Generalized Calabi-Yau manifolds(13)

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=

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