Generalized Calabi-Yau manifolds(2)

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)meansthatβislocal …… 此处隐藏：5548字，全部文档内容请下载后查看。喜欢就下载吧 ……