Calibrations on spaces with GxG-structure

0.07 MB·English

by  Florian Gmeiner

NIKHEF/2006-012
Calibrations on spaces with G G–structure
×
FlorianGmeiner1andFrederikWitt2
1 NIKHEF,Kruislaan409,1098SJAmsterdam,TheNetherlands
2 FreieUniversita¨tBerlin,Arminallee3,14195Berlin,Germany
7 InthesenoteswegiveanintroductiontotheconceptofspaceswithG×G–structureandtheirstructured
0
submanifolds.Theseobjectsgeneralisetheclassicalnotionofacalibratedsubmanifold.Therefore,theyare
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interestingfromastringtheoryviewpointastheyarerelevanttodescribeD–branesinstringcompactifica-
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tionsonbackgroundswithfluxes.

n
a
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ThisarticleisbasedonatalkgivenbythefirstauthorattheRTNWorkshop”Constituents,Fundamental

1 ForcesandSymmetriesoftheUniverse”,Naples,Italy,9-13Oct2006.
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1 Introduction
1
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7 Compactifications of type II string theory with D–branes are very interesting from the point of view of
0 string phenomenologyand have been studied extensivelyin recentyears. Generically, the effectivefour
/ dimensionaltheoriesobtainedthiswaycontainseveralmasslessscalars,theso-calledmoduli. Inorderto
h
t obtainasensibletheory,onewouldliketohaveapotentialtofixtheirvalues.Thiscanbeachievedbyintro-
-
p ducingbackgroundfluxes,non–trivialvaluesfortheNS–NSandR–Rfieldstrengthinthesix–dimensional
e manifoldusedtocompactifythetheory(forarecentreviewontypeIIcompactificationswithandwithout
h
fluxessee[1]). Introducinggeneralfluxes,oneleavestherealmofCalabi–Yaucompactificationsanddeals
:
v withbackgroundswithanSU(3) SU(3)–structure[2,3].
i ×
X ThesegeometriesareaspecialcaseofG G–structures,introducedin[4],andbasedonthenotionofso–
×
calledgeneralisedgeometry,whichgoesbacktoHitchin[5]. Possibleapplicationsinstringtheoryarenot
r
a limitedtocompactificationsoftypeII,butmightalsoberelevanttocompactificationsofM–theory[10].
Introducinghigher–dimensionalobjects,suchasD–branesintypeII,inspacesofgeneralisedgeometry
triggersnewquestions,in particularontheamountofpreservedsupersymmetryin thefour–dimensional
theory. In the classical case of SU(3)–compactifications, this question can be answered by considering
calibrations [6]. As it turns out, the concept of a calibration also makes sense in spaces with G G–
×
structure [7]. Moreover, this notion of a generalised calibration includes the classical case as a special
one. Inthecontextof D–branesintypeII stringtheory,suchcalibratedsubmanifoldshavebeenconsid-
eredin[8]. Thetopicissubjecttoactiveresearchandmanyaspectsarestillunexplored,inparticularwe
missexplicitrealisationsofthisconceptinstringtheory. Recentdevelopments[9] includetheinvestiga-
tion ofthe superpotentialofcompactifiedtypeII theoriesonSU(3) SU(3)–spaceswith D–branesand
×
deformationsofthecalibratedcycles(cf. alsothecontributionofL.Martuccitotheseproceedings).
This article is organisedas follows. First we will explainthe notion of spaces with G G–structure,
×
paving the ground for our main topic, the notion of calibrated submanifolds on these spaces which we
discussinSection3. Finally,anexplicitexampleshowinghowtoobtaintheclassicalcalibrationswillbe
discussedinthecaseofanSU(3) SU(3)–structure.
×
2 F.GmeinerandF.Witt:CalibrationsonspaceswithGxG–structure
2 G G–structures
×
Before defining G G–structures, let us briefly review the notion of a “classical” G–structureand what
we meanby topolo×gicalandgeometricalreductions1. Givena manifoldM ofdimensionn and a vector
bundle V of rank m over it, the transition functions generically take values in GL(m). A topological
reduction or G–structure is defined by requiringthe transition functionsto take values in G GL(m).
⊂
