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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),

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),