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

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

Example:Considerthecasep=0,sothatξisafunctionf.Wethenhave

[X+f,Y+g]=[X,Y]+Xg Yf.

ThisistheusualLiebracketonS1-invariantvector elds

X+f

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

3.2Spinors

ConsidertheexterioralgebraΛ V andtheactionofv+ξ∈V⊕V onitde nedby

(v+ξ)· =ι(v) +ξ∧

Wehave

(v+ξ)2· =ι(v)(ξ∧ )+ξ∧ι(v) =(ι(v)ξ) = (v+ξ,v+ξ)

whichmakesΛ V intoamoduleovertheCli ordalgebraofV⊕V .Thisde nesthespinrepresentationofthegroupSpin(V⊕V )ifwetensorwiththeone-dimensionalspace(ΛnV)1/2.Splittingintoevenandoddformswethenhavethetwoirreduciblehalf-spinrepresentations:

S+=ΛevV (ΛnV)1/2

S =ΛodV (ΛnV)1/2

IfwenowtakeB∈Λ2V so(V⊕V ),thenexponentiatingBtoexpBintheLiegroupSpin(V⊕V )givesfrom(3)thefollowingactiononspinors:

expB( )=(1+B+1(3)

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

3.3Purespinors

Given ∈S±,weconsideritsannihilator,thevectorspace

E ={v+ξ∈V⊕V :(v+ξ)· =0}

Sincev+ξ∈E satis es

0=(v+ξ)·(v+ξ)· = (v+ξ,v+ξ)

weseethatv+ξisnullandsoE isisotropic.Aspinor forwhichE ismaximallyisotropic(i.e.hasdimensionequaltodimV)iscalledapurespinor.AnytwopurespinorsarerelatedbyanactionofSpin(V⊕V ).Tobepureisanon-linearconditionwhich,inhigherdimensions,isquitecomplicated.Herearesomeexamples:Examples:

1.Thespinor1∈Λ0V ΛevV ispure,since(v+ξ)·1=ξandsotheannihilatorisde nedbyξ=0,themaximalisotropicsubspaceV V⊕V

2.ApplyinganyelementofSpin(V⊕V )to1givesanotherpurespinor.InparticularwecanexponentiateB∈Λ2V sothat

expB=1+B+1

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

¯=(T⊕T ) C E⊕E

thespaceofsectionsofEisclosedundertheCourantbracket

Eisisotropic

Therealversionofthisintegrability–amaximallyisotropicsubbundleofT⊕T withsectionsclosedunderCourantbracket–iscalledaDiracstructurein[7].AsymplecticorPoissonstructureonMde nesoneofthese.

OurmainconcerninthispaperwillbethenotionofageneralizedCalabi-Yaumanifoldwhichwede nenext.Gualtieri’sthesis[9]willcontainmoreresultsongeneralizedcomplexmanifolds.

De nition2AgeneralizedCalabi-YaustructureonasmoothmanifoldMofdi-mension2mis

aclosedform ∈ ev Cor od CwhichisacomplexpurespinorfortheorthogonalvectorbundleT⊕T andsuchthat

, ¯ =0ateachpoint.

ThefollowingpropositionshowsthatageneralizedCalabi-Yaumanifoldisaspecialcaseofageneralizedcomplexmanifold.

Proposition1If(M, )isageneralizedCalabi-YaumanifoldthentheannihilatorE (T⊕T ) Cde nesageneralizedcomplexstructureonM.

