  We develop Hodge theory for a Riemannian manifold $(M,g)$ with a background
closed 3-form, H. Precisely, we prove that if the metric connections with
torsion $\pm H$ have holonomy groups $G_\pm$, then the $d^H$-Laplacian
preserves the irreducible representations of the Lie algebras of the holonomy
groups on the space of forms.
