Pluripotential Theory and Special Holonomy: Hodge Decompositions and $\partial \bar{\partial }$ Lemmata

Let M be a compact torsion-free G2 7-manifold or Calabi-Yau 6-manifold. We prove Hodge decomposition theorems for the ddϕ operators, introduced by Harvey and Lawson [15], which generalize the i∂∂¯ operator used in classical pluripotential theory. We then obtain analogues of the ∂∂¯ 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 G2 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.