The spectrum of the scalar fields can then be obtained from the 4D scalar potential. In
fact, this computation is analogous to the one performed in [ 14] for the supersymmetric
N= 1 vacua on the coset spaces. As opposed to the models here, th e models in that
paper included orientifolds, which broke the supersymmetr y of the 4D effective theory
fromN= 2 toN= 1. However, also in the present case the N= 1 approach is applicable
and effectively we have used exactly the same procedure, i.e. u sing theN= 1 scalar
fluctuations and obtaining the scalar potential from the N= 1 superpotential and K¨ ahler
potential (see [55, 56, 57, 58]).7The reason is the following. The N= 2 scalar fluctuations
in the vector multiplets are
Jc=J−iB= (ki−ibi)ωi=tiωi, (4.2)
whereωispan the left-invariant two-forms of the coset manifold. Th e orientifold projection
of theN= 1 theory would then project out the scalar fluctuations comi ng from expanding
oneventwo-forms, which are absent for the N= 1 theory on the coset manifolds under
study. The scalar fluctuations in the N= 2 vector multiplets are thus exactly the same as
the scalars in the chiral multiplets of the K¨ ahler moduli se ctor of the N= 1 theory. The
6It is important to make the distinction between the number of supersymmetries of respectively the
4D effective theory, the 10D compactifications, and their 4D t runcation (which are the solutions of the
4D effective theory [36]). In the presence of one left-invari ant internal spinor, the effective theory will be
N= 2 since this same spinor can be used in the 4+ 6 decomposition of both ten-dimensional Majorana-
Weyl supersymmetry generators, but multiplied with indepe ndent four-dimensional spinors. On the other
hand, for a certain compactification to preserve the supersy mmetry, certain differential conditions, which
follow from putting the variations of the fermions to zero mu st be satisfied. In the presence of RR-fluxes,
these conditions mix both ten-dimensional Majorana-Weyl s pinors, putting the four-dimensional spinors in
both decompositions equal. A generic supersymmetric compa ctification therefore only preserves N= 1.
Theσ= 2 supersymmetric K¨ ahler-Einstein solution on CP3on the other hand is non-generic in that it
preserves N= 6, of which only one internal spinor is left-invariant unde r the action of Sp(2) and remains
after truncation to 4D.
7It is interesting to note that (in N= 1 language) all the D-terms vanish, so that the supersymmet ry
breaking is purely due to F-terms. Indeed, in [58] it is shown thatD= 0 is equivalent to d H(e2A−ΦReΨ1) =
0 in the generalized geometry formalism. For SU(3)-structu re this translates to d( e2A−ΦReΩ) = 0 and
H∧ReΩ = 0, which is satisfied for our ansatz, eq. (2.1) and (2.2).
– 12 –(a) Spectrum ofSp(2)
S(U(2)×U(1))
(b) Two extra modes of theSU(3)
U(1)×U(1)-model
Figure 3: Spectrum of left-invariant modes of the solutions onSp(2)
S(U(2)×U(1))andSU(3)
U(1)×U(1).
expansion forms can then be chosen to bethe same as the Y(2−)
iof [14]. Furthermore, there
is one tensor multiplet, which contains the dilaton Φ, the tw o-formBµνand two axions ξ
and˜ξcoming from the expansion of the RR-potential C3:
C3=ξα+˜ξβ, (4.3)
where a choice for αandβspanning the left-invariant three-forms would be Y(3−)and
Y(3+)of [14] respectively. In the presence of Romans mass or ˆF2-flux the two-form Bµν
becomes massive and cannot be dualized to a scalar. The dilat on and˜ξappear in a chiral
multiplet of the complex moduli sector of the N= 1 theory, while Bµνandξare projected
out by the orientifold. By using the N= 1 approach we thus loose the information on just
one scalarξ. A proper N= 2 analysis would however learn that ξdoes not appear in the
scalarpotential (seee.g.[36]), implyingthatitismassle ssandthusabovetheBreitenlohner-
Freedman bound. Moreover, the scalar potential should be th e same whether it is obtained
directly from reducing the 10D supergravity action (as in [5 9]) or whether it is obtained
usingN= 2 orN= 1 technology8. Furthermore we note that the massless scalar field ξ
not appearing in the potential is not a modulus, since it is ch arged [60, 61], and therefore
eaten by a vector field becoming massive.
Thespectraof left-invariant modesforSp(2)
S(U(2)×U(1))andSU(3)
U(1)×U(1)aredisplayed infigure
3. The Breitenlohner-Freedman bound is indicated as a horiz ontal dashed line. The Sp(2)-
model has six scalar fluctuations entering the potential: ki,biwithi= 1,2 from the two
vector multiplets, and Φ ,˜ξfrom the universal hypermultiplet, while the SU(3)-model h as
two more fluctuations from the extra vector multiplet. These two extra modes make a big
difference for the stability analysis since one of them tends t o be below the Breitenlohner-
Freedman bound for the purple and dark green solution. As a re sult, even though the
solutions for the Sp(2)- and SU(3)-model take the same form, the SU(3)-model has more
unstable solutions: compare figure 1 and 4.