Proof:Wesawfromthealgebraintheprevioussectionthattheannihilatorofapurespinorismaximallyisotropic,soE certainlysatis esthelastconditioninthede nitionofgeneralizedcomplexstructureandhasdimension2m.Moreover,since , ¯ =0,weknowthat¯0=E ∩E ¯=E ∩E

andso¯ =(T⊕T ) C.E ⊕E

ItremainstoshowthatsectionsofE areclosedundertheCourantbracket.SupposeX+ξandY+ηannihilate .Thenfrom(3)

ι(X) +ξ∧ =0=ι(Y) +η∧

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

Usingd =0andLX=dι(X)+ι(X)dweobtain

ι([X,Y]) =

=

=

=

=

=

=

=LX(ι(Y) ) ι(Y)LX LX(η∧ ) ι(Y)d(ι(X) ) LXη∧ η∧LX +ι(Y)d(ξ∧ ) LXη∧ η∧d(ι(X) )+ι(Y)(dξ∧ ) LXη∧ +η∧d(ξ∧ )+ι(Y)(dξ∧ ) LXη∧ +η∧dξ∧ +(ι(Y)dξ)∧ +dξ∧ι(Y) LXη∧ +η∧dξ∧ +(ι(Y)dξ)∧ dξ∧η∧

LXη∧ +(ι(Y)dξ)∧

1

( LXη∧ +(ι(Y)dξ)∧ +LYξ∧ (ι(X)dη)∧ )

1

12d(ι(X)η)]∧ andso,byskewsymmetry,ι([X,Y]) =2=[ι(Y)dξ+

=[LYξ LXη

m!

whichisnon-vanishing.Sincedω=0, =expiωde nesageneralizedCalabi-Yaumanifold.

ωm

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

3.Itisclearthattheproductoftwogeneralizedcomplexmanifoldsisageneralizedcomplexmanifold.Similarlyif(M1, 1),(M2, 2)aretwogeneralizedCalabi-Yaumanifolds,thenifp1,p2denotetheprojectionsfromtheproductM1×M2,

=p

1 1∧p2 2

de nesageneralizedCalabi-Yaustructureontheproduct.Theproductofanoddtypewithaneventypeisoddandtheproductoftwooddortwoeventypesiseven.

4.2TheB- eld

IfBisarealclosed2-form,and(M, )ageneralizedCalabi-Yaumanifoldthen

(expB) =(1+B+1

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

MultiplybytheconstanttkandwehaveafamilyofgeneralizedCalabi-Yaustructuresde nedby1 t=tkexp((ω1+iω2)/t)=tk+...+

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

whereβisacomplexclosed1-formandγacomplexclosed3-form.Theform mustde neacomplexpurespinorforT⊕T .HerewearelookingatthespinrepresentationS ofthecomplexi cationSpin(8,C)ofSpin(4,4).Ineightdimensionshowever,wehavethespecialfeatureoftriality–thevectorrepresentationandthetwospinrepresentationsarerelatedbyanouterautomorphismofSpin(8,C).ForusthismeansinparticularthatthetwospinspacesS±havethesamestructureasthevectorrepresentation–an8-dimensionalspacewithanon-degeneratequadraticform.Thepurespinorsarethenjustthenullvectorsinthisspace.

Itfollowsthat ispureif

0= , =β∧γ.

Wealsohavethecondition

¯∧γ=00= , ¯ =β∧γ¯+β(7)(6)

whichshowsinparticularthatβisnowherevanishing.Thusfrom(6),γ=β∧νforsome2-formν,well-de nedmoduloβ.Using(7)again,

¯∧(ν νβ∧β¯)=0(8)

andfromthiswecanseethatlocally,thestructureonMisde nedbyamapf:M→C(wheredf=β)de ninga brationoveranopenset,asymplecticstructure νandaB- eld νonthe bres.Aglobalexampleistheproductofanoddandaneven2-dimensionalgeneralizedCalabi-Yaumanifold.Tischler’stheorem[17]showsthatacompactmanifoldwithanon-vanishingclosed1-form bresoverthecircleandmoregenerallythatwithtwosuchformsliketherealandimaginarypartsofβ,itmust breoverT2.Inparticularthe rstBettinumberb1(M)isnon-zero.Forastructureofeventypewehave

=c+β+γ

foraconstantc,closed2-formβand4-formγ.For tobepureweneed

0= , =2cγ β2.

