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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)

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)