8The only potential difference between the latter two would be the contribution from the orientifold.
We have checked that this contribution vanishes in the scala r potentials of [14] in the limit of the orientifold
chargeµ→0.
– 13 –Figure 4:SU(3)
U(1)×U(1)-model: plot of aR4Din terms of the shape parameter σ. Unstable solutions
are indicated in red.
Inparticular, wenotethatthereductionoftheEnglert-typ esolutionisunstablefor σ=
2 in the Sp(2)-model, in agreement with [38], since the M-the ory lift of the corresponding
supersymmetric solution has eight Killing spinors. We inde ed find the same negative mass-
squaredM2=−(4/5)|Λ|for the unstable mode as in that paper. On the other hand,
forσ= 2/5 the Englert-type solution is stable against left-invaria nt fluctuations. This is
still in agreement with [38] which relied on the existence of at least two Killing spinors,
while the M-theory lift of the N= 1 supersymmetric solution at σ= 2/5 has only one
Killing-spinor. For the SU(3)-model, all Englert-type sol utions turn out to be unstable
(including the ones outside the condition (3.10)).
We also investigated the stability of the additional soluti ons at the special point σ= 2
found in [37]. We found that for the Sp(2)-model all these sol utions are stable against left-
invariant fluctuations. For the SU(3)-model on theother han dit turnsout that thediscrete
solutions ineqs.(3.16) and(3.17) ofthatreferenceareuns table, whilethecontinuous family
of eq. (3.18) becomes unstable for
γ2
β2>5(75∓16√
21)
8217, (4.4)
for the±sign choice in front of the square root in eq. (3.18) of that pa per respectively
(note that the supersymmetric solution corresponds to the p ointγ2/β2= 0 in this family).
Finally, we note that generically (i.e. unless an eigenvalu e is crossing zero at a special
value forσ) all the plotted modes are massive. For a range of values for σone of the
eigenvalues for the dark green and purple solution takes a sm all, but still non-zero value.
– 14 –5. Conclusions
In this paper we presented new families of non-supersymmetr ic AdS 4vacua. In fact,
extrapolating from our analysis on these specific coset mani folds and under the assumption
that a proper treatment of flux quantization does not kill muc h more vacua than in the
supersymmetric case, it would seem that there are more of the se non-supersymmetric
vacua than supersymmetric ones. This would imply that such v acua cannot be ignored
in landscape studies. We have moreover shown that many of the m are stable against a
specific set of fluctuations, namely the ones that can be expan ded in terms of left-invariant
forms. If these vacua turn out to be stable against all fluctua tions they should also have
a CFT-dual, which could be studied along the lines of [20], wh ere the three-dimensional
Chern-Simons-matter theory dual to a particular highly sym metric non-supersymmetric
vacuum was proposed. Furthermore, the nice property of some IIA vacua that all moduli
enter the superpotential and thus can be stabilized at a clas sical level [15] also extends to
our non-supersymmetric vacua.
A next step would be to relax the constraint that the solution s should have the same
geometry as the supersymmetric solution. It is also interes ting to investigate whether a
similar ansatz and techniques can be used to look for tree-le vel dS-vacua [62].
Acknowledgments
We thank Davide Cassani for useful email correspondence and proofreading, and further-
more Claudio Caviezel for active discussions and initial co llaboration. We would further
like to thank the Max-Planck-Institut f¨ ur Physik in Munich , where both of the authors
were affiliated during the bulk of the work on this paper. P.K. i s a Postdoctoral Fellow
of the FWO – Vlaanderen. The work of P.K. is further supported in part by the FWO –
Vlaanderen project G.0235.05 and in part by the Federal Office for Scientific, Technical and
Cultural Affairs through the ’Interuniversity Attraction Po les Programme Belgian Science
Policy’ P6/11-P. S.K. is supported by the SFB – Transregio 33 “The Dark Universe” by
the DFG.
A. SU(3)-structure
A real non-degenerate two-form Jand a complex decomposable three-form Ω define an
SU(3)-structure on the 6D manifold M6iff:
Ω∧J= 0, (A.1a)
Ω∧¯Ω =8i
3!J∧J∧J∝negationslash= 0, (A.1b)
and the associated metric is positive-definite. This metric is determined by Jand Ω as
follows:
gmn=−JmpIpn, (A.2)
withIthe complex structure associated (in the way of [63]) to Ω. Th e volume-form is
given by vol 6=1
3!J3=−(i/8)Ω∧¯Ω.
– 15 –Theintrinsictorsionofthemanifold M6decomposesintofivetorsionclasses W1,...,W5.
