Geometric structures on <i>G</i>² and Spin (7)-manifolds

Abstract

This article studies the geometry of moduli spaces of G 2 -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 G 2 -manifolds.We also discuss similar structures and transformations for Spin(7)manifolds.

Hodge theory for G2 -manifolds: intermediate Jacobians and Abel-Jacobi maps

“…Such connections have curvature 2-form F A contained in Ω 2 = g 2 . The deformed Donaldson-Thomas equation was introduced in[29]…”

MODULI SPACES OF G2MANIFOLDS

“…Expressions for the G 2 Yukawa coupling has been derived by different authors -in particular by Lee and Leung, [33], de Boer, Naqvi and Shomer [16], and Karigiannis [30]. Similarly, we can rewrite (4.53e) as As we have mentioned previously, by complexifying the G 2 moduli space, it is possible to turn the Hessian structure into a Kähler structure.…”

Conformally parallel Spin(7) structures on solvmanifolds

“…Finally, we mentioned different directions on G 2 and Spin(7) manifolds related to the geometric structures on these spaces. Starting from certain classes of G 2 -manifolds Y , conformally parallel Spin(7) metrics on Riemannian products associated to these manifolds with some special geometric properties should be studied [2,5,33]. All of them give also new research areas related to these exceptional geometries in dimensions d = 7 and d = 8 .…”