N=1 domain wall solutions of massive type II supergravity and the issue of mirror symmetry

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by  Silvia Vaula

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N=1 domain wall solutions of massive type II supergravity and
the issue of mirror symmetry
SilviaVaula` ∗
InstitutodeF´ısicaTeo´ricaUAM/CSIC
FacultaddeCienciasC-XVI,C.U.Cantoblanco,E-28049-Madrid,Spain
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0 Keywords Supergravity,Fluxcompactifications,DomainWalls,MirrorSymmetry
2 PACS 11.25.-w,12.10.-g,04.65.+e

n
WereportonDomainWallsolutionofCalabi–Yaucompactificationswithgeneral fluxesandtheir appli-
a
cationtothestudyofmirrorsymmetryingeneralizedbackgrounds. Weaddress,inparticular,totheissue
J
ofmagneticNSNSfluxes. WeshowthattheDomainWallgradientflowequationscanbeinterpretedasa

1 setofgeneralizedHitchin’sflowequationsofamanifoldwithSU(3)×SU(3)structurefiberedalongthe
3 directiontransversetotheDomainwall.

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v
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8 The equivalence between type II theories compactified on Calabi–Yau (CY) mirror pairs (Y, Y˜) is
2 a well–known property. In particular, compactification of type IIA and IIB supergravity on Y and Y˜
1 respectively, gives rise to the same N = 2 four dimensional ungauged supergravity. The possibility to
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considercompactificationsinthepresenceofp–formfluxesalongnontrivialp–cyclese = F raises
7 I γI p
thequestionwhetheritispossibletogeneralizethenotionofmirrorsymmetrytosuchmoregeHneralcases.
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/ CY compactifications of type II supergravities in the presence of fluxes yelds four dimensional gauged
h
supergravities, the gauge parameters being proportional to the fluxes [1]–[7]. It has been proved that
t
- type II supergravitiescompactified on mirror pairs (Y, Y˜) in the presence of RR fluxes, give rise to the
p
same gauged supergravity[2]. This shows that at the supergravitylevel type IIA and IIB RR fluxes are
e
h mappedontooneanotherundermirrorsymmetry,ascanbeexpectedfromthematchingofoddandeven
: cohomologyonmirrorpairs. ForthesamereasonitisunlikelythatasimilarmatchingholdsforNSNS3–
v
i formfluxes,whichresideintheoddcohomologybothfortypeIIAandIIB.InfactfortheNSNSsectorthe
X
3–formfluxescorrespondundermirrorsymmetryto adeformationoftheCY geometry. Moreprecisely,
r in the presence of electric fluxes, the SU(3) holonomyof the mirror manifold is relaxed in favor of an
a
SU(3) structure, the intrinsic torsion correspondingto the NSNS 3–formfluxes [8]. In particular, it has
beenproventhatCY compactificationof typeII supergravitiesin the presenceof electric NSNS 3–form
fluxescorrespondsundermirrorsymmetrytocompactificationsonahalf–flatmanifold[9,10],wherethe
CYcohomologyisdeformedaccordingto
dα =−e ωi; dα =0; dβA =0; dω =e β0; dωi =0. (1)
0 i a i i
In order to fix our notations let us consider e.g. type IIB supergravity compactified on a CY Y˜ in the
presence of NSNS 3–form fluxes. This correspondsto type IIA compactifiedon a half–flat manifoldYˆ
(1);inthiscaseA = (0, a),a = 1,...h(2,1) andi = 1,...h(1,1). Theparameterse correspondtothe
i
electricNSNSfluxesintypeIIBcompactificationonY˜,thatisH = ···−e βΛ withΛ = (0, i),while
3 Λ
thefluxcomponente ismappedintoitself.
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The descriptionof mirrorsymmetryin the presenceof magneticNSNS 3–formfluxesis more involved.
ConsiderindeedthegeneralcasewhereH = ···+mΛα −e βΛ. Thechoiceofelectricversusmag-
3 Λ Λ
netic fluxescorrespondsto a changeofbasisin (α , βΛ), the oddcohomologyofY˜, reflectingthe four
Λ
∗ Correspondingauthor E-mail:[email protected]
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4 SilviaVaula`:
