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