Self-duality in higher dimensions

Abstract

Let ω bea2-formon a 2n dimensional manifold. Inprevious work,we called ω “strongself-dual, if the eigenvalue sof its matrix with respect to an orthonormal frame areequal in absolute value. In a series of papers,wesh owed that strong self-duality agrees with previous definitions; inparticularif ω is strong self-dual,then, in 2n dimensions,ωn is proportional to its Hodge dual ω and in 4n dimensions,ωn is Hodgeself-dual.We also obtaineda local expression of the Bonan4-formon 8 mani folds with Spin7holonomy, as the sumof the squaresof any orthonormal basis of a maximal linear sub space of strong self-dual2-forms. In the present work we generalize the notion of strongself-duality to odddimension alman if old sand we express the dual of the Fundamental 3-form7 manifolds with G2 holonomy,as asumof the squaresofan orthonormal basis of amaximal linear subspace of strong self-dual2-forms.