Alternatively they correspond to the SU(3)-decomposition of the exterior derivatives of J
and Ω [64]. Intuitively, they parameterize the failure of th e manifold to be of special
holonomy, which can also be thought of as the deviation from c losure ofJand Ω. More
specifically we have:
dJ=3
2Im(W1¯Ω)+W4∧J+W3,
dΩ =W1J∧J+W2∧J+¯W5∧Ω,(A.3)
whereW1is a scalar, W2is a primitive (1,1)-form, W3is a real primitive (1 ,2)+(2,1)-form,
W4is a real one-form and W5a complex (1,0)-form. In this paper only the torsion classes
W1,W2are non-vanishing and they are purely imaginary, so it will b e convenient to define
W1,2so thatW1,2=iW1,2. A primitive (1,1)-form P(such asW2) transforms under the 8
of SU(3) and satisfies
P∧J∧J= 0. (A.4)
The Hodge dual is given by
⋆6P=−P∧J. (A.5)
A primitive (1 ,2)(or (2,1))-formQon the other hand transforms as a 6(or¯6) under SU(3)
and satisfies
Q∧J= 0. (A.6)
B. Type II supergravity
The bosonic content of type II supergravity consists of a met ricG, a dilaton Φ, an NSNS
three-form Hand RR-fields Fn. We use the democratic formalism of [65], in which the
number of RR-fields is doubled, so that nruns over 0 ,2,4,6,8,10 in type IIA and over
1,3,5,7,9 in IIB. We will often collectively denote the RR-fields with the polyform F=/summationtext
nFn. We have also doubled the RR-potentials, collectively deno ted byC=/summationtext
nC(n−1).
These potentials satisfy F= dHC+me−B= (d +H∧)C+me−B. In type IIB there is
of course no Romans mass m, so that the second term vanishes. In type IIA we find in
particularF0=m.
The bosonic part of the pseudo-action of the democratic form alism then simply reads
S=1
2κ2
10/integraldisplay
d10X√
−G/braceleftbigg
e−2Φ/bracketleftbigg
R+4(dΦ)2−1
2H2/bracketrightbigg
−1
4F2/bracerightbigg
, (B.1)
where we defined F2=/summationtext
nF2
nand the square of an l-formPas follows
P2=P·P=1
l!Pm1...mlPm1...ml, (B.2a)
where the indices are raised with the inverse of the metric Gmnor the internal metric gmn
(defined later on), depending on the context. In the followin g it will also be convenient to
define:
Pm·Pn=ιmP·ιnP=1
(l−1)!Pmm2...mlPnm2...ml. (B.2b)
– 16 –The extra degrees of freedom for the RR-fields in the democrat ic formalism have to be
removed by hand by imposing the following duality condition at the level of the equations
of motion after deriving them from the action (B.1):
Fn= (−1)(n−1)(n−2)
2⋆10F10−n. (B.3)
That is why (B.1) is only a pseudo-action.
The fermionic content consists of a doublet of gravitinos ψMand a doublet of dilatinos
λ. The components of the doublets are of different chirality in t ype IIA and of the same
chirality in type IIB.
In this paper we look for vacuum solutions that take the form A dS4×M6. In principle
there could also be a warp factor A, but it will always be constant for the solutions in this
paper. We can choose it to be zero. The compactification ansat z for the metric then reads
ds2
10=GmndXmdXn= ds2
4+gmndxmdxn, (B.4)
where ds2
4is the line-element for AdS 4andgmnis the metric on the internal space M6. For
the RR-fluxes the ansatz becomes
F=ˆF+vol4∧˜F, (B.5)
whereˆFand˜Fonly have internal indices. The duality constraint (B.3) im plies that ˜Fis
not independent of ˆF, and given by
˜Fn= (−1)(n−1)(n−2)
2⋆6ˆF6−n. (B.6)
What we need in this paper are the type II equations of motion, which can be found
from the pseudo-action (B.1). We use them as they are written down in [5] (originally they
were obtained for massive type IIA in [35]), but take some lin ear combinations in order
to further simplify then. Without source terms (i.e. we put jtotal= 0 in the equations of
motion of [5]), they then read:
dHF= 0 (Bianchi RR fields) , (B.7a)
d−H⋆10F= 0 (eom RR fields) , (B.7b)
dH= 0 (BianchiH), (B.7c)
d/parenleftbig
e−2Φ⋆10H/parenrightbig
−1
2/summationdisplay
n⋆10Fn∧Fn−2= 0 (eom H), (B.7d)
2R−H2+8/parenleftbig
∇2Φ−(∂Φ)2/parenrightbig
= 0 (dilaton eom) , (B.7e)
2(∂Φ)2−∇2Φ−1
2H2−e2Φ
8/summationdisplay
nnF2
n= 0 (trace Einstein/dilaton eom) ,(B.7f)
RMN+2∇M∂NΦ−1
2HM·HN−e2Φ
4/summationdisplay
nFnM·FnN= 0 (B.7g)
(Einstein eq./dilaton/trace) .
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