Generalized Calabi-Yau manifolds(6)

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)

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