Ifc=0,thisgivesγ=β2/2c.Thecondition0= , ¯ thengives

cc¯¯+c0=cγ¯ ββ¯γ=

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

whichisthetransformofasymplecticstructure.

Ifc=0,thepurityconditionisβ2=0,which(asintheequationoftheKleinquadric)meansthatβislocallydecomposable:β=θ1∧θ2.Wealsohave

¯=θ1∧θ2∧θ¯1∧θ¯20= , ¯ =β∧β

sothatθ1,θ2spanthespaceof(1,0)-formsforanalmostcomplexstructureandβisoftype(2,0).Sincedβ=0thestructureisintegrableandwehaveanordinaryCalabi-Yaumanifold.InthecompactcasethismustbeaK3surfaceoratorus.Theremaining4-formγistheresultofapplyinga(notnecessarilyclosed)B- eldtoβ.

4.5Structuregroupsandgeneralizations

Ourde nitionofageneralizedcomplexstructureyieldsacomplexstructureonT⊕T compatiblewithaninde nitemetric.ThisisareductionofthestructuregroupofT⊕T toU(m,m) SO(2m,2m),togetherwithanintegrabilitycondition.

Therearefurtherreductionspossiblewithinthissetting.FirstconsiderthecaseofageneralizedCalabi-Yaumanifold.Heretheform hastheproperty0= , ¯ so

1

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

SU(2,2)→SO(4,2)

Sp(1,1)→SO(4,1)

Sp(1)×Sp(1)→SO(4)

ThetwospinorsstabilizedbySU(2,2)aretherealandimaginarypartofapurespinor–thecomplexclosedform ofageneralizedCalabi-Yaustructure.ThegroupSp(1,1)isobtainedbyrequiringthreeformstobeclosedgivingageneralizedhyperk¨ahlerstructureandthelastoneinvolvesfourclosedforms.ThemodulispaceofsuchstructuresonaK3surfacehasbeenstudiedbyNahmandWendland[16].5

5.1Thesix-dimensionalcaseThequarticform

Weshallbegintostudythealgebrabyworkingoverthecomplexnumbers,usingSpin(12,C)insteadofSpin(6,6)andacomplexsix-dimensionalvectorspaceV.Inthisdimensionthebilinearformoneachofthe32-dimensionalspinspacesS±isskewsymmetric,andsothesearesymplecticrepresentations.ThelinearalgebraweshallbedoingisinsensitivetothechoiceoforientationwhichdistinguishesS+fromS ,butforvariousreasonstheisomorphism

S+～=ΛevV (Λ6V)1/2

willbeausefultool,soweshall xS=S+.

AsymplecticactionofaLiegroupGonavectorspaceSde nesamomentmap

µ:S→g

givenby

µ(ρ)(a)=1

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

1.Chooseabasisvectorνfor(Λ6V)1/2andconsiderthemomentmapforSpin(12,C)actingonSatν.Ifa=A+B+βinthedecompositionso(V⊕V )=EndV⊕Λ2V ⊕Λ2V,then

σ(a)ν,ν = (trA/2)ν+B∧ν,ν =0

sothemomentmapvanishesonνandhenceonanypurespinor.

2.Nowtake

Inthiscase

σ(a)ρ0,ρ0 = trA

andwe ndthat

µ(ρ0)(v+ξ)=( v+ξ)/4.

Themomentmapalsode nesaninvariant:

De nition3LetµbethemomentmapforthespinrepresentationSofSpin(12,C).Then

q(ρ)=trµ(ρ)2

isaninvariantquarticfunctiononS.

Thisquartichasacloserelationshipwithpurespinors:

Proposition2Forρ∈S,q(ρ)=0ifandonlyifρ=α+βwhereα,βarepurespinorsand α,β =0.Thespinorsα,βareuniqueuptoordering.

Proof:Considerasintheexampleρ0=ν+ν 1∈S:νispurewithisotropicsubspaceVandν 1withsubspaceV .

