Generalized Calabi-Yau manifolds(3)

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 ,whichiswhereweshallmake …… 此处隐藏：5297字，全部文档内容请下载后查看。喜欢就下载吧 ……