Geometric structures on G₂ and Spin(7)-manifolds

This article studies the geometry of moduli spaces of G₂-manifolds, associative cycles, coassociative cycles and deformed Donaldson-Thomas bundles. We introduce natural symmetric cubic tensors and differential forms on these moduli spaces. They correspond to Yukawa couplings and correlation functions in M-theory. We expect that the Yukawa coupling characterizes (co-)associative fibrations on these manifolds. We discuss the Fourier transformation along such fibrations and the analog of the Strominger-Yau-Zaslow mirror conjecture for G₂-manifolds. We also discuss similar structures and transformations for Spin(7)-manifolds.