Nowsupposethatαandβarepure.Because,uptoaconstant,Spin(12,C)actstransitivelyonpurespinors,wecanassumeα=kν.If α,β =0,weseefromthede nitionofthebilinearformthatβ6=0(wewriteαpforthedegreepcomponentofα).ByexponentiatinganelementofΛ2VintheLiealgebra,weobtainagroup withβ 4=0andβ 6=0.elementwhichleavesν xedbuttakesβtoanelementβ arepureandsothereisa6-dimensionalisotropicspaceofvectorsButβandhenceβ =0.Lookingatthedegree5term,thismeansthatv+ξsatisfying(v+ξ)·β 6+ξ∧β 4=ι(v)β 6sinceβ 4=0.Butthenv=0andthe6-dimensional0=ι(v)β = ν 1andα+βcanbetransformedtospaceisV .Thusβ

kν+ ν 1.

(9)ρ0=ν+ν 1∈(Λ6V)1/2⊕(Λ6V )(Λ6V)1/2 S.

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

From(9)weseethatq(ρ0)=3,andsobyinvarianceandhomogeneity

q(α+β)=q(kν+ ν 1)=3k2 2=3 α,β 2(10)

Inparticular,thequarticinvariantisnon-zeroforthesumoftwopurespinorswith α,β =0.

Atρ0=ν+ν 1wesawthatthemomentmapwasv+ξ→( v+ξ)/4.Henceµ(ρ0)2=I/16.If =kν+ ν 1thenµ(ρ)2=k2 2I/16andso

µ(ρ)2=1

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

Nowsupposethatρisreal.Proposition2saysthattherearecomplexpurespinorsα,βwithρ=α+β.Realityo erstwopossibilities:αandβarebothreal,orβ=α¯.Ifα,βarerealthensois α,β andsofrom(10)q(ρ)>0.Ifβ=α¯then α,α¯ isimaginaryandq(ρ)<0.FromProposition2,wededuce:

Proposition3Letρ∈Sbearealspinorwithq(ρ)<0.Thenρistherealpartofapurespinor with , ¯ =0.

Thesearepreciselythepurespinorsweneedinthede nitionofageneralizedCalabi-Yaumanifold.

WhenthevectorspaceVisreal,theopenset

U={ρ∈S:q(ρ)<0}

isactedontransitivelybytherealgroupR ×Spin(6,6).Weshallstudynextthegeometryofthisspace,followingcloselytheparalleldiscussionofthreeformsinsixdimensions,asin[10].Infact,whatwearedoinghereisadirectgeneralizationofthatwork.

5.2Thesymplecticgeometryofthespinrepresentation

De nition4OntheopensetU Sforwhichq(ρ)<0,de nethefunctionφ,homogeneousofdegree2,by φ(ρ)=

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

thederivativeDX:U→EndSde nesanintegrablealmostcomplexstructureJonU

Proof:Sinceiφ(ρ)= , ¯ ,di erentiatingalongacurveinU,

˙= ,˙ .iφ˙ ¯ + , ¯

Uptoascalar,thepurespinorsformanorbit,soateachpoint

˙=c +σ(a)

forsomec∈Canda∈so(12,C).Butthen

,˙ =c , + σ(a) , =0(12)

wherethe rsttermiszerobecausethebilinearformisskewandthesecondbecause,aswesawabove,ing(12)

˙˙ = ,˙ =iφ. ,¯ ˙+ ¯˙ ¯ + , ¯

Butthiscanbewrittenas˙= ρφ ,ρ˙

whichmeansthattheHamiltonianvector eldofφisX(ρ)=ρ .

Thecircleactioninrealtermsis

ρ→cosθρ+sinθρ

sothederivativeatθ=0isρ ,thevector eldX.

Sinceρ+iρ =2 ,ρ iρ= 2i andso

ρ = ρ.

Thus,asadi eomorphismofU,X X= idandthederivativeJ=DXthussatis esJ2=DX DX= Iandde nesanalmostcomplexstructureonU.Theproofthatitisintegrableisthesameasin[10]or[13]andholdsgenerallyforspecial(pseudo)-K¨ahlermanifolds,ofwhichUisanexample.