dimensionalelectric/magneticdualityofthevectorequationsofmotions/Bianchiidentities,whichholdsas
wellinthepresenceofgeneralfluxes[4,6]. Suchasymplecticstructurehasnoimmediatecounterpartin
theevencohomologyofY (1). Moreover,adeformationof(1)supportingamagneticindexwouldimply
thepresenceofaclosed5–formχ,suchthate.g. dωi ∝ miχ[11]. Thefollowingobservationgoesalong
thelineof[12,13]whereitisdiscussedthecorrespondenceundermirrorsymmetryofthemagneticNSNS
fluxesandcompactificationsonanSU(3)×SU(3)structuremanifoldYˇ.
Indeed, introducing a symplectic basis also for the even cohomology, which includes 0– and 6– forms,
(ω ,ωΛ), and indicating with (α , βΛ) a basis for the odd cohomology, which includes 1–, 3– and 5–
Λ Λ
forms,onecanwriteforYˇ
dα =mΛω −e ωΛ; dα =0; dβA =0; dω =e β0; dωΛ =mΛβ0. (2)
0 Λ Λ a Λ Λ
SinceverylittleisknownaboutthepropertiesofSU(3)×SU(3)structuremanifoldsitisnotstraightfor-
wardtocheck(2)byperforminganexplicitcompactificationonYˇ. Instead,following[14],wearegoing
tomakeuseofDomainWall(DW)solutions. ADWsolutionofthefourdimensionalgaugedsupergravity
correspondingtotypeIIBcompactificationinthepresenceofelectricNSNSfluxeswasconsideredin[14].
TakingadvantageoftheSO(1,2)× RsymmetryoftheDWsolution,onecanconsiderthegeometryof
w
Xˆ ≡Yˆ × R,whereRparametrizesthedirectiontransversetotheDW.IfYˆ isahalf–flatmanifoldthen
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Xˆ isaG holonomymanifold,whosegeometryischaracterizedbytheHitchin’sflowequations[16]
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dImΦ− =dImΦ+ =0; ∂yImΦ+ =−dReΦ−; ∂yImΦ− =dReΦ+ (3)
whereΦ±arethetwopurespinorcharacterizingYˆ,disthedifferentialalongYˆ while∂y isthederivative
alongthe(rescaled)directiontransversetotheDW.
Wearegoingtogeneralizetheresultof[14]andconsider[15]aDWsolutionofthegaugedsupergravity
corresponding to type IIB compactification in the presence of electric and magnetic NSNS fluxes. As-
sumingthatthegeometryofXˇ ≡ Yˇ × R isdescribedbyHitchin’s–likeflowequations[17], theflow
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equationsofthe DWsolutionprovidethe y–dependencein (3)allowingto checkif (2)issuitable to de-
scribeYˇ.
LetusthereforeconsiderthefollowingDWmetric:
g (xµ)dxµdxν =eU(z)gˆ (xm)dxmdxn −e−2pU(z)dzdz. (4)
µν mn
wheregˆ (xm), m = 0, 1, 2,isthemetricofathree-dimensionalspace-timewhichweassumetohave
mn
constantcurvatureandpisanarbitraryrealnumber.Usingµ=eU(z) insteadofzasthecoordinateofthe
transversespacewearriveat
dµdµ
g (xµ)dxµdxν =µ2gˆ (xm)dxmdxn− , (5)
µν mn µ2W2(z)
where
W =±epU(z)U′(z). (6)
In order to look for 1/2 BPS configurations we impose the following projector on the supersymmetry
parameters[14,18]
ε =hA γ εB . (7)
A AB 3
Here h(z) is a complexfunction while A is a constantmatrix. Consistency of (7) with its hermitian
AB
conjugateimplieshh=1whileAB ≡A ǫCB mustbeahermitianmatrixwhichinadditionsatisfies
A AC
ABAC =δC . (8)
A B A
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ThusAhastobeasuitablelinearcombinationof(1,~σ)where~σarethePaulimatrices.
As we are interested in the flow of the fields along the direction transverse to the Domain Wall, we
assume they do notdependon the coordinatesxm; moreoverwe set to zerothe vectorand tensor field–
strengthsFΛ =HI =0. Withthisassumptionthesupersymmetrytransformationlawsofthefermions
µν µνρ
simplifyandthe1/2BPSconditionreads[4]
δψ =D ε +iS γ εB =0 (9)
µA µ A AB µ
δλiA =i∂ tiγµεA+WiABε =0 (10)
µ B
δζ =iP ∂ quγµεA+NAε =0 (11)
α uAα µ α A
where(7)isimposed.ThematricesS ,WiABandNAarethefermionshifts;theyareduetothegauging
AB α
andencodeitsproperties. Forthepresentcase,andmoregenerallyforAbeliangaugings,theyarerelated
bygradientflowequations[19]
1 1
∇ S = g W¯ ; ∇ S = h PvαN (12)
i AB 2 i¯ AB u AB 2 uv (A B)α
Thispropertygivestotheconditions(10)and(11)thestructureofgradientflowequations
dti dqu
µ =−gi∇ lnW; µ =−guv∂ lnW (13)
dµ  dµ v
where we have used as a transverse coordinate µ(z) = eU(z). Equations (9)–(11) further imply that
