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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

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