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

5.3ThecomplexstructureJ

ThecomplexstructureJonU Sturnsouttobeimportantinthesubsequentdevelopment.RecallthatUisahomogeneousspaceofSpin(6,6)×R underthespinrepresentation.Thisisalinearaction,soeverytangentvectortotheopensetUatρisoftheformσ(a)ρforsomeaintheLiealgebra.Weshow

Proposition5Onthetangentvectorσ(a)ρ,thecomplexstructureJisde nedby

J(σ(a)ρ)=σ(a) ρ.

Thusthe(0,1)vectorsareoftheformσ(a) whereρ= + ¯.

Proof:Asρvariesσ(a)ρde nesavector eldYonU.IfaisintheLiealgebraofSpin(6,6),thensinceφisinvariantandXistheHamiltonianvector eldofφ,wehave[X,Y]=0.ThecentralfactorR inthegroupactsbyrescaling,soifa∈Rthevector eldYistheEulervector eld–thepositionvectorρ.Nowφishomogeneousofdegree2butsoisthesymplecticform,andthismeansthat[X,Y]=0also.SinceJ=DXand[X,Y]=0,

J(Y)=DX(Y)=DY(X)=σ(a)X=σ(a) ρ

whichprovestheproposition.

AlthoughJisde nedonthevectorspaceS,itde nesacomplexstructureonthetensorproductofSwithanyvectorspaceandinparticularΛev/odV ,whichiswhereweshallmakeuseofit.

Examples:

1.TaketheCalabi-Yaucasewhere = isa(3,0)form.Thespaceof(0,1)-vectorsinΛodV CisfromProposition5theimageof undertheactionoftheLiealgebraso(12,C)+C,andusingthedecompositionso(V⊕V )=EndV⊕Λ2V ⊕Λ2V,thisisthe16-dimensionalspaceofΛodV Cgivenby

Λ3,0⊕Λ2,1⊕Λ3,2⊕Λ1,0.

2.Inthesymplecticcase =expiω,andweobtainforthe(0,1)vectorsthe16-dimensionalspaceofΛevV Cgivenby

expiωC⊕expiω(Λ2 C).

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

6

6.1ThevariationalproblemThevolumefunctional

Wede nedabovethefunctionφonU Λev/odV (Λ6V)1/2.Untwistingbytheone-dimensionalvectorspace(Λ6V)1/2,thereisacorrespondingopenset,whichwestillcallU,inΛev/odV and,sinceφishomogeneousofdegree2,aninvariantfunction

φ:U→Λ6V .

ThebilinearsymplecticformnowtakesvaluesinΛ6V alsoandsothederivativeatρofφisalinearmapfromΛev/odV toΛ6V whichcanbewritten

Dφ(ρ˙)= ρ ,ρ˙ .(13)

SupposeMisacompactoriented6-manifold,andρisaform,eitheroddoreven,butingeneralofmixeddegree,whichliesateachpointofMintheopensubsetUdescribedabove.Following[11]weshallcallsuchaformstable.Wecanthende neavolumefunctional V(ρ)=φ(ρ).

M

Theorem6Aclosedstableformρ∈ ev/od(M)isacriticalpointofV(ρ)initscohomologyclassifandonlyifρ+iρ de nesageneralizedCalabi-YaustructureonM.

Proof:Takethe rstvariationofV(ρ):

δV(ρ˙)=Dφ(ρ˙)= ρ ,ρ˙

MM

from(13).Thevariationiswithina xedcohomologyclasssoρ˙=dα.Thus

δV(ρ˙)= ρ ,dα =σ( ρ)∧dα

MM

from(4).ByStokes’theoremthisis

±dσ( ρ)∧α=±σ(dρ )∧α=± dρ ,α

MMM

sincefromitsde nitionσcommuteswithd.

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

Thusthevariationvanishesforalldαifandonlyif

dρ =0.