AABS isproportionaltotheidentityorinotherwords
BC
1
iAABS = WδA , (14)
BC 2 C
wheretheproportionalityfactordefinesthesuperpotentialW in(13).Thecondition(14)imposesalgebraic
constraintsonthescalarfieldswhichplayanimportantroleinthederivationofthesolution[15].
Sincethenon–vanishingcomponentsofthespinconnectionωabaregivenby
µ
ωab =ωˆab, ωa3 =e(p+1)U(z)U′(z)eˆa , (15)
m m m m
equation(9)forµ=misaswellcontrolledbythesuperpotentialW.
1 1
= eUIm(hW); U′ =e−pURe(hW) (16)
ℓ 2
where 1 isthecosmologicalconstantalongtheDW.From(11)wealsoobtainthathW isrealandtherefore
ℓ
theDWisflat. Furthermore,wecanidentify
W =±|W|. (17)
Using(13)and(16)onecanfindtheexplicitU(z)dependenceoftherelevantscalarfields[15].
SinceouraimistogiveageometricinterpretationintermoftypeIIAcompactification,fromnowonwe
willinterpretallthefieldsinthisframework.Inparticularwehave[15]
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eφA =C; e−KV =4e2U; e−KH =16C2e−6U; W = eU(XΛe −F mΛ) (18)
Λ Λ
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NotethattheIIAdilatonisconstant.TheKa¨hlerpotentialsofthevector–andhyper–multipletsaredefined
intermsofthemoduliofYˆ intypeIIAoncompactificationsas
e−KV ≡i X¯ΛF −F¯ XΛ ; e−KH ≡i Z¯AW −W¯ ZA (19)
Λ Λ A A
(cid:2) (cid:3) (cid:2) (cid:3)
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whereΛ = 0,...n andA = 0,...n beingn andn thenumberofvector–andhyper–multiplets,
V H V H
respectively. They parametrize the Ka¨hler class and the complex structure deformations n = h(1,1),
V
n =h(1,2).
H
The complexscalars in (10) are the special coordinatesof the vector multipletsspecial Ka¨hler manifold
ti = Xi. The scalars za = Za span the special Ka¨hler submanifold inside the quaternionic manifold
X0 Z0
spannedbythequ(11).
In order to describe the geometry of the internal manifold Yˇ, we can convenientlyrewrite equations
(13)and(16).Beforeweperformachangeofcoordinates∂ =e(p+3)U∂ anddefinearescaledsection
z w
(ZA, W )=|c|(ZA, W ) , |c|2 ≡eKV−KH (20)
A A η
Wecannowrewrite(13)as
ImXΛ mΛ
∂ =− (21)
w(cid:18)ImFΛ(cid:19) (cid:18)eΛ(cid:19)
ImZA 0
∂wImWa=−|c| 0  . (22)
ImW ReXΛe −ReF mΛ
 0η  Λ Λ 
andthezerocosmologicalconstatcondition(16)as
ImXΛe −ImF mΛ =0. (23)
Λ Λ
Letusnowdefinetwopurespinorsaccordingtothesymplecticinvariantbasis(2)
Φ+ =XΛωΛ−FΛωΛ, Φ− =ZηAαA−WηAβA . (24)
ThederivativealongthedirectiontransversetotheDWiseasilycomputedusing(21),(22)
∂yImΦ+ =eΛωΛ−mΛωΛ; ∂yImΦ− =−|c|(ReXΛeΛ−ReFΛmΛ)β0. (25)
Using(2)weobtain
dΦ+ =(XΛeΛ−FΛmΛ)β0; dΦ− =|c|−1(mΛωΛ−eΛωΛ) (26)
Up to a change of coordinatesdy = |c|−1dw, equations(25), (26) are equivalentto (3), where now the
purespinorsΦ± aredefinedin(24).
Finally, letusdiscussthe propertiesoftheseven-dimensionalmanifoldXˇ . Asthemetriconthe DWis
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flatandthebackgroundM × Xˇ solvesthestringequationofmotion,weexpectXˇ tobeRicciflat.
(1,2) w 7 7
Forhalf-flatmanifoldsthiswasindeedshowninrefs.[14,16,20]. Inordertodiscussthegeneralizationat
handletusintroducethesevendimensionalexteriorderivativeby
dˆ=d+dy∂ , (27)
y
where d acts on Yˇ and ∂ is the derivativewith respectto the coordinateof R. Furthermore,following
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[17]onecandefinethegeneralizedformsρand∗ρonXˇ7whicharegivenintermsofΦ±by
ρ=−ReΦ+∧dy−ImΦ− , ∗ρ=ReΦ−∧dy+ImΦ+. (28)
∗ρistheHodgedualofρwithrespecttothegeneralizedmetric. Asnotedin[17]theequations(25)–(26)
thencorrespondto
dρ=∗d∗ρ=0, (29)
andimplythatXˇ hasanintegrableG ×G structureandisindeedRicci-flat.
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Acknowledgements ThisworkhasbeensupportedbytheSpanishMinistryofScienceandEducationgrantBFM2003-
01090,theComunidaddeMadridgrantHEPHACOSP-ESP-00346andbytheEUResearchTrainingNetworkCon-
stituents,FundamentalForcesandSymmetriesoftheUniverseMRTN-CT-2004-005104
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