Reduced holonomy and Hodge theory

TL;DR

This paper extends Hodge theory to Riemannian manifolds with a closed 3-form background, showing that certain Laplacians preserve irreducible holonomy representations, revealing new geometric structures.

Contribution

It develops a novel Hodge theory framework for manifolds with torsion-related holonomy groups, connecting Laplacian operators with irreducible representations of holonomy Lie algebras.

Findings

The $d^H$-Laplacian preserves irreducible holonomy representations.

Holonomy groups $G_ ext{±}$ influence the structure of the Laplacian.

New relations between torsion, holonomy, and Hodge theory are established.

Abstract

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.