Acriticalpointthereforeimpliesd =0where2 =ρ+iρ .FromDe nition2wehaveageneralizedCalabi-Yaumanifold.

6.2TheHessian

WeshallinvestigatetheHessianofthefunctionalVatacriticalpointnext.SinceXistheHamiltonianvector eldforφ,andJ=DXitisclearthatJisessentiallythesecondderivativeD2φ.Moreprecisely,wehave

D2φ(ρ˙1,ρ˙2)= DXρ˙1,ρ˙2 = Jρ˙1,ρ˙2

Thus,atacriticalpointofV,theHessianHis

H(ρ˙1,ρ˙2)=D2φ(ρ˙1,ρ˙2)= Jρ˙1,ρ˙2

MM(14)(15)

wherewearerestrictingthevariationtotakeplaceina xedcohomologyclass,sothatρ˙1,ρ˙2areexactforms.

BecauseoftheinvariancepropertiesofthefunctionalV,anycriticalpointliesonanorbitofcriticalpoints,sotheHessianisnevernon-degenerate.Whatisthenaturalgroupofinvariants?

FirstlyVisinvariantunderdi eomorphismsandthosewhicharehomotopictotheidentitypreservethedeRhamcohomologyclassofρandsotheclassofformsforthevariationalproblem.TheintegrandisalsoinvariantunderthefullgroupSpin(6,6),soexponentiatingsectionsofthecomponentsoftheLiealgebraisomorphictoΛ2T andΛ2Tgivefurtherinvariantactions.Ourvariationalproblemisbasedonρbeingclosedhowever,andthisconditionwillnotbepreservedundertheactionofsectionsofΛ2T.TheactionofB∈C∞(Λ2T )istheB- eldaction

ρ→expB∧ρ.

WhenBisclosed,thistakesclosedformstoclosedforms,butto xthecohomologyclassweneedBingeneraltobeexact,forthenB=dξand

(expdξ)∧ρ=ρ+d(ξ∧ρ+1

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

liesinthesamecohomologyclass.

ThenaturalsymmetrygroupoftheproblemisthenthegroupextensionG

2exact→G→Di 0(M).

WewanttodeterminewhenageneralizedCalabi-Yaumanifoldisde nedaccordingtoTheorem6byaMorse-Bottcriticalpoint–non-degeneratetransversetotheorbitsofthegroupG.Weconsiderthetangentspacetothisorbitnext.

Theactionofavector eldonρisjusttheLiederivative

LXρ=dι(X)ρ+ι(X)dρ=d(ι(X)ρ)

sinceρisclosed.Thein nitesimalactionofanexactB- eldB=dξis

dξ∧ρ=d(ξ∧ρ).

ThusthetangentstoanorbitofGatρareforms

ρ˙=d(ι(X)ρ+ξ∧ρ)=d((X+ξ)·ρ)(16)

Becauseoftheinvarianceofthefunctional,ifαisexactandβ=d(ι(X)ρ+ξ∧ρ),thenH(α,β)=0.Supposeconverselythattheexactformβ=dτhasthepropertythatH(α,β)=0forallexactformsα=dψ,thenfrom(15), Jdψ,dτ =± ψ,dJdτ =0

MM

forallψsothat

dJdτ=0.

Thustransversenondegeneracyisequivalenttothefollowingproperty:

De nition5AgeneralizedCalabi-YaumanifoldissaidtosatisfytheddJ-lemmaif

dJdτ=0 dτ=d(ι(X)ρ+ξ∧ρ)

foravector eldXand1-formξ.

Thisconditionmaynotalwaysbesatis ed.Herearetwocaseswhenitis:

Proposition7TheddJ-lemmaholdsif:

a)thegeneralizedCalabi-Yaumanifoldisacomplex3-manifoldwithanonvanishing¯-lemma,orholomorphic3-formandwhichsatis esthe

b)itisasymplectic6-manifoldsatisfyingthestrongLefschetzcondition.

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