Generalized Calabi-Yau manifolds(22)

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).

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