Generalized Calabi-Yau manifolds(2)

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

aims,then: rsttointroducethegeneralconcept,andthentolookatthevariationalandmodulispaceprobleminthespecialcaseofsixdimensions.

Webeginwiththede nitioninalldimensionsofwhatwecallgeneralizedcomplexmanifoldsandgeneralizedCalabi-Yaumanifolds1.Therearetwonovelfeaturesin-volved.The rstistheuseoftheCourantbracket,ageneralizationoftheLiebracketonsectionsofthetangentbundleTtosectionsofthebundleT⊕T ,andwhichcomestousfromthestudyofconstrainedmechanicalsystems[7].ThesecondistheB- eld(thisistheterminologyofthephysicists,butitisassuredlythesamemathematicalobject).Itturnsoutthatthegeometrywedescribetransformsnaturallynotonlyunderthedi eomorphismgroup,butalsobytheactionofaclosed2-formB.

Tode neageneralizedcomplexmanifoldweimitateonede nitionofaK¨ahlerman-ifold.Insteadofaskingforthe(1,0)vectorstobede nedbyanisotropicsubbundleE T C,whosespaceofsectionsisclosedundertheLiebracket,weinsteadaskforasubbundleE (T⊕T ) C,isotropicwithrespecttotheinde nitemetriconT⊕T de nedbythenaturalpairingbetweenTandT ,andmoreoverwhosespaceofsectionsisclosedundertheCourantbracket.Forthede nitionofagen-eralizedCalabi-Yaumanifoldweasknotforaclosed(n,0)-form,butinsteadforaclosedcomplexform ofmixeddegreeandofacertainalgebraictype.ThistypeisobtainedbythinkingofaformasaspinorfortheorthogonalvectorbundleT⊕T andthenrequiringthespinortobepure.Thewell-knowncorrespondencebetweenmaximallyisotropicsubspacesandpurespinorsmeansthatsuchaformde nesasubbundleE (T⊕T ) CandweshowthatsectionsofEareclosedundertheCourantbracketifd =0.Therearetwoclassesofsuchstructures,dependingonwhetherthedegreeof isevenorodd.

Therearetwomotivatingexamples:anordinaryCalabi-Yaumanifoldandasym-plecticmanifold.ACalabi-Yaumanifoldwithholomorphic(n,0)form de nesageneralizedCalabi-Yaustructurebytaking = .Asymplecticmanifoldwithsymplecticformωde nesageneralizedCalabi-Yaustructurebytaking =expiω.Transformingwithaclosed2-formBmeansreplacing by(expB)∧ .Incertaincases,asweshallsee,theB- eldinterpolatesbetweensymplecticandCalabi-Yaustructures.

ThespecialroleofsixdimensionsarisesfromthefactthatthegroupR ×Spin(6,6)hasanopenorbitineitherofits32-dimensionalspinrepresentations.Moreoveraspinorinthisopensetistherealpartofacomplexpurespinor .Wecande nefromthisalgebraaninvariant“volume”functionalde nedonrealforms,andweconsidercriticalpointsofthisfunctionalontheclosedformsinacohomologyclass

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