Generalized Calabi-Yau manifolds(7)

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

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