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Alternatively they correspond to the SU(3)-decomposition of the exterior derivatives of J
and Ω [64]. Intuitively, they parameterize the failure of th e manifold to be of special
holonomy, which can also be thought of as the deviation from c losure ofJand Ω. More
speciﬁcally we have:
dJ=3
2Im(W1¯Ω)+W4∧J+W3,
dΩ =W1J∧J+W2∧J+¯W5∧Ω,(A.3)
whereW1is a scalar, W2is a primitive (1,1)-form, W3is a real primitive (1 ,2)+(2,1)-form,
W4is a real one-form and W5a complex (1,0)-form. In this paper only the torsion classes
W1,W2are non-vanishing and they are purely imaginary, so it will b e convenient to deﬁne
W1,2so thatW1,2=iW1,2. A primitive (1,1)-form P(such asW2) transforms under the 8
of SU(3) and satisﬁes
P∧J∧J= 0. (A.4)
The Hodge dual is given by
⋆6P=−P∧J. (A.5)
A primitive (1 ,2)(or (2,1))-formQon the other hand transforms as a 6(or¯6) under SU(3)
and satisﬁes
Q∧J= 0. (A.6)
B. Type II supergravity
The bosonic content of type II supergravity consists of a met ricG, a dilaton Φ, an NSNS
three-form Hand RR-ﬁelds Fn. We use the democratic formalism of [65], in which the
number of RR-ﬁelds is doubled, so that nruns over 0 ,2,4,6,8,10 in type IIA and over
1,3,5,7,9 in IIB. We will often collectively denote the RR-ﬁelds with the polyform F=/summationtext
nFn. We have also doubled the RR-potentials, collectively deno ted byC=/summationtext
nC(n−1).
These potentials satisfy F= dHC+me−B= (d +H∧)C+me−B. In type IIB there is
of course no Romans mass m, so that the second term vanishes. In type IIA we ﬁnd in
particularF0=m.
The bosonic part of the pseudo-action of the democratic form alism then simply reads
S=1
2κ2
10/integraldisplay
d10X√
−G/braceleftbigg
e−2Φ/bracketleftbigg
R+4(dΦ)2−1
2H2/bracketrightbigg
−1
4F2/bracerightbigg
, (B.1)
where we deﬁned F2=/summationtext
nF2
nand the square of an l-formPas follows
P2=P·P=1
l!Pm1...mlPm1...ml, (B.2a)
where the indices are raised with the inverse of the metric Gmnor the internal metric gmn
(deﬁned later on), depending on the context. In the followin g it will also be convenient to
deﬁne:
Pm·Pn=ιmP·ιnP=1
(l−1)!Pmm2...mlPnm2...ml. (B.2b)
– 16 –The extra degrees of freedom for the RR-ﬁelds in the democrat ic formalism have to be
removed by hand by imposing the following duality condition at the level of the equations
of motion after deriving them from the action (B.1):
Fn= (−1)(n−1)(n−2)
2⋆10F10−n. (B.3)
That is why (B.1) is only a pseudo-action.
The fermionic content consists of a doublet of gravitinos ψMand a doublet of dilatinos
λ. The components of the doublets are of diﬀerent chirality in t ype IIA and of the same
chirality in type IIB.
In this paper we look for vacuum solutions that take the form A dS4×M6. In principle
there could also be a warp factor A, but it will always be constant for the solutions in this
paper. We can choose it to be zero. The compactiﬁcation ansat z for the metric then reads
ds2
10=GmndXmdXn= ds2
4+gmndxmdxn, (B.4)
where ds2
4is the line-element for AdS 4andgmnis the metric on the internal space M6. For
the RR-ﬂuxes the ansatz becomes
F=ˆF+vol4∧˜F, (B.5)
whereˆFand˜Fonly have internal indices. The duality constraint (B.3) im plies that ˜Fis
not independent of ˆF, and given by
˜Fn= (−1)(n−1)(n−2)
2⋆6ˆF6−n. (B.6)
What we need in this paper are the type II equations of motion, which can be found
from the pseudo-action (B.1). We use them as they are written down in [5] (originally they
were obtained for massive type IIA in [35]), but take some lin ear combinations in order
to further simplify then. Without source terms (i.e. we put jtotal= 0 in the equations of
motion of [5]), they then read:
dHF= 0 (Bianchi RR ﬁelds) , (B.7a)
d−H⋆10F= 0 (eom RR ﬁelds) , (B.7b)
dH= 0 (BianchiH), (B.7c)
d/parenleftbig
e−2Φ⋆10H/parenrightbig
−1
2/summationdisplay
n⋆10Fn∧Fn−2= 0 (eom H), (B.7d)
2R−H2+8/parenleftbig
∇2Φ−(∂Φ)2/parenrightbig
= 0 (dilaton eom) , (B.7e)
2(∂Φ)2−∇2Φ−1
2H2−e2Φ
8/summationdisplay
nnF2
n= 0 (trace Einstein/dilaton eom) ,(B.7f)
RMN+2∇M∂NΦ−1
2HM·HN−e2Φ
4/summationdisplay
nFnM·FnN= 0 (B.7g)
(Einstein eq./dilaton/trace) .
– 17 –References

