Generalized Calabi-Yau manifolds

A geometrical structure on even-dimensional manifolds is defined which generalizes the notion of a Calabi-Yau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action of both diffeomorphis

arXiv:math/0209099v1 [math.DG] 10 Sep 2002GeneralizedCalabi-YaumanifoldsNigelHitchinMathematicalInstitute24-29StGilesOxfordOX13LBUKhitchin@maths.ox.ac.ukFebruary1,2008AbstractAgeometricalstructureoneven-dimensionalmanifoldsisde nedwhichgener-alizesthenotionofaCalabi-Yaumanifoldandalsoasymplecticmanifold.Suchstructuresareofeitheroddoreventypeandcanbetransformedbytheactionofbothdi eomorphismsandclosed2-forms.Inthespecialcaseofsixdimen-sionswecharacterizethemascriticalpointsofanaturalvariationalproblemonclosedforms,andprovethatalocalmodulispaceisprovidedbyanopensetineithertheoddorevencohomology.1Introduction

WeintroduceinthispaperageometricalstructureonamanifoldwhichgeneralizesboththeconceptofaCalabi-Yaumanifold–acomplexmanifoldwithtrivialcanonicalbundle–andthatofasymplecticmanifold.Thisispossiblyausefulsettingforthebackgroundgeometryofrecentdevelopmentsinstringtheory,butthiswasnottheoriginalmotivationfortheauthor’s rstencounterwiththisstructure:itaroseinsteadaspartofaprogramme(followingthepapersforcharacterizingspecialgeometryinlowdimensionsbymeansofinvariantfunctionalsofdi erentialforms.Inthisrespect,thedimensionsixisparticularlyimportant.Thispaperhastwo

A geometrical structure on even-dimensional manifolds is defined which generalizes the notion of a Calabi-Yau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action of both diffeomorphis

aims,then: rsttointroducethegeneralconcept,andthentolookatthevariationalandmodulispaceprobleminthespecialcaseofsixdimensions.

Webeginwiththede nitioninalldimensionsofwhatwecallgeneralizedcomplexmanifoldsandgeneralizedCalabi-Yaumanifolds1.Therearetwonovelfeaturesin-volved.The rstistheuseoftheCourantbracket,ageneralizationoftheLiebracketonsectionsofthetangentbundleTtosectionsofthebundleT⊕T ,andwhichcomestousfromthestudyofconstrainedmechanicalsystems[7].ThesecondistheB- eld(thisistheterminologyofthephysicists,butitisassuredlythesamemathematicalobject).Itturnsoutthatthegeometrywedescribetransformsnaturallynotonlyunderthedi eomorphismgroup,butalsobytheactionofaclosed2-formB.

Tode neageneralizedcomplexmanifoldweimitateonede nitionofaK¨ahlerman-ifold.Insteadofaskingforthe(1,0)vectorstobede nedbyanisotropicsubbundleE T C,whosespaceofsectionsisclosedundertheLiebracket,weinsteadaskforasubbundleE (T⊕T ) C,isotropicwithrespecttotheinde nitemetriconT⊕T de nedbythenaturalpairingbetweenTandT ,andmoreoverwhosespaceofsectionsisclosedundertheCourantbracket.Forthede nitionofagen-eralizedCalabi-Yaumanifoldweasknotforaclosed(n,0)-form,butinsteadforaclosedcomplexform ofmixeddegreeandofacertainalgebraictype.ThistypeisobtainedbythinkingofaformasaspinorfortheorthogonalvectorbundleT⊕T andthenrequiringthespinortobepure.Thewell-knowncorrespondencebetweenmaximallyisotropicsubspacesandpurespinorsmeansthatsuchaformde nesasubbundleE (T⊕T ) CandweshowthatsectionsofEareclosedundertheCourantbracketifd =0.Therearetwoclassesofsuchstructures,dependingonwhetherthedegreeof isevenorodd.

Therearetwomotivatingexamples:anordinaryCalabi-Yaumanifoldandasym-plecticmanifold.ACalabi-Yaumanifoldwithholomorphic(n,0)form de nesageneralizedCalabi-Yaustructurebytaking = .Asymplecticmanifoldwithsymplecticformωde nesageneralizedCalabi-Yaustructurebytaking =expiω.Transformingwithaclosed2-formBmeansreplacing by(expB)∧ .Incertaincases,asweshallsee,theB- eldinterpolatesbetweensymplecticandCalabi-Yaustructures.

ThespecialroleofsixdimensionsarisesfromthefactthatthegroupR ×Spin(6,6)hasanopenorbitineitherofits32-dimensionalspinrepresentations.Moreoveraspinorinthisopensetistherealpartofacomplexpurespinor .Wecande nefromthisalgebraaninvariant“volume”functionalde nedonrealforms,andweconsidercriticalpointsofthisfunctionalontheclosedformsinacohomologyclass

A geometrical structure on even-dimensional manifolds is defined which generalizes the notion of a Calabi-Yau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action of both diffeomorphis

ineithertheevenoroddpartofH (M,R)foracompact6-manifoldM.IftheylieintheopenorbitateachpointofM,thesecriticalpointsarepreciselygeneralizedCalabi-Yaumanifolds.Imitating[10]wethenshowthat,underacertaincondition,alocalmodulispaceforthesestructuresisanopensetinthecorrespondingcohomology¯-lemmaforgroupofevenorodddegree.Therequiredconditionisimpliedbythe

complexmanifoldsandthestrongLefschetztheoremforsymplecticones.Weshouldnotethatthisapproachforcesustoconsidertwostructurestobeequivalentiftheyarerelatednotjustbythegroupofdi eomorphismsisotopictotheidentity,butbyitsextensionbytheactionofexactB- elds.

Thereisaspecialpseudo-K¨ahlerstructureonthemodulispaceinducedasacon-sequenceofthisapproach.Intheevencaseitisthestructuredeterminedbytheintersectionform–“withoutquantumcorrections”inthephysicists’language.

Finally,byreturningtotheoriginsoftheCourantbracket,weobservethatthewholestructurecanbetwistedbyaclosedthree-form,ormorenaturallybyagerbewithconnection.

TheauthorwishestothanktheUniversidadAut´onoma,MadridandtheProgramaCat`edraFundaci´onBancodeBilbaoyVizcayaforsupportduringpartoftheprepa-rationofthispaper.

2TheCourantbracket

Weshallbeginbysettingupthelessfamiliarpiecesofdi erentialgeometry.The rstisthebracketoperationintroducedbyT.Courant(forp=1)in[7].Thisisanoperationde nedonpairs(X,ξ)=X+ξofavector eldXandap-formξonamanifoldM.TakeX+ξ,Y+η∈C∞(T⊕ΛpT )andde ne

[X+ξ,Y+η]=[X,Y]+LXη LYξ 1

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