Generalized Calabi-Yau manifolds(4)

Thusthevariationvanishesforalldαifandonlyif

dρ =0.

Acriticalpointthereforeimpliesd =0where2 =ρ+iρ .FromDe nition2wehaveageneralizedCalabi-Yaumanifold.

6.2TheHessian

WeshallinvestigatetheHessianofthefunctionalVatacriticalpointnext.SinceXistheHamiltonianvector eldforφ,andJ=DXitisclearthatJisessentiallythesecondderivativeD2φ.Moreprecisely,wehave

D2φ(ρ˙1,ρ˙2)= DXρ˙1,ρ˙2 = Jρ˙1,ρ˙2

Thus,atacriticalpointofV,theHessianHis

H(ρ˙1,ρ˙2)=D2φ(ρ˙1,ρ˙2)= Jρ˙1,ρ˙2

MM(14)(15)

wherewearerestrictingthevariationtotakeplaceina xedcohomologyclass,sothatρ˙1,ρ˙2areexactforms.

BecauseoftheinvariancepropertiesofthefunctionalV,anycriticalpointliesonanorbitofcriticalpoints,sotheHessianisnevernon-degenerate.Whatisthenaturalgroupofinvariants?

FirstlyVisinvariantunderdi eomorphismsandthosewhicharehomotopictotheidentitypreservethedeRhamcohomologyclassofρandsotheclassofformsforthevariationalproblem.TheintegrandisalsoinvariantunderthefullgroupSpin(6,6),soexponentiatingsectionsofthecomponentsoftheLiealgebraisomorphictoΛ2T andΛ2Tgivefurtherinvariantactions.Ourvariationalproblemisbasedonρbeingclosedhowever,andthisconditionwillnotbepreservedundertheactionofsectionsofΛ2T.TheactionofB∈C∞(Λ2T )istheB- eldaction

ρ→expB∧ρ.

WhenBisclosed,thistakesclosedformstoclosedforms,butto xthecohomologyclassweneedBingeneraltobeexact,forthenB=dξand

(expdξ)∧ρ=ρ+d(ξ∧ρ+1

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

liesinthesamecohomologyclass.

ThenaturalsymmetrygroupoftheproblemisthenthegroupextensionG

2exact→G→Di 0(M).

WewanttodeterminewhenageneralizedCalabi-Yaumanifoldisde nedaccordingtoTheorem6byaMorse-Bottcriticalpoint–non-degeneratetransversetotheorbitsofthegroupG.Weconsiderthetangentspacetothisorbitnext.

Theactionofavector eldonρisjusttheLiederivative

LXρ=dι(X)ρ+ι(X)dρ=d(ι(X)ρ)

sinceρisclosed.Thein nitesimalactionofanexactB- eldB=dξis

dξ∧ρ=d(ξ∧ρ).

ThusthetangentstoanorbitofGatρareforms

ρ˙=d(ι(X)ρ+ξ∧ρ)=d((X+ξ)·ρ)(16)

Becauseoftheinvarianceofthefunctional,ifαisexactandβ=d(ι(X)ρ+ξ∧ρ),thenH(α,β)=0.Supposeconverselythattheexactformβ=dτhasthepropertythatH(α,β)=0forallexactformsα=dψ,thenfrom(15), Jdψ,dτ =± ψ,dJdτ =0

MM

forallψsothat

dJdτ=0.

Thustransversenondegeneracyisequivalenttothefollowingproperty:

De nition5AgeneralizedCalabi-YaumanifoldissaidtosatisfytheddJ-lemmaif

dJdτ=0 dτ=d(ι(X)ρ+ξ∧ρ)

foravector eldXand1-formξ.

Thisconditionmaynotalwaysbesatis ed.Herearetwocaseswhenitis:

Proposition7TheddJ-lemmaholdsif:

a)thegeneralizedCalabi-Yaumanifoldisacomplex3-manifoldwithanonvanishing¯-lemma,orholomorphic3-formandwhichsatis esthe

b)itisasymplectic6-manifoldsatisfyingthestrongLefschetzcondition.