ObjectsQ ,whicharestabilisedbytheactionofG,i. e. theyareinvariantunderG–transformations,can
i
thereforebegloballydefinedonM. ThepossiblechoicesforembeddingGintoGL(m)areparametrised
byGL(m)/G.ThisalsogivesrisetotopologicalobstructionagainsttheexistenceofaG–structure,namely
we need to have at least one globally defined section of the associated bundle with fibre GL(m)/G. In
thefollowingwewillalsospeakofareductionofthestructuregroupGL(m)toG. AG–structureonthe
tangentbundleT isreferedtoasclassical.
As a simple example consider the case of a manifold of dimension n with an SO(n)–structure. The
reductionofthestructuregroupfromGL(n)toO(n)stabilisesasymmetrictwo–tensor,whichweidentify
astheRiemannianmetricg. A furtherreductionfromO(n)toSO(n) isequivalenttotheexistenceofa
globallydefinedvolumeform,thatisanorientationonM. Herewefindatopologicalobstructionagainst
the existence of an SO(n)–structure, namely M has to be orientable, which is equivalentto w (M) =
1
0, the vanishing of the first Stiefel–Whitney class. On M exists a prefered connection, the Levi-Cevita
connection, which is compatible with the SO(n)–structure, meaning LCg = 0. This links into the
∇
conceptofageometricalreduction.Anarbitraryconnection onT canbelocallywrittenasd+ω,where
d is usual differentiation on Rn and ω a 1–form taking val∇ues in the Lie algebra gl(n). A topological
reduction to G is compatible with if the locally defined matrices actually take values in g, the Lie
∇
algebraofG. ThisisequivalenttothestatementthatthestabilisedobjectsQ arecovariantlyconstant,i.e.
i
Q =0,asithappens,forinstance,forgandtheLevi–CivitaconnectioninthecaseofanO(n)–structure.
i
∇
OnourwaytoG G–structureswewilltaketwomoreintermediatesteps.Firstly,weconsiderso–called
×
generalisedstructures,whichareassociatedwithasubgroupofSO(n,n).Thisisthestructuregroupofthe
bundleE =T T∗,whichnaturallyexistsonanymanifoldM.Itisarank2nvectorbundlewithanatural
⊕
orientationandinnerproductofsignature(n,n). TheSO(n,n)–structureonE isalwaysspinnable,for
w (E) = w (T)+w (T) w (T∗)+w (T∗) = 0 (using w (T∗) = w (T) and w (T∗) = w (T)),
2 2 1 1 2 1 1 2 2
∪
hence we obtaina correspondingspin structure (which is actually canonic). The correspondingspinors,
sectionsofthespinbundlesS±ofSpin(n,n),arealmostisomorphictoevenoroddformsonM. Almost,
becauseitturnsoutthatwehave
S± =Λev,odT∗ √ΛnT,
⊗
soinordertoidentifyspinorswithdifferentialformswehavetochooseanowherevanishingn–vectoron
M. ThechoicesforthisareparametrisedbysectionsinGL(n)/SL(n),thatis,byascalarfunctione−φ,
φ C∞(M). Thisscalarfieldcan beidentifiedwith thewell–knowndilatonfieldin stringtheory. The
Cli∈ffordactionofelementsx ξ T T∗onspinorsρ Γ(S±)isgivenby
⊕ ∈ ⊕ ∈
(x ξ) ρ= xxρ+ξ ρ.
⊕ • − ∧
Fortwospinorsρ,σ Γ(S±),wecandefineaninvariantbilinearform(aninnerproduct)as
∈
ρ,σ =[ρ σˆ]n,
h i ∧
where ρˆ = ρ is the anti–automorphismof the Clifford algebrathat givesa plus sign for p–formswith
±
p 0,3 mod 4andaminussignotherwise.
≡
To finally defineG G–structures, we look atspecific subgroupsof the generalisedstructure defined
byE =T T∗,name×lythosewhichinduceanadditionalmetricg(besidesthenaturalinnerproductthat
⊕
existsonE)andatwo–formB onT. Thepair(g,B)issometimesalsocalledageneralisedRiemannian
metric. The associated H–flux of the generalised Riemannian metric is then H = dB. One can also
1 Foramathematicallymoresophisticatedtreatmentofthecontentofthissectionseee.g.[11].
3
incorporateclosed,butnotgloballyexactH–fluxesbyusinggerbes,butwewillnotneedthathere. The
existenceofthedatum(g,B)reducesthestructuregrouptoSO(n) SO(n). AgeneralisedG–structure
×
is a further reduction of this group to G G. We can describe this reduction in terms of the stabilised
×
objectsweobtain. InourcaseeachGpreserves,asasubgroupofSpin(n),achiralspinorΨ ,andthe
1,2
invariantSpin(n,n)–spinorisρ=eB[Ψ Ψ ].
1 2
⊗
3 Calibrations
Todescribecalibratedsubmanifoldsonspaceswithageneralisedstructureasintroducedinthelastsection
