Pluripotential theory and special holonomy, I: Hodge decompositions and $\mathrm{\partial }\bar{\mathrm{\partial }}$ lemmata

Let $M$ be a compact torsion-free $G_2$ 7-manifold or Calabi-Yau 6-manifold. We prove Hodge decomposition theorems for the $d{d}^{\phi }$ operators, introduced by Harvey and Lawson, which generalize the $i\mathrm{\partial }\bar{\mathrm{\partial }}$ operator used in classical pluripotential theory. We then obtain analogues of the $\mathrm{\partial }\bar{\mathrm{\partial }}$ lemma in this context. We formalize this by defining cohomology spaces analogous to Bott-Chern cohomology and we relate them to harmonic forms on $M$. In the $G_2$ case we provide a geometric interpretation of the corresponding cohomology classes in terms of coassociative submanifolds and gerbes: this is analogous to the classical interpretation of Bott-Chern cohomology classes in terms of divisors and holomorphic line bundles.