Alternatively they correspond to the SU(3)-decomposition of the exterior derivatives of J

and Ω [64]. Intuitively, they parameterize the failure of th e manifold to be of special

holonomy, which can also be thought of as the deviation from c losure ofJand Ω. More

speciﬁcally we have:

dJ=3

2Im(W1¯Ω)+W4∧J+W3,

dΩ =W1J∧J+W2∧J+¯W5∧Ω,(A.3)

whereW1is a scalar, W2is a primitive (1,1)-form, W3is a real primitive (1 ,2)+(2,1)-form,

W4is a real one-form and W5a complex (1,0)-form. In this paper only the torsion classes

W1,W2are non-vanishing and they are purely imaginary, so it will b e convenient to deﬁne

W1,2so thatW1,2=iW1,2. A primitive (1,1)-form P(such asW2) transforms under the 8

of SU(3) and satisﬁes

P∧J∧J= 0. (A.4)

The Hodge dual is given by

⋆6P=−P∧J. (A.5)

A primitive (1 ,2)(or (2,1))-formQon the other hand transforms as a 6(or¯6) under SU(3)

and satisﬁes

Q∧J= 0. (A.6)

B. Type II supergravity

The bosonic content of type II supergravity consists of a met ricG, a dilaton Φ, an NSNS

three-form Hand RR-ﬁelds Fn. We use the democratic formalism of [65], in which the

number of RR-ﬁelds is doubled, so that nruns over 0 ,2,4,6,8,10 in type IIA and over

1,3,5,7,9 in IIB. We will often collectively denote the RR-ﬁelds with the polyform F=/summationtext

nFn. We have also doubled the RR-potentials, collectively deno ted byC=/summationtext

nC(n−1).

These potentials satisfy F= dHC+me−B= (d +H∧)C+me−B. In type IIB there is

of course no Romans mass m, so that the second term vanishes. In type IIA we ﬁnd in

particularF0=m.

The bosonic part of the pseudo-action of the democratic form alism then simply reads

S=1

2κ2

10/integraldisplay

d10X√

−G/braceleftbigg

e−2Φ/bracketleftbigg

R+4(dΦ)2−1

2H2/bracketrightbigg

−1

4F2/bracerightbigg

, (B.1)

where we deﬁned F2=/summationtext

nF2

nand the square of an l-formPas follows

P2=P·P=1

l!Pm1...mlPm1...ml, (B.2a)

where the indices are raised with the inverse of the metric Gmnor the internal metric gmn

(deﬁned later on), depending on the context. In the followin g it will also be convenient to

deﬁne:

Pm·Pn=ιmP·ιnP=1

(l−1)!Pmm2...mlPnm2...ml. (B.2b)

– 16 –The extra degrees of freedom for the RR-ﬁelds in the democrat ic formalism have to be

removed by hand by imposing the following duality condition at the level of the equations

of motion after deriving them from the action (B.1):

Fn= (−1)(n−1)(n−2)

2⋆10F10−n. (B.3)

That is why (B.1) is only a pseudo-action.

The fermionic content consists of a doublet of gravitinos ψMand a doublet of dilatinos

λ. The components of the doublets are of diﬀerent chirality in t ype IIA and of the same

chirality in type IIB.

In this paper we look for vacuum solutions that take the form A dS4×M6. In principle

there could also be a warp factor A, but it will always be constant for the solutions in this

paper. We can choose it to be zero. The compactiﬁcation ansat z for the metric then reads

ds2

10=GmndXmdXn= ds2

4+gmndxmdxn, (B.4)

where ds2

4is the line-element for AdS 4andgmnis the metric on the internal space M6. For

the RR-ﬂuxes the ansatz becomes

F=ˆF+vol4∧˜F, (B.5)

whereˆFand˜Fonly have internal indices. The duality constraint (B.3) im plies that ˜Fis

not independent of ˆF, and given by

˜Fn= (−1)(n−1)(n−2)

2⋆6ˆF6−n. (B.6)

What we need in this paper are the type II equations of motion, which can be found

from the pseudo-action (B.1). We use them as they are written down in [5] (originally they

were obtained for massive type IIA in [35]), but take some lin ear combinations in order

to further simplify then. Without source terms (i.e. we put jtotal= 0 in the equations of

motion of [5]), they then read:

dHF= 0 (Bianchi RR ﬁelds) , (B.7a)

d−H⋆10F= 0 (eom RR ﬁelds) , (B.7b)

dH= 0 (BianchiH), (B.7c)

d/parenleftbig

e−2Φ⋆10H/parenrightbig

−1

2/summationdisplay

n⋆10Fn∧Fn−2= 0 (eom H), (B.7d)

2R−H2+8/parenleftbig

∇2Φ−(∂Φ)2/parenrightbig

= 0 (dilaton eom) , (B.7e)

2(∂Φ)2−∇2Φ−1

2H2−e2Φ

8/summationdisplay

nnF2

n= 0 (trace Einstein/dilaton eom) ,(B.7f)

RMN+2∇M∂NΦ−1

2HM·HN−e2Φ

4/summationdisplay

nFnM·FnN= 0 (B.7g)

(Einstein eq./dilaton/trace) .

– 17 –References