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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

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