Generalized Calabi-Yau manifolds(3)

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

ineithertheevenoroddpartofH (M,R)foracompact6-manifoldM.IftheylieintheopenorbitateachpointofM,thesecriticalpointsarepreciselygeneralizedCalabi-Yaumanifolds.Imitating[10]wethenshowthat,underacertaincondition,alocalmodulispaceforthesestructuresisanopensetinthecorrespondingcohomology¯-lemmaforgroupofevenorodddegree.Therequiredconditionisimpliedbythe

complexmanifoldsandthestrongLefschetztheoremforsymplecticones.Weshouldnotethatthisapproachforcesustoconsidertwostructurestobeequivalentiftheyarerelatednotjustbythegroupofdi eomorphismsisotopictotheidentity,butbyitsextensionbytheactionofexactB- elds.

Thereisaspecialpseudo-K¨ahlerstructureonthemodulispaceinducedasacon-sequenceofthisapproach.Intheevencaseitisthestructuredeterminedbytheintersectionform–“withoutquantumcorrections”inthephysicists’language.

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