Reduced holonomy and Hodge theory

Gil R. Cavalcanti

Published 2012-08-17, updated 2013-09-09Version 2

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.