Manifolds with parallel differential forms and Kähler identities for G2G2-manifolds

Let MM be a compact Riemannian manifold equipped with a parallel differential form ωω. We prove a version of the Kähler identities in this setting. This is used to show that the de Rham algebra of MM is weakly equivalent to its subquotient (Hc∗(M),d), called the pseudo-cohomology   of MM. When MM is compact and Kähler, and ωω is its Kähler form, (Hc∗(M),d) is isomorphic to the cohomology algebra of MM. This gives another proof of homotopy formality for Kähler manifolds, originally shown by Deligne, Griffiths, Morgan and Sullivan. We compute Hc∗(M) for a compact G2G2-manifold, showing that Hci(M)≅Hi(M) unless i=3,4i=3,4. For i=3,4i=3,4, we compute Hc∗(M) explicitly in terms of the first-order differential operator ∗d:Λ3(M)⟶Λ3(M).