we have to extend our usual notion of a subspace. Before doing so, let us briefly recall the definition
of calibrated subspaces in the classical case [6]. A k–form ρ on an oriented vector space W is called
a calibration, if ρ vol for some k–subplane U, where vol is the volume form induced by the
U U U
| ≤
RiemannianmetriconW.Theequalityshouldholdforatleastonesubplane,whichissaidtobecalibrated
withrespecttoρ.
Forgeneralisedgeometries,we haveto considernotonlya subspaceU, butrathera pair(U,F) with
F Λ2U∗. Later on, this 2–formwill accountfor a possible abelian2 gaugefield on the D–brane. We
∈
canassociateaso–calledpurespinortothedatum(U,F),namelythespinorannihilatedbytheisotropic
subspaceeF(U N∗U). Purityreferstothefactthattheannihilatorisofmaximaldimension,namelyn.
⊕ \
Thisspinorisgiven,uptoamultiple,byeF ⋆vol . Normalising,wedefine
U
•
\
eF ⋆vol
U
ρ := • .
(U,F) eF (⋆vol )
U
k • k
Inanalogytotheclassicalcase,wespeakofacalibrationformρev,od Λev,od if
∈
ρ,ρ 1
(U,F)
h i≤
for any ρ and if the bound is met for at least one ”‘generalised”’ subplane (U,F). The calibration
U,F
conditioncanalsobecastinaslightlymorefamiliarformforphysicists(seeProp. 2.71in[7]),namely
[eF ρ ]k det((g+B) F)vol .
U U U
∧ | ≤ | −
p
Everythingwehavedonesofarforvectorspacesextendsgloballyinanobviousway,forinstancethecal-
ibrationcondition.DescribingD–branesinstringcompactificationsrequiresthereductionofthestructure
groupnotonlyonthetopologicallevel,butalsoimposessomeintegrabilityconditioninanalogytothege-
ometricalreductionsintheclassicalcase. Inparticular,wewantconditionsforthebranestopreservesome
partoftheoriginalsupersymmetrywhichisequivalenttodemandingthatthecorrespondingsubmanifolds
are volume minimising within their homology class. In the presence of F–fields or fluxes, the volume
functionalgetsreplacedbytheBorn–Infeldfunctionalcontainingcontributionsfromtheadditionaldatum.
Ashasbeenprovenin[7],everycalibratedsubmanifoldwillminimisethefunctional
e−φ det((g+B)U F)
Z | −
U p
if the calibration form is closed. To include the RR–potentials C = C(k) of type II string theory
k
intothissetupamountstoconsideringanadditionalterminthefunctionPalandchangingtheintegrability
condition.
Inthemostgeneralcaseweconsidertheintegrabilitycondition
(d+H) e−φρ=(d+H) (eB C),
∧ ∧ ∧
underwhichcalibratedsubmanifoldsminimisethefunctional,
e−φ det((g+B) F) eB−F C .
ZU(cid:16) p |U − − ∧ (cid:17)
2 Theconceptpresentedhereneedsextensioninordertoincorporatenonabeliangaugegroups.
4 F.GmeinerandF.Witt:CalibrationsonspaceswithGxG–structure
4 Example
TomakecontactwithcalibratedsubspacesusedintypeIIstringcompactifications,letuscheckthegeneral
conditionforthecaseofn=6andG=SU(3). AnSU(3) SU(3)–structurestabilisestwoSpin(6,6)–
×
spinorsρod,ev,whichcanbedecomposedintothetensorproductoftwochiralSpin(6)spinorsΨ asat
L/R
theendofSection2. Inthelimitingcasewherethetwospinorsareequal,weobtainthewell–knowncase
ofaclassicalSU(3)–structure.
WecancalibratewithrespecttoReρev =1 ω2/2orReρod =ReΩ,whereωistheKa¨hlerformand
−
Ωtheholomorphic(3,0)–form. ForsimplicityletB = F = 0. Thetworespectivecalibrationconditions
thendescribepreciselytheholomorphiccyclesofB–branesandtheLagrangiancyclesofA–branes. For
non–zero F, we obtain an additional solution which corresponds to coisotropic A–branes (for a more
detaileddescriptionofthisandotherexamples,see[7]).
ForageneralSU(3) SU(3)–structure,submanifoldsofanyevenorodddimension(dependingonthe
×
choiceofthecalibrationform)canoccur. Inparticular,wecouldobtainisotropicA–branesofdimension
one.
Acknowledgements F.G.wouldliketothanktheorganisersoftheRTNnetworkconferenceinNaplesforcreating
suchanicemeeting. TheworkofF.G.issupportedbytheFoundationforFundamentalResearchofMatter(FOM)
andtheNationalOrganisationforScientificResearch(NWO).TheworkofF.W.issupportedbytheSFB647ofthe
GermanResearchCouncil(DFG).
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