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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.

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.